source stringclasses 9
values | problem stringlengths 18 4.91k | solution stringlengths 37 7.79k | messages listlengths 2 2 |
|---|---|---|---|
orca_math | 2/3 of the boys and a certain fraction of the girls of a school participate in a function. The number of participating students is 550, out of which 150 are girls. The total number of students in the school is 800. What fraction of the girls participated in the function? | Let's denote the total number of boys in the school as B and the total number of girls as G. We know that the total number of students in the school is 800, so:
B + G = 800
We are told that 2/3 of the boys participate in the function. Therefore, the number of boys participating is (2/3)B.
We are also told that the t... | [
{
"content": "2/3 of the boys and a certain fraction of the girls of a school participate in a function. The number of participating students is 550, out of which 150 are girls. The total number of students in the school is 800. What fraction of the girls participated in the function?",
"role": "user"
},
... |
olympiads | A small fish is holding 17 cards, labeled 1 through 17, which he shuffles into a random order. Then, he notices that although the cards are not currently sorted in ascending order, he can sort them into ascending order by removing one card and putting it back in a different position (at the beginning, between some two ... |
To solve this problem, we will consider the various possible ways to backtrack one move starting from an ordered sequence to understand how the cards could have originally been arranged.
1. **Start with the ordered sequence:**
Consider the sequence of cards in ascending order: \(1, 2, 3, \ldots, 17\).
2. **Remov... | [
{
"content": "A small fish is holding 17 cards, labeled 1 through 17, which he shuffles into a random order. Then, he notices that although the cards are not currently sorted in ascending order, he can sort them into ascending order by removing one card and putting it back in a different position (at the beginn... |
cn_k12 | Given $a=2^{2\ln 3}$, $b=3^{3\ln 2}$, $c=5^{\ln 5}$, then ()
A: $a \lt b \lt c$
B: $b \lt a \lt c$
C: $a \lt c \lt b$
D: $b \lt c \lt a$ | Given $a=2^{2\ln 3}$, $b=3^{3\ln 2}$, $c=5^{\ln 5}$, we proceed to compare these values step by step.
First, let's simplify the logarithmic expressions for $a$ and $b$:
- For $a$, we have:
\[
\ln a = 2\ln 2\ln 3 = \ln 2^2\ln 3 = \ln 4\ln 3 = \ln 2\ln 9
\]
- For $b$, we have:
\[
\ln b = 3\ln 2\ln 3 = \ln 3^3\ln 2 = \... | [
{
"content": "Given $a=2^{2\\ln 3}$, $b=3^{3\\ln 2}$, $c=5^{\\ln 5}$, then ()\n\nA: $a \\lt b \\lt c$\n\nB: $b \\lt a \\lt c$\n\nC: $a \\lt c \\lt b$\n\nD: $b \\lt c \\lt a$",
"role": "user"
},
{
"content": "Given $a=2^{2\\ln 3}$, $b=3^{3\\ln 2}$, $c=5^{\\ln 5}$, we proceed to compare these values s... |
cn_k12 | In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $9\sin ^{2}B=4\sin ^{2}A$ and $\cos C=\frac{1}{4}$, then $\frac{c}{a}=$
A: $\frac{{\sqrt{11}}}{4}$
B: $\frac{{\sqrt{10}}}{4}$
C: $\frac{{\sqrt{11}}}{3}$
D: $\frac{{\sqrt{10}}}{3}$ | Given the problem, we start by analyzing the given conditions and translating them into mathematical equations step by step.
1. We are given that $9\sin^2B = 4\sin^2A$. This can be related to the sides of the triangle using the Law of Sines, which states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. S... | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $9\\sin ^{2}B=4\\sin ^{2}A$ and $\\cos C=\\frac{1}{4}$, then $\\frac{c}{a}=$\n\nA: $\\frac{{\\sqrt{11}}}{4}$\n\nB: $\\frac{{\\sqrt{10}}}{4}$\n\nC: $\\frac{{\\sqrt{11}... |
orca_math | 8 people decided to split the restaurant bill evenly . if the bill was $ 314.12 dollars , how much money did they 1 cent is the smallest unit ? | To split the bill evenly among 8 people, we need to divide the total bill by the number of people.
Total bill: $314.12
Number of people: 8
$314.12 ÷ 8 = $39.265
Since 1 cent is the smallest unit, we cannot have a fraction of a cent. Therefore, we need to round to the nearest cent. The amount $39.265 would be rounded... | [
{
"content": "8 people decided to split the restaurant bill evenly . if the bill was $ 314.12 dollars , how much money did they 1 cent is the smallest unit ?",
"role": "user"
},
{
"content": "To split the bill evenly among 8 people, we need to divide the total bill by the number of people.\n\nTotal ... |
cn_k12 | The function $f(x)= \begin{cases}1-x^{2}, & x\leqslant 1 \\ x^{2}-x-3, & x > 1\end{cases}$, then the value of $f\left( \dfrac {1}{f(3)}\right)$ is ______. | When $x > 1$, $f(x)=x^{2}-x-3$, so $f(3)=3^{2}-3-3=3$,
$\therefore \dfrac {1}{f(3)}= \dfrac {1}{3}$,
When $x\leqslant 1$, $f(x)=1-x^{2}$,
$\therefore f\left( \dfrac {1}{f(3)}\right)=f\left( \dfrac {1}{3}\right)=1- \dfrac {1}{9}= \dfrac {8}{9}$.
Therefore, the answer is: $\boxed{\dfrac {8}{9}}$.
When $x > 1$, $f(x)... | [
{
"content": "The function $f(x)= \\begin{cases}1-x^{2}, & x\\leqslant 1 \\\\ x^{2}-x-3, & x > 1\\end{cases}$, then the value of $f\\left( \\dfrac {1}{f(3)}\\right)$ is ______.",
"role": "user"
},
{
"content": "When $x > 1$, $f(x)=x^{2}-x-3$, so $f(3)=3^{2}-3-3=3$,\n\n$\\therefore \\dfrac {1}{f(3)}=... |
synthetic_math | Let $f(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$. What is the smallest positive integer $n$ for which $f(n) = 4$? | To solve for $f(n) = 4$, we need to find $n$ for which there are exactly four distinct ordered pairs $(a, b)$ such that $a^2 + b^2 = n$. We first observe that these pairs must all be different, as there are no repeated pairs like $(m, m)$ that can add to the count without changing $n$.
1. **Testing small values**:
... | [
{
"content": "Let $f(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$. What is the smallest positive integer $n$ for which $f(n) = 4$?",
"role": "user"
},
{
"content": "To solve for $f(n) = 4$, we need to find $n$ for whic... |
synthetic_math | Let $A$, $B$, and $C$ be points such that $\frac{AC}{AB} = \frac{BC}{AB} = \frac{1}{3}$. If $B = (2, 3)$ and $C = (5, 12)$, then what is the sum of the coordinates of $A$? | From the problem, $AC = BC$ and $AC + BC = AB$. This implies that $A$, $B$, and $C$ form a degenerate triangle, meaning $C$ is the midpoint of $AB$. Thus, we will use the vector from $B$ to $C$ (which is $(5-2, 12-3) = (3, 9)$) to extend similarly from $C$ to $A$.
Since $C$ is the midpoint, the same vector that leads ... | [
{
"content": "Let $A$, $B$, and $C$ be points such that $\\frac{AC}{AB} = \\frac{BC}{AB} = \\frac{1}{3}$. If $B = (2, 3)$ and $C = (5, 12)$, then what is the sum of the coordinates of $A$?",
"role": "user"
},
{
"content": "From the problem, $AC = BC$ and $AC + BC = AB$. This implies that $A$, $B$, a... |
cn_k12 | The monotonic decreasing interval of the function $y=\frac{\sqrt{x}}{x+1}$ is ______, and the monotonic increasing interval is ______. | **Analysis**
This problem examines the use of derivatives to determine the monotonicity of a function, which is a basic question.
To solve it, we differentiate the function and use the relationship between the monotonicity of the function and the sign of its derivative.
**Solution**
Given $y= \frac{\sqrt{x}}{x+1}$,... | [
{
"content": "The monotonic decreasing interval of the function $y=\\frac{\\sqrt{x}}{x+1}$ is ______, and the monotonic increasing interval is ______.",
"role": "user"
},
{
"content": "**Analysis**\n\nThis problem examines the use of derivatives to determine the monotonicity of a function, which is ... |
orca_math | When a new person joins a group of 15 people, the average weight of the entire group increases by 3.7 kg after one of them weighing 90 kg leaves. What is the weight of the new person? | Let's denote the total weight of the original group of 15 people as W. The average weight of this group is W/15.
When the person weighing 90 kg leaves, the total weight becomes W - 90, and the number of people in the group becomes 14. The average weight of this new group is (W - 90)/14.
A new person joins the group, ... | [
{
"content": "When a new person joins a group of 15 people, the average weight of the entire group increases by 3.7 kg after one of them weighing 90 kg leaves. What is the weight of the new person?",
"role": "user"
},
{
"content": "Let's denote the total weight of the original group of 15 people as ... |
orca_math | Mikaela earns $10 an hour tutoring. For the first month, she tutored for 35 hours and in the second month, she tutored 5 hours more than the first month. She spent 4/5 of her total earnings on her personal needs and saved the rest of the money. How much did she save? | First, let's calculate Mikaela's earnings for the first month. She earns $10 an hour and tutored for 35 hours, so her earnings for the first month are:
35 hours * $10/hour = $350
In the second month, she tutored for 5 hours more than the first month, so she tutored for:
35 hours + 5 hours = 40 hours
Her earnings fo... | [
{
"content": "Mikaela earns $10 an hour tutoring. For the first month, she tutored for 35 hours and in the second month, she tutored 5 hours more than the first month. She spent 4/5 of her total earnings on her personal needs and saved the rest of the money. How much did she save?",
"role": "user"
},
{
... |
orca_math | The ratio of 3 numbers is 5 : 1 : 4. The last number of the 3 numbers is 400. What is the sum of the 3 numbers? | If the ratio of the three numbers is 5 : 1 : 4, and the last number is 400, we can set up the following proportion to find the value of the first two numbers:
Let the first number be 5x, the second number be 1x, and the third number be 4x.
According to the given information, 4x = 400.
To find the value of x, we divi... | [
{
"content": "The ratio of 3 numbers is 5 : 1 : 4. The last number of the 3 numbers is 400. What is the sum of the 3 numbers?",
"role": "user"
},
{
"content": "If the ratio of the three numbers is 5 : 1 : 4, and the last number is 400, we can set up the following proportion to find the value of the ... |
synthetic_amc | Michael subscribes to an internet service that charges a fixed monthly fee plus a per-hour fee for usage. In February, his bill was $18.60, and in March, his bill rose to $30.90 because he used three times as much internet time as in February. What is the fixed monthly fee?
(A) $12.45$
(B) $15.30$
(C) $18.60$
(D) $10.2... | Let:
- \( f \) be the fixed monthly fee.
- \( h \) be the hourly charge for the internet usage in February.
The equations based on the problem statement are:
1. \( f + h = 18.60 \) (February bill)
2. \( f + 3h = 30.90 \) (March bill, with three times the usage)
**Step 1: Subtract the first equation from the second:**... | [
{
"content": "Michael subscribes to an internet service that charges a fixed monthly fee plus a per-hour fee for usage. In February, his bill was $18.60, and in March, his bill rose to $30.90 because he used three times as much internet time as in February. What is the fixed monthly fee?\n(A) $12.45$\n(B) $15.3... |
olympiads | Prove that when \( n \) and \( k \) are given positive integers, and \( n > 2 \) and \( k > 2 \), \( n(n-1)^{k-1} \) can be written as the sum of \( n \) consecutive even numbers. |
To show that $n(n-1)^{k-1}$ can be written as the sum of $n$ consecutive even numbers for given integers $n > 2$ and $k > 2$:
1. **Expression involving sum of consecutive even numbers**:
Consider the sum of $n$ consecutive even numbers starting from a certain even number \(2m\).
The sum of these $n$ numbe... | [
{
"content": "Prove that when \\( n \\) and \\( k \\) are given positive integers, and \\( n > 2 \\) and \\( k > 2 \\), \\( n(n-1)^{k-1} \\) can be written as the sum of \\( n \\) consecutive even numbers.",
"role": "user"
},
{
"content": "\nTo show that $n(n-1)^{k-1}$ can be written as the sum of $... |
cn_k12 | If the line $ax + 2y + 6 = 0$ is parallel but not coincident with the line $x + (a - 1)y + (a^2 - 1) = 0$, then $a$ equals | **Answer**
For two lines to be parallel but not coincident, their slopes must be equal, and their intercepts must be different. The slope of a line in the form $Ax + By + C = 0$ is $-\frac{A}{B}$.
For the first line, $ax + 2y + 6 = 0$, the slope is $-\frac{a}{2}$.
For the second line, $x + (a - 1)y + (a^2 - 1) = 0... | [
{
"content": "If the line $ax + 2y + 6 = 0$ is parallel but not coincident with the line $x + (a - 1)y + (a^2 - 1) = 0$, then $a$ equals",
"role": "user"
},
{
"content": "**Answer**\n\nFor two lines to be parallel but not coincident, their slopes must be equal, and their intercepts must be different... |
synthetic_math | How many three-digit perfect cubes are divisible by $8$? | A cube is divisible by $8$ if the number being cubed is divisible by $\sqrt[3]{8} = 2$. This implies the number being cubed should be a multiple of $2$, so the cube will be of the form $(2m)^3 = 8m^3$. To find how many such cubes are three-digit numbers, we need to find $m$ such that $100 \leq 8m^3 \leq 999$.
We simpl... | [
{
"content": "How many three-digit perfect cubes are divisible by $8$?",
"role": "user"
},
{
"content": "A cube is divisible by $8$ if the number being cubed is divisible by $\\sqrt[3]{8} = 2$. This implies the number being cubed should be a multiple of $2$, so the cube will be of the form $(2m)^3 =... |
synthetic_math | Add $321_8 + 127_8$. Express your answer in base 8. | We line up the two octal numbers and add them column by column, following the rules of addition in base 8 (i.e., octal):
$$
\begin{array}{c@{}c@{}c@{}c@{}c}
& 3 & 2 & 1_8 \\
+ & 1 & 2 & 7_8 \\
\hline
\end{array}
$$
1. In the rightmost column: $1_8 + 7_8 = 10_{8}$, which translates to $0_8$ in this column and carryi... | [
{
"content": "Add $321_8 + 127_8$. Express your answer in base 8.",
"role": "user"
},
{
"content": "We line up the two octal numbers and add them column by column, following the rules of addition in base 8 (i.e., octal):\n\n$$\n\\begin{array}{c@{}c@{}c@{}c@{}c}\n & 3 & 2 & 1_8 \\\\\n+ & 1 & 2 & 7_8... |
olympiads | The integers $a$, $b$, and $c$ are such that $a^3 + b^3 + c^3$ is divisible by 7. Prove that $abc$ is divisible by 7. | To prove that if \(a^3 + b^3 + c^3\) is divisible by 7, then \(abc\) must also be divisible by 7, we use a proof by contradiction. Suppose \(abc\) is **not** divisible by 7.
1. **Assume \(a\), \(b\), and \(c\) are not divisible by 7:**
Since \(a\) is not divisible by 7, by Fermat's Little Theorem, we have:
\[
... | [
{
"content": "The integers $a$, $b$, and $c$ are such that $a^3 + b^3 + c^3$ is divisible by 7. Prove that $abc$ is divisible by 7.",
"role": "user"
},
{
"content": "To prove that if \\(a^3 + b^3 + c^3\\) is divisible by 7, then \\(abc\\) must also be divisible by 7, we use a proof by contradiction.... |
cn_k12 | A factory produces a certain product, and the defective rate $p$ is related to the daily output $x$ (in ten thousand units) as follows: $P= \begin{cases} \dfrac {1}{6-x}, & 0 < x\leqslant c \\ \dfrac {2}{3}, & x > c\end{cases}$ ($c$ is a constant, and $0 < c < 6$). It is known that for every qualified product produced,... | Solution:
$(1)$ When $x > c$, $p= \dfrac {2}{3}$,
$\therefore y= \dfrac {1}{3} \cdot x \cdot 3 - \dfrac {2}{3} \cdot x \cdot \dfrac {3}{2} = 0$
When $0 < x\leqslant c$, $p= \dfrac {1}{6-x}$
$\therefore y=(1- \dfrac {1}{6-x}) \cdot x \cdot 3 - \dfrac {1}{6-x} \cdot x \cdot \dfrac {3}{2} = \dfrac {3}{2} \cdot \df... | [
{
"content": "A factory produces a certain product, and the defective rate $p$ is related to the daily output $x$ (in ten thousand units) as follows: $P= \\begin{cases} \\dfrac {1}{6-x}, & 0 < x\\leqslant c \\\\ \\dfrac {2}{3}, & x > c\\end{cases}$ ($c$ is a constant, and $0 < c < 6$). It is known that for ever... |
orca_math | A 340-liter solution of kola is made from 80% water, a certain percentage of concentrated kola, and the rest is made from sugar. If 3.2 liters of sugar, 10 liters of water, and 6.8 liters of concentrated kola were added to the solution, 14.111111111111112% of the solution is made from sugar. What percentage of the solu... | Let's start by calculating the initial amounts of water, concentrated kola, and sugar in the 340-liter solution.
Since the solution is 80% water, the initial amount of water is:
\( 0.80 \times 340 \) liters = \( 272 \) liters
Let \( c \) be the percentage of concentrated kola in the solution initially. Then the initi... | [
{
"content": "A 340-liter solution of kola is made from 80% water, a certain percentage of concentrated kola, and the rest is made from sugar. If 3.2 liters of sugar, 10 liters of water, and 6.8 liters of concentrated kola were added to the solution, 14.111111111111112% of the solution is made from sugar. What ... |
orca_math | A train 100 meters long completely crosses a bridge of a certain length in 15 seconds. The speed of the train is 96 km/h. What is the length of the bridge? | First, we need to convert the speed of the train from kilometers per hour (km/h) to meters per second (m/s) because the length of the train and the time taken to cross the bridge are given in meters and seconds, respectively.
To convert km/h to m/s, we use the conversion factor:
1 km/h = 1000 meters / 3600 seconds
So... | [
{
"content": "A train 100 meters long completely crosses a bridge of a certain length in 15 seconds. The speed of the train is 96 km/h. What is the length of the bridge?",
"role": "user"
},
{
"content": "First, we need to convert the speed of the train from kilometers per hour (km/h) to meters per s... |
orca_math | the length of the bridge , which a train 475 metres long and travelling at 90 km / hr can cross in 30 seconds , is : | First, we need to convert the speed of the train from km/hr to m/s because the length of the train is given in meters and the time in seconds.
To convert km/hr to m/s, we use the conversion factor: 1 km/hr = 1000 m / 3600 s
So, the speed of the train in m/s is:
90 km/hr * (1000 m / 3600 s) = 25 m/s
Now, we can calc... | [
{
"content": "the length of the bridge , which a train 475 metres long and travelling at 90 km / hr can cross in 30 seconds , is :",
"role": "user"
},
{
"content": "First, we need to convert the speed of the train from km/hr to m/s because the length of the train is given in meters and the time in s... |
aops_forum | <details><summary>D stands for Dedekind, Z stands for Zermelo</summary>they had two problem sets under those two names</details>
<u>Set 4</u>**D16.** The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least t... | To solve the problem, we need to determine the digits \(a\) and \(b\) such that the number \(12ab9876543\) is divisible by 101. Let's break down the steps:
1. **Express the number in terms of \(a\) and \(b\):**
\[
N = \overline{12ab9876543} = 12009876543 + (10a + b) \times 10^7
\]
Since \(N\) is divisible ... | [
{
"content": "<details><summary>D stands for Dedekind, Z stands for Zermelo</summary>they had two problem sets under those two names</details> \n\n\n<u>Set 4</u>**D16.** The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one... |
synthetic_math | A rectangular prism has a length of $l = 5$ meters, a height of $h = 13$ meters, and the length of its diagonal is $15$ meters. Determine the width $w$ of the prism. | We know from the geometry of a rectangular prism that the diagonal $d$ can be calculated using the formula:
$$ d = \sqrt{l^2 + w^2 + h^2} $$
Given $l = 5$, $h = 13$, and $d = 15$, we substitute these values into the equation:
$$ 15 = \sqrt{5^2 + w^2 + 13^2} $$
$$ 15 = \sqrt{25 + w^2 + 169} $$
$$ 15 = \sqrt{194 + w^2} $... | [
{
"content": "A rectangular prism has a length of $l = 5$ meters, a height of $h = 13$ meters, and the length of its diagonal is $15$ meters. Determine the width $w$ of the prism.",
"role": "user"
},
{
"content": "We know from the geometry of a rectangular prism that the diagonal $d$ can be calculat... |
synthetic_math | The volume of a sphere is \(150\pi\) cm\(^3\). How many cubic centimeters are in the volume of a hemisphere with the same radius as the sphere? Express your answer in terms of \(\pi\). | 1. **Volume of a Sphere**: The formula for the volume of a sphere is \(\frac{4}{3}\pi r^3\). Given that the volume is \(150\pi\), we set up the equation:
\[
\frac{4}{3}\pi r^3 = 150\pi
\]
Solving for \(r^3\), we get:
\[
r^3 = \frac{150\pi \times 3}{4\pi} = 112.5
\]
2. **Volume of a Hemisphere**: T... | [
{
"content": "The volume of a sphere is \\(150\\pi\\) cm\\(^3\\). How many cubic centimeters are in the volume of a hemisphere with the same radius as the sphere? Express your answer in terms of \\(\\pi\\).",
"role": "user"
},
{
"content": "1. **Volume of a Sphere**: The formula for the volume of a ... |
synthetic_amc | Alicia earns 25 dollars per hour, of which 2% is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
A) 40 cents
B) 45 cents
C) 50 cents
D) 55 cents
E) 60 cents | 1. **Convert Alicia's hourly wage to cents**: Alicia earns $25$ dollars per hour. Since there are $100$ cents in a dollar, her hourly wage in cents is:
\[
25 \text{ dollars} \times 100 \text{ cents per dollar} = 2500 \text{ cents}
\]
2. **Calculate the tax deduction in cents**: The new local tax rate is $2\%$... | [
{
"content": "Alicia earns 25 dollars per hour, of which 2% is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?\nA) 40 cents\nB) 45 cents\nC) 50 cents\nD) 55 cents\nE) 60 cents",
"role": "user"
},
{
"content": "1. **Convert Alicia's hourly wage to c... |
orca_math | The average (arithmetic mean) of 20, 40, and 60 is 5 more than the average of 10, 50, and what number? | First, let's find the average of 20, 40, and 60. The average is the sum of the numbers divided by the count of the numbers.
Average of 20, 40, and 60:
(20 + 40 + 60) / 3 = 120 / 3 = 40
Now, let's denote the unknown number as x. We are told that the average of 20, 40, and 60 is 5 more than the average of 10, 50, and x... | [
{
"content": "The average (arithmetic mean) of 20, 40, and 60 is 5 more than the average of 10, 50, and what number?",
"role": "user"
},
{
"content": "First, let's find the average of 20, 40, and 60. The average is the sum of the numbers divided by the count of the numbers.\n\nAverage of 20, 40, and... |
orca_math | Find the mass percentage of each element (C, H, N, and O) in the molecule C20H25N3O. | To find the mass percentage of each element in the molecule C20H25N3O, we first need to calculate the molar mass of the entire molecule and then the molar mass contributed by each element. Then we can find the mass percentage of each element by dividing the mass contributed by the element by the total molar mass of the... | [
{
"content": "Find the mass percentage of each element (C, H, N, and O) in the molecule C20H25N3O.",
"role": "user"
},
{
"content": "To find the mass percentage of each element in the molecule C20H25N3O, we first need to calculate the molar mass of the entire molecule and then the molar mass contrib... |
synthetic_math | How many different four-letter sets of initials are possible using the letters $A$ through $J$? | 1. **Determine the range of letters:** There are ten choices for each initial, representing the letters $A$ to $J$.
2. **Independence of choices:** Each of the four positions in the set of initials can be filled independently from these 10 letters.
3. **Calculate total combinations:** The total number of different sets... | [
{
"content": "How many different four-letter sets of initials are possible using the letters $A$ through $J$?",
"role": "user"
},
{
"content": "1. **Determine the range of letters:** There are ten choices for each initial, representing the letters $A$ to $J$.\n2. **Independence of choices:** Each of... |
olympiads | Solve for all complex numbers \( z \) such that \( z^{4} + 4z^{2} + 6 = z \). |
We are given the problem of solving for all complex numbers \( z \) such that
\[ z^4 + 4z^2 + 6 = z. \]
1. **Rewrite the given equation**: Initially, we can rewrite the given equation in a form that helps us look for solutions more effectively. We want to manipulate the equation to find possible factors.
\[
z... | [
{
"content": "Solve for all complex numbers \\( z \\) such that \\( z^{4} + 4z^{2} + 6 = z \\).",
"role": "user"
},
{
"content": "\nWe are given the problem of solving for all complex numbers \\( z \\) such that\n\n\\[ z^4 + 4z^2 + 6 = z. \\]\n\n1. **Rewrite the given equation**: Initially, we can r... |
aops_forum | What is the smallest positive integer $k$ such that the number $\textstyle\binom{2k}k$ ends in two zeros? $\textbf{(A) }3\hspace{14em}\textbf{(B) }4\hspace{14em}\textbf{(C) }5$ $\textbf{(D) }6\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }8$ $\textbf{(G) }9\hspace{14em}\textbf{(H) }10\hspace{13.3em}\textbf{... | To determine the smallest positive integer \( k \) such that the number \( \binom{2k}{k} \) ends in two zeros, we need to ensure that \( \binom{2k}{k} \) is divisible by \( 100 \). This means that \( \binom{2k}{k} \) must have at least two factors of 2 and two factors of 5 in its prime factorization.
Recall that the b... | [
{
"content": "What is the smallest positive integer $k$ such that the number $\\textstyle\\binom{2k}k$ ends in two zeros? $\\textbf{(A) }3\\hspace{14em}\\textbf{(B) }4\\hspace{14em}\\textbf{(C) }5$ $\\textbf{(D) }6\\hspace{14em}\\textbf{(E) }7\\hspace{14em}\\textbf{(F) }8$ $\\textbf{(G) }9\\hspace{14em}\\... |
orca_math | A alone can do a piece of work in 6 days and B alone in some days. A and B undertook to do it for Rs. 3200. With the help of C, they completed the work in 3 days. C is to be paid Rs. 400.0000000000002. How many days does it take for B alone to do the work? | Let's denote the number of days B alone takes to do the work as \( B \) days.
A's one day work is \( \frac{1}{6} \) of the total work since A can complete the work in 6 days.
B's one day work is \( \frac{1}{B} \) of the total work.
When A, B, and C work together, they complete the work in 3 days. So, their combined o... | [
{
"content": "A alone can do a piece of work in 6 days and B alone in some days. A and B undertook to do it for Rs. 3200. With the help of C, they completed the work in 3 days. C is to be paid Rs. 400.0000000000002. How many days does it take for B alone to do the work?",
"role": "user"
},
{
"conten... |
cn_k12 | Among the following options, which set of three side lengths cannot form a right triangle?
A: $5$, $7$, $10$
B: $3$, $4$, $5$
C: $1$, $3$, $2$
D: $7$, $24$, $25$ | To determine which set of three side lengths cannot form a right triangle, we apply the Pythagorean theorem for right triangles and the triangle inequality theorem for general triangles. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angl... | [
{
"content": "Among the following options, which set of three side lengths cannot form a right triangle?\n\nA: $5$, $7$, $10$\n\nB: $3$, $4$, $5$\n\nC: $1$, $3$, $2$\n\nD: $7$, $24$, $25$",
"role": "user"
},
{
"content": "To determine which set of three side lengths cannot form a right triangle, we ... |
cn_k12 | In electronic communication and the internet, signal transmission, processing, and Fourier transform are related. Fourier transform can represent a function that meets certain conditions as a linear combination of trigonometric functions (sine and cosine functions). For example, the graph of the function $f(x)=\frac{{\... | To analyze the function $f(x)=\frac{\sin x}{1}+\frac{\sin 3x}{3}+\frac{\sin 5x}{5}+\cdots+\frac{\sin 13x}{13}$ and the given statements, we proceed as follows:
**For Statement A: Periodicity of $f(x)$**
To check if $\pi$ is the period of $f(x)$, we evaluate $f(x+\pi)$:
\begin{align*}
f(x+\pi) &= \frac{\sin(x+\pi)}{1}... | [
{
"content": "In electronic communication and the internet, signal transmission, processing, and Fourier transform are related. Fourier transform can represent a function that meets certain conditions as a linear combination of trigonometric functions (sine and cosine functions). For example, the graph of the f... |
synthetic_amc | Several students are seated at a large circular table. They pass around a bag containing $120$ pieces of candy. Each person receives the bag, takes one piece of candy, and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be?
$\text{... | 1. **Understanding the distribution**: The problem states that the bag containing 120 candies starts with Chris and also ends with him. After each student takes one candy, the cycle completes when Chris receives the bag again after the last candy is taken.
2. **Calculating total rounds**: Chris taking the first and la... | [
{
"content": "Several students are seated at a large circular table. They pass around a bag containing $120$ pieces of candy. Each person receives the bag, takes one piece of candy, and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the t... |
olympiads | Through the vertex \(C\) of the base of a regular triangular pyramid \(SABC\), a plane is drawn perpendicular to the lateral edge \(SA\). This plane forms an angle with the base plane, the cosine of which is \( \frac{2}{3} \). Find the cosine of the angle between two lateral faces. |
1. **Identify the Configuration**:
- Consider a regular triangular pyramid \( SABC \) with the plane passing through vertex \( C \) and perpendicular to lateral edge \( SA \).
- Let \( CK \) and \( BK \) denote the intersections of the plane with edges \( SC \) and \( SB \) respectively, forming a triangle \(... | [
{
"content": "Through the vertex \\(C\\) of the base of a regular triangular pyramid \\(SABC\\), a plane is drawn perpendicular to the lateral edge \\(SA\\). This plane forms an angle with the base plane, the cosine of which is \\( \\frac{2}{3} \\). Find the cosine of the angle between two lateral faces.",
... |
olympiads | Find all composite natural numbers \( n \) that possess the following property: every natural divisor of \( n \) (in particular, \( n \) itself), decreased by 1, is a square of an integer. | To find all composite natural numbers \( n \) such that for every natural divisor \( d \) of \( n \), \( d-1 \) is a perfect square, we proceed as follows:
1. **Assume the Form**: Assume \( n \) is a composite number divisible by the square of a prime number, say \( p \). Then, \( n \) must have a divisor \( p^2 \). A... | [
{
"content": "Find all composite natural numbers \\( n \\) that possess the following property: every natural divisor of \\( n \\) (in particular, \\( n \\) itself), decreased by 1, is a square of an integer.",
"role": "user"
},
{
"content": "To find all composite natural numbers \\( n \\) such that... |
cn_k12 | In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that $$\frac {sin2B}{ \sqrt {3}cos(B+C)-cosCsinB}= \frac {2b}{c}$$.
(I) Find the measure of angle A.
(II) If $$a= \sqrt {3}$$, find the maximum area of triangle ABC. | (I) Since $$\frac {sin2B}{ \sqrt {3}cos(B+C)-cosCsinB}= \frac {2b}{c}$$, we can derive: $$\frac {2sinBcosB}{- \sqrt {3}cosA-cosCsinB}= \frac {2sinB}{sinC}$$.
As sinB ≠ 0, we can simplify to get: sinCcosB = -$$\sqrt {3}$$cosA - cosCsinB.
This further simplifies to: -$$\sqrt {3}$$cosA = sinCcosB + cosCsinB = sin(B+C) = s... | [
{
"content": "In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that $$\\frac {sin2B}{ \\sqrt {3}cos(B+C)-cosCsinB}= \\frac {2b}{c}$$.\n(I) Find the measure of angle A.\n(II) If $$a= \\sqrt {3}$$, find the maximum area of triangle ABC.",
"role": "user"
},
{
... |
olympiads | The lines containing the bisectors of the exterior angles of a triangle with angle measures of 42 and 59 degrees intersect pairwise to form a new triangle. Find the degree measure of its largest angle. |
1. **Identify Given Elements and Setup**
Given a triangle with angles 42°, 59°, and 79° (since the sum of angles in a triangle is 180°).
Let's denote the vertices of the given triangle as \( A, B, \) and \( C \), and the angles at these vertices as \( \alpha = 42^\circ \), \( \beta = 59^\circ \), and \( \gam... | [
{
"content": "The lines containing the bisectors of the exterior angles of a triangle with angle measures of 42 and 59 degrees intersect pairwise to form a new triangle. Find the degree measure of its largest angle.",
"role": "user"
},
{
"content": "\n1. **Identify Given Elements and Setup** \n ... |
synthetic_amc | Given the equations $z^2=1+3\sqrt{15}i$ and $z^2=3+5\sqrt{3}i$, where $i=\sqrt{-1}$, calculate the area of the parallelogram formed by the roots of these equations in the complex plane.
A) $24\sqrt{17}-40\sqrt{3}$
B) $40\sqrt{17}-24\sqrt{3}$
C) $24\sqrt{3}-40\sqrt{17}$
D) $84\sqrt{21}-16\sqrt{5}$ | ### Solving $z^2 = 1 + 3\sqrt{15}i$:
Using polar form $z = r\operatorname{cis}\theta$, we need $r^2\operatorname{cis}(2\theta) = 1+3\sqrt{15}i$. This gives us:
\[ r^2 = |1 + 3\sqrt{15}i| = \sqrt{1^2+(3\sqrt{15})^2} = \sqrt{136}, \]
so $r = \sqrt{34}$.
\[ \operatorname{cis}(2\theta) = \frac{1}{\sqrt{34}} + \frac{3\sqrt{... | [
{
"content": "Given the equations $z^2=1+3\\sqrt{15}i$ and $z^2=3+5\\sqrt{3}i$, where $i=\\sqrt{-1}$, calculate the area of the parallelogram formed by the roots of these equations in the complex plane.\nA) $24\\sqrt{17}-40\\sqrt{3}$ \nB) $40\\sqrt{17}-24\\sqrt{3}$ \nC) $24\\sqrt{3}-40\\sqrt{17}$ \nD) $84\\s... |
cn_k12 | Given the function $f(x) = \sin(\omega x - \frac{\pi}{6}) + \frac{1}{2}$, where $x \in \mathbb{R}$, and $f(\alpha) = -\frac{1}{2}$, $f(\beta) = \frac{1}{2}$. If the minimum value of $|\alpha - \beta|$ is $\frac{3\pi}{4}$, find the value of $\omega$.
Options:
A) $\frac{4}{3}$
B) $\frac{2}{3}$
C) $1$
D) $\frac{8}{3}$ | Since $f(\alpha) = -\frac{1}{2}$ and $f(\beta) = \frac{1}{2}$, and given that the minimum value of $|\alpha - \beta|$ is $\frac{3\pi}{4}$, we can infer that $\alpha$ and $\beta$ correspond to points on the sine curve that are $\frac{1}{4}$ of the period apart, since these are the points where the sine function obtains ... | [
{
"content": "Given the function $f(x) = \\sin(\\omega x - \\frac{\\pi}{6}) + \\frac{1}{2}$, where $x \\in \\mathbb{R}$, and $f(\\alpha) = -\\frac{1}{2}$, $f(\\beta) = \\frac{1}{2}$. If the minimum value of $|\\alpha - \\beta|$ is $\\frac{3\\pi}{4}$, find the value of $\\omega$.\nOptions:\nA) $\\frac{4}{3}$\nB)... |
synthetic_math | Let $g_1(x) = \frac{1}{2} - \frac{4}{4x+2},$ and for $n \ge 2,$ define $g_n(x) = g_1(g_{n-1}(x)).$ Find the value of $x$ that satisfies $g_{1002}(x) = x-4.$ | Firstly, simplify $g_1(x)$:
\[ g_1(x) = \frac{1}{2} - \frac{4}{4x+2} = \frac{4x+2 - 8}{4(4x+2)} = \frac{4x-6}{16x+8} = \frac{2x-3}{8x+4}. \]
Compute $g_2(x)$:
\[ g_2(x) = g_1\left(\frac{2x-3}{8x+4}\right) = \frac{2 \cdot \frac{2x-3}{8x+4} - 3}{8 \cdot \frac{2x-3}{8x+4} + 4} = \frac{4x-12}{32x+28} = \frac{x-3}{8x+7}. \]... | [
{
"content": "Let $g_1(x) = \\frac{1}{2} - \\frac{4}{4x+2},$ and for $n \\ge 2,$ define $g_n(x) = g_1(g_{n-1}(x)).$ Find the value of $x$ that satisfies $g_{1002}(x) = x-4.$",
"role": "user"
},
{
"content": "Firstly, simplify $g_1(x)$:\n\\[ g_1(x) = \\frac{1}{2} - \\frac{4}{4x+2} = \\frac{4x+2 - 8}{... |
cn_k12 | Given that curve C is the locus of a moving point whose distances to two fixed points A(1, 0) and B(4, 0) have a ratio of $\frac{1}{2}$.
(1) Find the equation of curve C;
(2) Find the minimum and maximum values of the distance $d$ from points on curve C to the line $l: x - y + 3 = 0$. | Solution:
(1) Let the coordinates of the moving point be $(x, y)$. Then, $\frac{\sqrt{(x-1)^2+y^2}}{\sqrt{(x-4)^2+y^2}} = \frac{1}{2}$,
which simplifies to $x^2+y^2=4$;
(2) The distance from the center of the circle to the line is $\frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} > 2$,
therefore, the minimum value of $... | [
{
"content": "Given that curve C is the locus of a moving point whose distances to two fixed points A(1, 0) and B(4, 0) have a ratio of $\\frac{1}{2}$.\n\n(1) Find the equation of curve C; \n(2) Find the minimum and maximum values of the distance $d$ from points on curve C to the line $l: x - y + 3 = 0$.",
... |
orca_math | A train of length l is traveling at a constant velocity and passes a pole in a certain amount of time. If the same train traveling at the same velocity passes a platform in 4 times that amount of time, and the length of the platform is 3 times the length of the train, what is the time it takes for the train to pass the... | Let's denote the length of the train as \( l \) and the velocity of the train as \( v \). Let \( t \) be the time it takes for the train to pass the pole.
When the train passes the pole, it only needs to cover its own length \( l \) at velocity \( v \). Therefore, we can write the equation:
\[ l = v \cdot t \]
\[ t =... | [
{
"content": "A train of length l is traveling at a constant velocity and passes a pole in a certain amount of time. If the same train traveling at the same velocity passes a platform in 4 times that amount of time, and the length of the platform is 3 times the length of the train, what is the time it takes for... |
cn_k12 | Select 2 digits from the odd numbers 1, 3, 5, 7, 9, and 3 digits from the even numbers 0, 2, 4, 6, 8, to form as many unique five-digit numbers as possible. How many such numbers can be formed? | **Analysis**
First, select 2 out of the 5 odd numbers, and then select 3 out of the 4 even numbers (excluding 0, so only 2, 4, 6, 8 are considered) to arrange into a five-digit number. This can be done in $C_5^2 \cdot C_4^3 \cdot A_5^5 = 4,800$ ways.
Next, select 2 out of the 5 odd numbers, and then select 2 out of t... | [
{
"content": "Select 2 digits from the odd numbers 1, 3, 5, 7, 9, and 3 digits from the even numbers 0, 2, 4, 6, 8, to form as many unique five-digit numbers as possible. How many such numbers can be formed?",
"role": "user"
},
{
"content": "**Analysis**\n\nFirst, select 2 out of the 5 odd numbers, ... |
cn_k12 | Given real numbers $a = \log_2{3}$, $b = \log_{\frac{1}{3}}{\frac{1}{2}}$, $c = \frac{1}{\int_{0}^{\pi} x \, dx}$, determine the correct order of these three numbers. | First, let's evaluate each term.
For a, we know that
$$a = \log_2{3} > \log_2{2} = 1,$$
since 3 is greater than 2 and the logarithm function is strictly increasing for base larger than 1.
Now, for b, we need to be careful with the base:
$$
b = \log_{\frac{1}{3}}{\frac{1}{2}} = \frac{\log_{3}{2}}{\log_{3}{\frac{1}{3... | [
{
"content": "Given real numbers $a = \\log_2{3}$, $b = \\log_{\\frac{1}{3}}{\\frac{1}{2}}$, $c = \\frac{1}{\\int_{0}^{\\pi} x \\, dx}$, determine the correct order of these three numbers.",
"role": "user"
},
{
"content": "First, let's evaluate each term.\n\nFor a, we know that \n$$a = \\log_2{3} > ... |
cn_k12 | The equation of the tangent line to the graph of the function $y=f(x)$ at the point $x=5$ is $y=-x+8$. Find the value of $f(5)+f′(5)$, which is equal to ( ).
A: $1$
B: $2$
C: $0$
D: $\frac {1}{2}$ | First, we are given the equation of the tangent line to the graph of the function at $x=5$, which is $y=-x+8$. The slope of this tangent line is $-1$. Since the derivative of a function at a specific point gives the slope of the tangent line at that point, we know that $f'(5) = -1$.
To find $f(5)$, we need to find the... | [
{
"content": "The equation of the tangent line to the graph of the function $y=f(x)$ at the point $x=5$ is $y=-x+8$. Find the value of $f(5)+f′(5)$, which is equal to ( ).\nA: $1$\nB: $2$\nC: $0$\nD: $\\frac {1}{2}$",
"role": "user"
},
{
"content": "First, we are given the equation of the tangent ... |
synthetic_math | If $x^2 + bx + 16$ has two non-real roots, find all real possible values of $b$. Express your answer in interval notation. | Consider the quadratic equation $x^2 + bx + 16 = 0$. To determine the condition for non-real roots, we look at the discriminant:
\[
\Delta = b^2 - 4ac
\]
Here, $a = 1$ and $c = 16$. For the roots to be non-real, we need $\Delta < 0$:
\[
b^2 - 4(1)(16) < 0 \implies b^2 - 64 < 0
\]
Solving this inequality:
\[
b^2 < 64 \i... | [
{
"content": "If $x^2 + bx + 16$ has two non-real roots, find all real possible values of $b$. Express your answer in interval notation.",
"role": "user"
},
{
"content": "Consider the quadratic equation $x^2 + bx + 16 = 0$. To determine the condition for non-real roots, we look at the discriminant:\... |
cn_k12 | Simplify, and then evaluate.
(1) Simplify $-( -a^2 + 2ab + b^2 ) + ( -a^2 - ab + b^2 )$;
(2) When $a$ and $b$ are reciprocals of each other, evaluate the expression above. | Let's tackle this problem step by step:
**Step 1: Simplify the Expression**
First, we simplify the given expression in part (1):
\[
\begin{align*}
& -( -a^2 + 2ab + b^2 ) + ( -a^2 - ab + b^2 ) \\
&= a^2 - 2ab - b^2 - a^2 - ab + b^2 \\
&= -3ab + a^2 - a^2 + b^2 - b^2 \\
&= \boxed{-3ab}.
\end{align*}
\]
Here, we used... | [
{
"content": "Simplify, and then evaluate.\n(1) Simplify $-( -a^2 + 2ab + b^2 ) + ( -a^2 - ab + b^2 )$;\n(2) When $a$ and $b$ are reciprocals of each other, evaluate the expression above.",
"role": "user"
},
{
"content": "Let's tackle this problem step by step:\n\n**Step 1: Simplify the Expression**... |
orca_math | Jeremy loves to play Scrabble. He once played a three-letter word on a triple word score to earn thirty points. If before the word score was tripled, the first and third letters were each valued at one point apiece, how much was the middle letter valued at before the word score was tripled? | Let's denote the value of the middle letter as x.
The total score before tripling is 1 (first letter) + x (middle letter) + 1 (third letter) = 2 + x.
Since the total score after tripling is 30 points, we have:
3 * (2 + x) = 30
Dividing both sides by 3 to solve for x:
2 + x = 30 / 3
2 + x = 10
Subtracting 2 from b... | [
{
"content": "Jeremy loves to play Scrabble. He once played a three-letter word on a triple word score to earn thirty points. If before the word score was tripled, the first and third letters were each valued at one point apiece, how much was the middle letter valued at before the word score was tripled?",
... |
orca_math | A number subtracted from 821 is 267. What is the value when 267 is subtracted from the number? | Let's call the number we are looking for "x". According to the information given:
821 - x = 267
To find the value of x, we add x to both sides and subtract 267 from both sides:
821 = x + 267
821 - 267 = x
Now we calculate the value of x:
x = 821 - 267
x = 554
Now, we need to find the value when 267 is subtracted ... | [
{
"content": "A number subtracted from 821 is 267. What is the value when 267 is subtracted from the number?",
"role": "user"
},
{
"content": "Let's call the number we are looking for \"x\". According to the information given:\n\n821 - x = 267\n\nTo find the value of x, we add x to both sides and su... |
aops_forum | A pen costs $\mathrm{Rs.}\, 13$ and a note book costs $\mathrm{Rs.}\, 17$ . A school spends exactly $\mathrm{Rs.}\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$ , the school spends a l... | 1. We start with the given diophantine equation:
\[
17x + 13y = 10000
\]
We need to find integer solutions \((x, y)\) such that \(|x - y|\) is minimized.
2. By observation, we find one particular solution:
\[
17 \cdot 589 - 13 \cdot 1 = 10000
\]
This gives us the solution \((x, y) = (589, 1)\).... | [
{
"content": "A pen costs $\\mathrm{Rs.}\\, 13$ and a note book costs $\\mathrm{Rs.}\\, 17$ . A school spends exactly $\\mathrm{Rs.}\\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018... |
cn_k12 | A certain university's mathematics department has a total of 5000 undergraduate students, with the first, second, third, and fourth years having a ratio of their numbers as 4:3:2:1. If stratified sampling method is employed to select a sample of 200 students from all undergraduates, the number of third-year students th... | Since the total number of students is 5000 and the ratio of first, second, third, and fourth-year students is 4:3:2:1, let's determine the total ratio units:
$$ 4 + 3 + 2 + 1 = 10. $$
Using the ratio, we calculate the number of third-year students:
$$ \text{Number of third-year students} = \frac{2}{10} \times 5000. ... | [
{
"content": "A certain university's mathematics department has a total of 5000 undergraduate students, with the first, second, third, and fourth years having a ratio of their numbers as 4:3:2:1. If stratified sampling method is employed to select a sample of 200 students from all undergraduates, the number of ... |
synthetic_amc | An inverted cone with a base radius of $15 \, \text{cm}$ and height $24 \, \text{cm}$ is full of water. This water is then poured into a tall cylinder with a base radius of $30 \, \text{cm}$. Determine the height of the water in the cylinder.
$\textbf{(A)} ~1.5 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~2.5 \qquad\textb... | 1. **Calculate the volume of water initially in the cone**:
The volume $V$ of a cone is provided by:
\[
V = \frac{1}{3} \pi r^2 h
\]
Plugging in $r = 15 \text{ cm}$ and $h = 24 \text{ cm}$, the volume becomes:
\[
V = \frac{1}{3} \pi (15)^2 (24) = \frac{1}{3} \pi \cdot 225 \cdot 24 = 1800 \pi \tex... | [
{
"content": "An inverted cone with a base radius of $15 \\, \\text{cm}$ and height $24 \\, \\text{cm}$ is full of water. This water is then poured into a tall cylinder with a base radius of $30 \\, \\text{cm}$. Determine the height of the water in the cylinder.\n$\\textbf{(A)} ~1.5 \\qquad\\textbf{(B)} ~2 \\qq... |
olympiads | Find the limit $\lim _{x \rightarrow 1} \frac{3 x^{2}-x-2}{4 x^{2}-5 x+1}$. |
1. First, we identify the function to be simplified:
\[ f(x) = \frac{3x^2 - x - 2}{4x^2 - 5x + 1} \]
2. We substitute \( x = 1 \):
\[
f(1) = \frac{3(1)^2 - 1 - 2}{4(1)^2 - 5(1) + 1} = \frac{3 - 1 - 2}{4 - 5 + 1} = \frac{0}{0}
\]
This is an indeterminate form of the type \(\frac{0}{0}\).
3. To resolve the indetermi... | [
{
"content": "Find the limit $\\lim _{x \\rightarrow 1} \\frac{3 x^{2}-x-2}{4 x^{2}-5 x+1}$.",
"role": "user"
},
{
"content": "\n1. First, we identify the function to be simplified:\n\\[ f(x) = \\frac{3x^2 - x - 2}{4x^2 - 5x + 1} \\]\n\n2. We substitute \\( x = 1 \\):\n\\[ \nf(1) = \\frac{3(1)^2 - 1... |
cn_k12 | Given a rectangular prism, the lengths of three edges that meet at one vertex are $1$, $\sqrt{10}$, and $5$, respectively. If all the vertices of the rectangular prism lie on the surface of a sphere, what is the volume of this sphere? ( )
A: $288π$
B: $144π$
C: $108π$
D: $36π$ | **Analysis**
This problem primarily tests the ability to solve for the volume of a sphere circumscribed about a rectangular prism. This is a fundamental question. To solve this problem, the key is to use the space diagonal of the rectangular prism as the diameter of the sphere, combine it with the lengths of the edges... | [
{
"content": "Given a rectangular prism, the lengths of three edges that meet at one vertex are $1$, $\\sqrt{10}$, and $5$, respectively. If all the vertices of the rectangular prism lie on the surface of a sphere, what is the volume of this sphere? ( )\nA: $288π$\nB: $144π$\nC: $108π$\nD: $36π$",
"role": "... |
olympiads | A container in the shape of a cube has edge length 20 cm and contains some water. A solid gold cube, with edge length 15 cm, sinks to the bottom of this container, causing the water level to rise just to the top of the solid cube. Which of the following is closest to the original depth of the water?
(A) 6.56 cm
(B) 8.2... | 1. Define $d$ as the original depth of the water in the container before the gold cube is added.
2. Calculate the initial volume of water in the container:
\[
\text{Volume of water} = \text{base area} \times \text{height} = 20 \, \mathrm{cm} \times 20 \, \mathrm{cm} \times d = 400d \, \mathrm{cm^3}
\]
3. Cal... | [
{
"content": "A container in the shape of a cube has edge length 20 cm and contains some water. A solid gold cube, with edge length 15 cm, sinks to the bottom of this container, causing the water level to rise just to the top of the solid cube. Which of the following is closest to the original depth of the wate... |
cn_k12 | If the terminal side of angle $α$ passes through point $P(1,-2)$, then the value of $\tan 2α$ is ______. | Since the terminal side of angle $α$ passes through point $P(1,-2)$,
$\therefore \tan α= \frac{-2}{1}=-2 \Rightarrow \tan 2α= \frac{2\tan α}{1-\tan^{2}α} = \frac{4}{3}$
Therefore, the answer is: $\boxed{\frac{4}{3}}$.
Based on the terminal side of angle $α$ passing through point $P(1,-2)$, we can first find the valu... | [
{
"content": "If the terminal side of angle $α$ passes through point $P(1,-2)$, then the value of $\\tan 2α$ is ______.",
"role": "user"
},
{
"content": "Since the terminal side of angle $α$ passes through point $P(1,-2)$,\n\n$\\therefore \\tan α= \\frac{-2}{1}=-2 \\Rightarrow \\tan 2α= \\frac{2\\ta... |
aops_forum | Every bacterium has a horizontal body with natural length and some nonnegative number of vertical feet, each with nonnegative (!) natural length, that lie below its body. In how many ways can these bacteria fill an $m\times n$ table such that no two of them overlap?
*proposed by Mahyar Sefidgaran* | To solve the problem of determining the number of ways bacteria can fill an \( m \times n \) table such that no two of them overlap, we need to consider the placement of the bacteria's bodies and their feet. Here is a detailed step-by-step solution:
1. **Understanding the Bacteria Structure**:
- Each bacterium has ... | [
{
"content": "Every bacterium has a horizontal body with natural length and some nonnegative number of vertical feet, each with nonnegative (!) natural length, that lie below its body. In how many ways can these bacteria fill an $m\\times n$ table such that no two of them overlap?\n\n*proposed by Mahyar Sefi... |
cn_k12 | The equation of the asymptotes of the hyperbola \\( \dfrac {x^{2}}{9}- \dfrac {y^{2}}{4}=1 \\) is \_\_\_\_\_. | Since the equation of the hyperbola is \\( \dfrac {x^{2}}{9}- \dfrac {y^{2}}{4}=1 \\), the equation of its asymptotes is obtained by replacing the \\(1\\) in the standard equation of the hyperbola with \\(0\\), which gives \\( \dfrac {x^{2}}{9}- \dfrac {y^{2}}{4}=0 \\). Simplifying this, we get \\(y=± \dfrac {2}{3}x\\)... | [
{
"content": "The equation of the asymptotes of the hyperbola \\\\( \\dfrac {x^{2}}{9}- \\dfrac {y^{2}}{4}=1 \\\\) is \\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "Since the equation of the hyperbola is \\\\( \\dfrac {x^{2}}{9}- \\dfrac {y^{2}}{4}=1 \\\\), the equation of its asymptotes is obta... |
gsm8k | John starts climbing a very steep hill. He travels 1 foot vertically for every two feet horizontally. His elevation increases from 100 feet to 1450 feet. How far does he move horizontally, in feet? | To solve this problem, we need to calculate the total vertical distance John travels and then use the given ratio to find the horizontal distance.
1. First, we find the total vertical distance John climbs by subtracting his starting elevation from his final elevation:
\[1450 \text{ feet} - 100 \text{ feet} = 1350 \te... | [
{
"content": "John starts climbing a very steep hill. He travels 1 foot vertically for every two feet horizontally. His elevation increases from 100 feet to 1450 feet. How far does he move horizontally, in feet?",
"role": "user"
},
{
"content": "To solve this problem, we need to calculate the tot... |
olympiads | The court astrologer of King Gorokh calls the time of day good if, on a clock with a central second hand, the minute hand intersects after the hour hand and before the second hand while instantaneously moving around the clock face in the direction of the clock. Which time is greater in a day: good or bad? (The hands of... |
1. Let’s analyze the positions of the clock hands at any given time, \( t \). Consider two time instances: \( t \) after noon and \( t \) before noon.
2. Given that the positions of clock hands are symmetrical around the vertical diameter of the clock:
- The minute hand moves 6 degrees per minute.
- The hour... | [
{
"content": "The court astrologer of King Gorokh calls the time of day good if, on a clock with a central second hand, the minute hand intersects after the hour hand and before the second hand while instantaneously moving around the clock face in the direction of the clock. Which time is greater in a day: good... |
synthetic_amc | Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Determine the least positive period $p$ for which this property holds.
A. 12
B. 18
C. 24
D. 36
E. 72 | 1. **Original Functional Equation**: We start with $f(x+6) + f(x-6) = f(x)$.
2. **Substitute and Find New Relationships**: Substitute $x + 6$ for $x$:
\[
f(x+12) + f(x) = f(x+6).
\]
Hence,
\[
f(x+12) + f(x) - f(x+6) = 0.
\]
Again, substitute $x+12$ for $x$:
\[
f(x+18) + f(x+6) = f(x+12).
... | [
{
"content": "Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Determine the least positive period $p$ for which this property holds.\nA. 12\nB. 18\nC. 24\nD. 36\nE. 72",
"role": "user"
},
{
"content": "1. **Original Functional Equation**: We start with $f(x+6) + ... |
orca_math | A renovation project required some truck-load of sand, 0.3333333333333333 truck-load of dirt, and 0.16666666666666666 truck-load of cement. In all, 0.6666666666666666 truck-loads of material were needed. How many truck-loads of sand were required? | To find out how many truck-loads of sand were required, we can subtract the truck-loads of dirt and cement from the total truck-loads of material needed.
Total truck-loads of material needed = 0.6666666666666666
Truck-loads of dirt = 0.3333333333333333
Truck-loads of cement = 0.16666666666666666
Truck-loads of sand =... | [
{
"content": "A renovation project required some truck-load of sand, 0.3333333333333333 truck-load of dirt, and 0.16666666666666666 truck-load of cement. In all, 0.6666666666666666 truck-loads of material were needed. How many truck-loads of sand were required?",
"role": "user"
},
{
"content": "To f... |
cn_k12 | In the ancient Chinese mathematical classic "Commentary on the Mathematical Classic of the Nine Chapters," there is a discussion on the method of cutting circles: "The more finely we cut, the less we lose, and if we continue to cut until it cannot be cut anymore, it will merge with the circumference of the circle witho... | **Analysis**
This question examines the method of analogical reasoning, focusing on analogy in terms of method, and is a basic question.
The method to find the value is derived from the known: first, substitute, then set up an equation, solve the equation (discarding the negative root), and apply this method. Note th... | [
{
"content": "In the ancient Chinese mathematical classic \"Commentary on the Mathematical Classic of the Nine Chapters,\" there is a discussion on the method of cutting circles: \"The more finely we cut, the less we lose, and if we continue to cut until it cannot be cut anymore, it will merge with the circumfe... |
cn_k12 | Let $U = \mathbb{R}$, $M = \{x | x < 1\}$, and $N = \{x | -1 < x < 2\}$. Then $\{x | x \geqslant 2\} = (\ \ )$
A: $\complement _{U}(M\cup N)$
B: $N\cup \complement _{U}M$
C: $\complement _{U}(M\cap N)$
D: $M\cup \complement _{U}N$ | To solve the problem, let's analyze the given sets and the operations step by step:
1. **Define the Sets**:
- $M = \{x | x < 1\}$
- $N = \{x | -1 < x < 2\}$
- $U = \mathbb{R}$, the universal set.
2. **Find $M \cup N$**:
- Since $M$ includes all numbers less than 1, and $N$ includes all numbers between $-1... | [
{
"content": "Let $U = \\mathbb{R}$, $M = \\{x | x < 1\\}$, and $N = \\{x | -1 < x < 2\\}$. Then $\\{x | x \\geqslant 2\\} = (\\ \\ )$\n\nA: $\\complement _{U}(M\\cup N)$\n\nB: $N\\cup \\complement _{U}M$\n\nC: $\\complement _{U}(M\\cap N)$\n\nD: $M\\cup \\complement _{U}N$",
"role": "user"
},
{
"co... |
cn_k12 | In a regular polygon, each interior angle is $90^{\circ}$ larger than each exterior angle. The number of sides of this polygon is ______. | To solve for the number of sides $n$ of a regular polygon where each interior angle is $90^{\circ}$ larger than each exterior angle, we start by expressing the interior and exterior angles in terms of $n$.
1. The sum of the interior angles of a polygon is given by $(n-2) \cdot 180^{\circ}$. Therefore, each interior an... | [
{
"content": "In a regular polygon, each interior angle is $90^{\\circ}$ larger than each exterior angle. The number of sides of this polygon is ______.",
"role": "user"
},
{
"content": "To solve for the number of sides $n$ of a regular polygon where each interior angle is $90^{\\circ}$ larger than ... |
synthetic_math | $A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at point $P$, such that $AP=16$, $PC=2$, and $BD=12$. Find $BP$, given that $3 < BP < DP.$ | Let $BP = x$ and $PD = 12 - x$. According to the problem condition, we know:
- $3 < BP < DP$
- $BP < DP \implies x < 6$
From the power of the point theorem at point $P$, we derive:
$$AP \cdot PC = BP \cdot PD \implies 16 \cdot 2 = x \cdot (12 - x) \implies 32 = 12x - x^2.$$
Rearranging gives us a quadratic equation:
$... | [
{
"content": "$A$, $B$, $C$, and $D$ are points on a circle, and segments $\\overline{AC}$ and $\\overline{BD}$ intersect at point $P$, such that $AP=16$, $PC=2$, and $BD=12$. Find $BP$, given that $3 < BP < DP.$",
"role": "user"
},
{
"content": "Let $BP = x$ and $PD = 12 - x$. According to the prob... |
synthetic_amc | How many 3-digit whole numbers, whose digit-sum is 27, are divisible by 3 and even?
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$ | To determine how many 3-digit numbers have a digit-sum of 27, are even, and divisible by 3, consider the following constraints:
1. **Digit-Sum Requirement**: The sum of the digits must be 27.
2. **Even Number Requirement**: The number must be even, meaning the last digit could be 0, 2, 4, 6, or 8.
3. **Divisibility by... | [
{
"content": "How many 3-digit whole numbers, whose digit-sum is 27, are divisible by 3 and even?\n$\\text{(A)}\\ 0 \\qquad \\text{(B)}\\ 1 \\qquad \\text{(C)}\\ 2 \\qquad \\text{(D)}\\ 3 \\qquad \\text{(E)}\\ 4$",
"role": "user"
},
{
"content": "To determine how many 3-digit numbers have a digit-su... |
synthetic_math | Twelve distinct points are placed on the circumference of a circle. How many different convex quadrilaterals can be formed if each vertex must be one of these 12 points? | With twelve points on the circumference of a circle, any set of 4 of these points will form a convex (cyclic) quadrilateral. The number of different convex quadrilaterals that can be formed is given by choosing 4 points out of 12. This can be calculated using the combination formula:
\[
{12 \choose 4} = \frac{12 \cdot... | [
{
"content": "Twelve distinct points are placed on the circumference of a circle. How many different convex quadrilaterals can be formed if each vertex must be one of these 12 points?",
"role": "user"
},
{
"content": "With twelve points on the circumference of a circle, any set of 4 of these points ... |
synthetic_math | Express \(\frac{59}{2^2 \cdot 5^7}\) as a terminating decimal. | Given the fraction \(\frac{59}{2^2 \cdot 5^7}\), we need it to have a denominator of the form \(10^b = 2^b \cdot 5^b\). The given denominator is \(2^2 \cdot 5^7\), so we need to equalize the powers of 2 and 5 by multiplying both the numerator and the denominator by \(2^5\), since the power of 5 in the denominator is hi... | [
{
"content": "Express \\(\\frac{59}{2^2 \\cdot 5^7}\\) as a terminating decimal.",
"role": "user"
},
{
"content": "Given the fraction \\(\\frac{59}{2^2 \\cdot 5^7}\\), we need it to have a denominator of the form \\(10^b = 2^b \\cdot 5^b\\). The given denominator is \\(2^2 \\cdot 5^7\\), so we need ... |
gsm8k | Last week, Arman worked 35 hours for 5 days at $10 per hour. This week, he will receive an increase of $0.5 per hour if he can work for 40 hours. Suppose he can work for 40 hours this week, how much will Arman receive for working two weeks? | To calculate Arman's earnings over two weeks, we break down the calculation into two parts: earnings from last week and earnings from this week.
**Last Week:**
- Arman worked for 35 hours at a rate of $10 per hour.
- His earnings for last week can be calculated as follows:
\[ \text{Earnings for last week} = 35 \, \tex... | [
{
"content": "Last week, Arman worked 35 hours for 5 days at $10 per hour. This week, he will receive an increase of $0.5 per hour if he can work for 40 hours. Suppose he can work for 40 hours this week, how much will Arman receive for working two weeks?",
"role": "user"
},
{
"content": "To calculat... |
synthetic_math | Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 6,$ and
\[\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}.\] Compute $\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}.$ | Given $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$, we have:
\[(\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{a} + \mathbf{b} + \mathbf{c}) = 0.\]
Expanding, we get:
\[\mathbf{a} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} + \mathbf{c} \cdot \mathbf{c} + 2 (\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdo... | [
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = 2,$ $\\|\\mathbf{b}\\| = 3,$ and $\\|\\mathbf{c}\\| = 6,$ and\n\\[\\mathbf{a} + \\mathbf{b} + \\mathbf{c} = \\mathbf{0}.\\] Compute $\\mathbf{a} \\cdot \\mathbf{b} + \\mathbf{a} \\cdot \\mathbf{c} + \\mat... |
orca_math | In a division sum, the remainder is 0. A student mistook the divisor by some number instead of 21 and obtained 49 as quotient. The correct quotient is 28. What was the mistaken divisor? | Let's denote the mistaken divisor as "x".
Since the remainder is 0, the dividend (the number being divided) is exactly divisible by the divisor.
According to the student's mistake, the dividend divided by x gives a quotient of 49:
Dividend = 49 * x
According to the correct division, the dividend divided by 21 give... | [
{
"content": "In a division sum, the remainder is 0. A student mistook the divisor by some number instead of 21 and obtained 49 as quotient. The correct quotient is 28. What was the mistaken divisor?",
"role": "user"
},
{
"content": "Let's denote the mistaken divisor as \"x\". \n\nSince the remainde... |
olympiads | If $\cos \theta > 0$ and $\sin 2\theta < 0$, then the quadrant in which the terminal side of angle $\theta$ lies is:
A. First quadrant
B. Second quadrant
C. Third quadrant
D. Fourth quadrant |
To determine the quadrant in which the angle $\theta$ lies, we need to analyze the given conditions: $\cos \theta > 0$ and $\sin 2\theta < 0$.
1. **Analyze $\cos \theta > 0$:**
- The cosine function is positive in the first and fourth quadrants.
- Thus, $\theta$ could be either in the first quadrant or the four... | [
{
"content": "If $\\cos \\theta > 0$ and $\\sin 2\\theta < 0$, then the quadrant in which the terminal side of angle $\\theta$ lies is:\nA. First quadrant\nB. Second quadrant\nC. Third quadrant\nD. Fourth quadrant",
"role": "user"
},
{
"content": "\nTo determine the quadrant in which the angle $\\th... |
olympiads | Given an infinite triangular grid on a plane composed of equilateral triangles with side length 1, where the vertices of the triangles are called lattice points and points at a distance of 1 are called adjacent lattice points.
Two frogs, A and B, are playing a jumping game. "One jump" means a frog jumps from its curre... | Given the problem, the key is to show the impossibility of achieving the final targeted position where $A$ and $B$ perfectly swap their positions after a series of movements as per the rules (I) and (II).
1. **Initial Setup and Assumptions**:
- Let the grid points be generated by $1$ and $\omega = \frac{1 + \sqrt{... | [
{
"content": "Given an infinite triangular grid on a plane composed of equilateral triangles with side length 1, where the vertices of the triangles are called lattice points and points at a distance of 1 are called adjacent lattice points.\n\nTwo frogs, A and B, are playing a jumping game. \"One jump\" means a... |
orca_math | A courtyard is some meters long and 14 meters wide and is to be paved with bricks of dimensions 25 cm by 15 cm. The total number of bricks required is 8960. How long is the courtyard? | First, let's convert the dimensions of the bricks from centimeters to meters to match the units of the courtyard's width. Since 1 meter is equal to 100 centimeters, the dimensions of the bricks in meters are:
Length of brick = 25 cm / 100 = 0.25 meters
Width of brick = 15 cm / 100 = 0.15 meters
The area covered by on... | [
{
"content": "A courtyard is some meters long and 14 meters wide and is to be paved with bricks of dimensions 25 cm by 15 cm. The total number of bricks required is 8960. How long is the courtyard?",
"role": "user"
},
{
"content": "First, let's convert the dimensions of the bricks from centimeters t... |
cn_k12 | On a straight road, there are an odd number of warehouses. The distance between adjacent warehouses is 1 kilometer, and each warehouse contains 8 tons of goods. A truck with a load capacity of 8 tons starts from the warehouse on the far right and needs to collect all the goods into the warehouse in the middle. It is kn... | From the far right to the middle, then from the far left to the middle, and back to the far right... This cycle repeats.
Let's assume there are $x$ warehouses on each side. According to the problem, we have $4x + 4(x-1) + 4(x-2) + \ldots + 4 - x = 300$,
Therefore, $2x(x+1) - x = 300$, thus $2x^2 + x - 300 = 0$, so ... | [
{
"content": "On a straight road, there are an odd number of warehouses. The distance between adjacent warehouses is 1 kilometer, and each warehouse contains 8 tons of goods. A truck with a load capacity of 8 tons starts from the warehouse on the far right and needs to collect all the goods into the warehouse i... |
olympiads | In a triangle, the lengths of the three sides are integers $l, m, n$, with $l > m > n$. It is known that $\left\{\frac{3^{l}}{10^{4}}\right\}=\left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\}$, where $\{x\}=x-[x]$, and $[x]$ denotes the greatest integer not exceeding $x$. Find the minimum perimete... |
Given the conditions $l > m > n$ and $\left\{\frac{3^l}{10^4}\right\} = \left\{\frac{3^m}{10^4}\right\} = \left\{\frac{3^n}{10^4}\right\}$, we deduce:
1. **Congruence Relation:**
\[
3^l \equiv 3^m \equiv 3^n \pmod{10^4}
\]
This congruence relation implies that:
\[
3^{m-n} \equiv 1 \pmod{10^4}
\]
... | [
{
"content": "In a triangle, the lengths of the three sides are integers $l, m, n$, with $l > m > n$. It is known that $\\left\\{\\frac{3^{l}}{10^{4}}\\right\\}=\\left\\{\\frac{3^{m}}{10^{4}}\\right\\}=\\left\\{\\frac{3^{n}}{10^{4}}\\right\\}$, where $\\{x\\}=x-[x]$, and $[x]$ denotes the greatest integer not e... |
olympiads | On the lower base $AB$ of trapezoid $ABCD$, there are two fixed points $M$ and $N$. On the upper base $CD$, there is a moving point $P$. Let $E = DN \cap AP$, $F = DN \cap MC$, $G = MC \cap PB$. Given that $DP = \lambda DC$, for what value of $\lambda$ is the area of quadrilateral $PEFG$ maximized? |
1. **Assume Dimensions and Variables from the Problem Statement:**
- Let \( DC = 1 \), hence \( DP = \lambda \) and \( PC = 1 - \lambda \).
- Define the segments: \( AM = a \), \( MN = b \), \( NB = c \).
- The height of the trapezoid \( h \).
2. **Construct Parallel Line and Quadrilateral:**
- Draw \( ... | [
{
"content": "On the lower base $AB$ of trapezoid $ABCD$, there are two fixed points $M$ and $N$. On the upper base $CD$, there is a moving point $P$. Let $E = DN \\cap AP$, $F = DN \\cap MC$, $G = MC \\cap PB$. Given that $DP = \\lambda DC$, for what value of $\\lambda$ is the area of quadrilateral $PEFG$ maxi... |
olympiads | Points \( A_1, B_1, \) and \( C_1 \) are taken on the sides \( BC, CA, \) and \( AB \) of triangle \( ABC \), respectively, such that \( AC_1 = AB_1 \), \( BA_1 = BC_1 \), and \( CA_1 = CB_1 \). Prove that \( A_1, B_1, \) and \( C_1 \) are the points of tangency of the incircle with the sides. |
1. Let's define \( AC_1 = AB_1 = x \), \( BA_1 = BC_1 = y \), and \( CA_1 = CB_1 = z \). These conditions imply that:
\[
AC_1 = AB_1 = x, \quad BA_1 = BC_1 = y, \quad CA_1 = CB_1 = z
\]
2. Next, we can express the side lengths of the triangle \( ABC \) in terms of \( x, y, z \):
\[
a = y + z, \quad b = ... | [
{
"content": "Points \\( A_1, B_1, \\) and \\( C_1 \\) are taken on the sides \\( BC, CA, \\) and \\( AB \\) of triangle \\( ABC \\), respectively, such that \\( AC_1 = AB_1 \\), \\( BA_1 = BC_1 \\), and \\( CA_1 = CB_1 \\). Prove that \\( A_1, B_1, \\) and \\( C_1 \\) are the points of tangency of the incircle... |
orca_math | There are one-third as many Ford trucks as Dodge trucks in the parking lot of the Taco Castle. But there are twice as many Ford trucks as Toyota trucks in this parking lot, and there are half as many Volkswagen Bugs as there are Toyota trucks in this same parking lot. If there are 5 Volkswagon Bugs in the parking lot... | Let's denote the number of Volkswagen Bugs as V, Toyota trucks as T, Ford trucks as F, and Dodge trucks as D.
According to the information given:
V = 5 (There are 5 Volkswagen Bugs)
Since there are half as many Volkswagen Bugs as Toyota trucks:
T = 2 * V
T = 2 * 5
T = 10 (There are 10 Toyota trucks)
Since there are ... | [
{
"content": "There are one-third as many Ford trucks as Dodge trucks in the parking lot of the Taco Castle. But there are twice as many Ford trucks as Toyota trucks in this parking lot, and there are half as many Volkswagen Bugs as there are Toyota trucks in this same parking lot. If there are 5 Volkswagon B... |
synthetic_amc | Part of an "n-pointed regular star" is depicted. It is a simple closed polygon where all $2n$ edges are congruent, angles $A_1, A_2, \ldots, A_n$ are congruent, and angles $B_1, B_2, \ldots, B_n$ are congruent. If the acute angle at $A_1$ is $15^\circ$ less than the acute angle at $B_1$, and furthermore if the sum of a... | 1. **Set Up the Angles**: Let the angle at each $B_i$ be $\theta$. Then, the angle at each $A_i$ is $\theta - 15^\circ$.
2. **Sum of Angles Difference**:
\[
\sum B_i - \sum A_i = n \cdot \theta - n \cdot (\theta - 15^\circ) = 180^\circ
\]
Simplifying, we find:
\[
n \cdot \theta - n \cdot \theta + 15n... | [
{
"content": "Part of an \"n-pointed regular star\" is depicted. It is a simple closed polygon where all $2n$ edges are congruent, angles $A_1, A_2, \\ldots, A_n$ are congruent, and angles $B_1, B_2, \\ldots, B_n$ are congruent. If the acute angle at $A_1$ is $15^\\circ$ less than the acute angle at $B_1$, and ... |
cn_k12 | The distance from the focus of the parabola $y=2x^{2}$ to its directrix is $\_\_\_\_\_\_\_.$ | First, we convert the given parabola equation into the standard form. By doing so, we get $x^{2}= \frac{1}{2}y$.
Since the coefficient of $y$ is positive, the parabola opens upwards. In the standard form $x^2 = 2py$, we have $2p=\frac{1}{2}$, which gives $p=\frac{1}{4}$. Thus, the coordinate of the focus is $F(0, \fra... | [
{
"content": "The distance from the focus of the parabola $y=2x^{2}$ to its directrix is $\\_\\_\\_\\_\\_\\_\\_.$",
"role": "user"
},
{
"content": "First, we convert the given parabola equation into the standard form. By doing so, we get $x^{2}= \\frac{1}{2}y$.\n\nSince the coefficient of $y$ is pos... |
cn_k12 | Given $\triangle ABC \sim \triangle DEF$, the perimeter of $\triangle ABC$ is $3$, and the perimeter of $\triangle DEF$ is $1$. Then the ratio of the area of $\triangle ABC$ to the area of $\triangle DEF$ is \_\_\_\_\_\_. | **Analysis**
This question examines the properties of similar triangles, which is relatively easy. According to the property that the ratio of the perimeters of similar triangles equals the similarity ratio, and the ratio of the areas equals the square of the similarity ratio, the result can be easily obtained.
**Sol... | [
{
"content": "Given $\\triangle ABC \\sim \\triangle DEF$, the perimeter of $\\triangle ABC$ is $3$, and the perimeter of $\\triangle DEF$ is $1$. Then the ratio of the area of $\\triangle ABC$ to the area of $\\triangle DEF$ is \\_\\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "**Analysis**\n\nT... |
olympiads | On a 100×100 grid paper, several non-overlapping cardboard isosceles right-angled triangles with leg length 1 were placed; each triangle occupies exactly half of one cell. It turned out that each unit segment of the grid (including the boundary segments) is covered by exactly one leg of a triangle. Find the maximum pos... |
Given a 100 × 100 grid of cells, we need to place $1 \times 1$ right-angled isosceles triangular pieces so that they cover exactly one half of any cell and that each edge of the grid is covered exactly once by one of these triangles. We are to find out the maximum number of uncovered cells.
1. **Subdivide the Problem... | [
{
"content": "On a 100×100 grid paper, several non-overlapping cardboard isosceles right-angled triangles with leg length 1 were placed; each triangle occupies exactly half of one cell. It turned out that each unit segment of the grid (including the boundary segments) is covered by exactly one leg of a triangle... |
orca_math | A camping site recorded a total of 150 campers for the past three weeks. Two weeks ago, there were 40 campers which was 10 more than the number of campers three weeks ago. How many campers were there last week? | Let's denote the number of campers three weeks ago as \( C \).
According to the information given, two weeks ago there were 10 more campers than three weeks ago. So, the number of campers two weeks ago was \( C + 10 \).
We are also told that the total number of campers for the past three weeks is 150. So, we can writ... | [
{
"content": "A camping site recorded a total of 150 campers for the past three weeks. Two weeks ago, there were 40 campers which was 10 more than the number of campers three weeks ago. How many campers were there last week?",
"role": "user"
},
{
"content": "Let's denote the number of campers three ... |
olympiads | Find points $D$ and $E$ on the sides $AB$ and $AC$ of the triangle $ABC$ such that $AD = DE = EC$. |
1. **Identify and place points \( D \) and \( E \) on \( \triangle ABC \):**
We need to find points \( D \) on side \( AB \) and \( E \) on side \( AC \) of the triangle \( ABC \) such that \( AD = DE = EC \).
2. **Show that \(\triangle ADE\) is an isosceles triangle:**
Since \(AD = DE\), \(\triangle ADE\) is a... | [
{
"content": "Find points $D$ and $E$ on the sides $AB$ and $AC$ of the triangle $ABC$ such that $AD = DE = EC$.",
"role": "user"
},
{
"content": "\n1. **Identify and place points \\( D \\) and \\( E \\) on \\( \\triangle ABC \\):**\n We need to find points \\( D \\) on side \\( AB \\) and \\( E \... |
orca_math | A certain protective additive increases the time between required maintenance checks on an industrial vehicle from 25 days to 30 days. By what percent is the time between maintenance checks increased by using the additive? | To find the percentage increase, we can use the formula:
Percentage Increase = [(New Value - Old Value) / Old Value] * 100
In this case, the old value is the time between maintenance checks without the additive, which is 25 days, and the new value is the time between maintenance checks with the additive, which is 30 ... | [
{
"content": "A certain protective additive increases the time between required maintenance checks on an industrial vehicle from 25 days to 30 days. By what percent is the time between maintenance checks increased by using the additive?",
"role": "user"
},
{
"content": "To find the percentage increa... |
orca_math | A 1200 m long train crosses a tree in 120 sec. How much time will it take to pass a platform 1000 m long? | To calculate the time it will take for the train to pass a platform 1000 m long, we first need to determine the speed of the train.
The train crosses a tree (which we can consider as a point) in 120 seconds. The length of the train is 1200 m. Therefore, the speed of the train (v) can be calculated using the formula:
... | [
{
"content": "A 1200 m long train crosses a tree in 120 sec. How much time will it take to pass a platform 1000 m long?",
"role": "user"
},
{
"content": "To calculate the time it will take for the train to pass a platform 1000 m long, we first need to determine the speed of the train.\n\nThe train c... |
olympiads | Investigate the continuity and plot the function
$$
f(x)=\left\{\begin{array}{l}
x+1.5 \quad \text{for} \quad x<-2 \\
\frac{1}{x} \quad \text{for} \quad -2 \leq x<0 \\
2x \quad \text{for} \quad x \ge 0
\end{array}\right.
$$ |
1. **Function Definition and Initial Interval Analysis:**
The function \( f(x) \) is defined on the entire set of real numbers \(\mathbb{R}\), and it is given by three different expressions corresponding to three intervals of the domain:
\[
f(x) = \begin{cases}
x + 1.5 & \text{if } x < -2 \\
\fr... | [
{
"content": "Investigate the continuity and plot the function\n\n$$\nf(x)=\\left\\{\\begin{array}{l}\nx+1.5 \\quad \\text{for} \\quad x<-2 \\\\\n\\frac{1}{x} \\quad \\text{for} \\quad -2 \\leq x<0 \\\\\n2x \\quad \\text{for} \\quad x \\ge 0\n\\end{array}\\right.\n$$",
"role": "user"
},
{
"content":... |
olympiads | Let $\mathscr{A}=\left\{A_{1}, A_{2}, \cdots, A_{t}\right\}$ be a family of $r$-element subsets of set $X$. If the intersection of every $r+1$ subsets in $\mathscr{A}$ is non-empty, prove that the intersection $A_{1} \cap A_{2} \cap \cdots \cap A_{t}$ is non-empty. | 1. Let’s denote $\mathscr{A} = \{A_1, A_2, \ldots, A_t\}$ as a family of $r$-element subsets of a set $X$.
2. Given that the intersection of any $r+1$ sets in $\mathscr{A}$ is non-empty, we need to prove that the intersection of all sets in the family $\mathscr{A}$ is non-empty:
\[
A_1 \cap A_2 \cap \cdots \cap A_t \n... | [
{
"content": "Let $\\mathscr{A}=\\left\\{A_{1}, A_{2}, \\cdots, A_{t}\\right\\}$ be a family of $r$-element subsets of set $X$. If the intersection of every $r+1$ subsets in $\\mathscr{A}$ is non-empty, prove that the intersection $A_{1} \\cap A_{2} \\cap \\cdots \\cap A_{t}$ is non-empty.",
"role": "user"
... |
aops_forum | In a triangle $ ABC$ with $ AB\equal{}20, AC\equal{}21$ and $ BC\equal{}29$ , points $ D$ and $ E$ are taken on the segment $ BC$ such that $ BD\equal{}8$ and $ EC\equal{}9$ . Calculate the angle $ \angle DAE$ . | 1. Given a triangle \(ABC\) with sides \(AB = 20\), \(AC = 21\), and \(BC = 29\). We need to calculate the angle \(\angle DAE\) where points \(D\) and \(E\) are on segment \(BC\) such that \(BD = 8\) and \(EC = 9\).
2. First, we verify that \(\triangle ABC\) is a right triangle. We use the Pythagorean theorem:
\[
... | [
{
"content": "In a triangle $ ABC$ with $ AB\\equal{}20, AC\\equal{}21$ and $ BC\\equal{}29$ , points $ D$ and $ E$ are taken on the segment $ BC$ such that $ BD\\equal{}8$ and $ EC\\equal{}9$ . Calculate the angle $ \\angle DAE$ .",
"role": "user"
},
{
"content": "1. Given a triangle... |
synthetic_math | What is the shortest distance between the circles defined by $x^2 - 6x + y^2 + 10y + 9 = 0$ and $x^2 + 8x + y^2 - 2y + 16 = 0$? | Start by completing the square for both circles to find their centers and radii.
For the first equation:
\[
x^2 - 6x + y^2 + 10y + 9 = 0 \implies (x^2 - 6x + 9) + (y^2 + 10y + 25) = 0 + 9 + 25 - 9 - 25 \implies (x - 3)^2 + (y + 5)^2 = 25
\]
Hence, the center is $(3, -5)$ and the radius is $5$.
For the second equation... | [
{
"content": "What is the shortest distance between the circles defined by $x^2 - 6x + y^2 + 10y + 9 = 0$ and $x^2 + 8x + y^2 - 2y + 16 = 0$?",
"role": "user"
},
{
"content": "Start by completing the square for both circles to find their centers and radii.\n\nFor the first equation:\n\\[\nx^2 - 6x +... |
olympiads | Let \( n \) be an integer and \( x \) be a real number. Show that:
\[ 1 + x + x^2 + \ldots + x^n = \frac{x^{n+1} - 1}{x - 1} \] |
1. **Base case**:
Verify the formula for \( n = 0 \).
\[
1 + x^0 = 1
\quad \text{and} \quad
\frac{x^{0+1} - 1}{x - 1} = \frac{x - 1}{x - 1} = 1.
\]
Thus, for \( n = 0 \), the formula holds.
2. **Inductive step**:
Assume that the formula holds for some \( n = k \). That is, assume:
\[
1 ... | [
{
"content": "Let \\( n \\) be an integer and \\( x \\) be a real number. Show that: \n\\[ 1 + x + x^2 + \\ldots + x^n = \\frac{x^{n+1} - 1}{x - 1} \\]",
"role": "user"
},
{
"content": "\n1. **Base case**: \n Verify the formula for \\( n = 0 \\).\n \\[\n 1 + x^0 = 1\n \\quad \\text{and} \\qu... |
olympiads | When adding two integers, a student mistakenly appended an extra zero to the end of the second addend and obtained a sum of 7182 instead of 3132. Determine the addends. |
1. Let's denote the first integer as \(a\) and the second integer as \(b\).
2. The student, by mistake, placed an extra zero at the end of the second integer. Hence, the erroneous second integer can be written as \(10b\).
3. According to the problem, the erroneous sum is given as 7182 instead of 3132. Thus, we can f... | [
{
"content": "When adding two integers, a student mistakenly appended an extra zero to the end of the second addend and obtained a sum of 7182 instead of 3132. Determine the addends.",
"role": "user"
},
{
"content": "\n1. Let's denote the first integer as \\(a\\) and the second integer as \\(b\\).\n... |
cn_k12 | Given that $i$ is the imaginary unit, the imaginary part of the complex number $z = (1 + 2i)(2 - i)$ is ( )
A: $-3$
B: $-3i$
C: $3$
D: $3i$ | The complex number $z = (1 + 2i)(2 - i) = 3 + 3i$.
The imaginary part of the complex number is: $3$.
Therefore, the correct answer is: $\boxed{C}$.
**Analysis:** Directly simplify the multiplication of complex numbers to find the solution. | [
{
"content": "Given that $i$ is the imaginary unit, the imaginary part of the complex number $z = (1 + 2i)(2 - i)$ is ( )\nA: $-3$\nB: $-3i$\nC: $3$\nD: $3i$",
"role": "user"
},
{
"content": "The complex number $z = (1 + 2i)(2 - i) = 3 + 3i$. \nThe imaginary part of the complex number is: $3$. \nT... |
orca_math | A person can do a work in some days and q can do the same work in 20 days. They worked together for 4 days. The fraction of work left is 0.5333333333333333. How many days does it take for the first person to do the work alone? | Let's denote the number of days it takes for the first person to do the work alone as \( p \) days.
The work done by the first person in one day is \( \frac{1}{p} \) and the work done by the second person (q) in one day is \( \frac{1}{20} \).
When they work together for 4 days, the fraction of work they complete is:
... | [
{
"content": "A person can do a work in some days and q can do the same work in 20 days. They worked together for 4 days. The fraction of work left is 0.5333333333333333. How many days does it take for the first person to do the work alone?",
"role": "user"
},
{
"content": "Let's denote the number o... |
synthetic_amc | Calculate how many zeros immediately follow the decimal point in the decimal representation of
\[
\frac{1}{(6 \cdot 10)^{10}}.
\]
A) 15
B) 16
C) 17
D) 18
E) 19 | 1. **Express $\frac{1}{(6 \cdot 10)^{10}}$ in terms of powers of 10**:
\[
(6 \cdot 10)^{10} = 6^{10} \cdot 10^{10}.
\]
Hence,
\[
\frac{1}{(6 \cdot 10)^{10}} = \frac{1}{6^{10}} \cdot \frac{1}{10^{10}}.
\]
2. **Explicit calculation by approximating $6^{10}$**:
\[
6^{10} = 60,466,176.
\]
... | [
{
"content": "Calculate how many zeros immediately follow the decimal point in the decimal representation of \n\\[\n\\frac{1}{(6 \\cdot 10)^{10}}.\n\\]\nA) 15\nB) 16\nC) 17\nD) 18\nE) 19",
"role": "user"
},
{
"content": "1. **Express $\\frac{1}{(6 \\cdot 10)^{10}}$ in terms of powers of 10**: \n \... |
synthetic_math | In the figure below, which is partially a rectangular shape, all angles that appear as right angles are indeed right angles, and each of the ten congruent sides marked with a tick mark have length 2 feet. If the region has an area of 140 square feet, what is the number of feet in the perimeter of the modified region? A... | Visualize the region as a large rectangle, from which a smaller, inverted staircase-shaped portion is subtracted. Extend the two discussing sides of this staircase to visualize the large rectangle:
- Assume the rectangle's width remains 18 feet as given.
- Dissect the small inversed staircase. If each of the ten sectio... | [
{
"content": "In the figure below, which is partially a rectangular shape, all angles that appear as right angles are indeed right angles, and each of the ten congruent sides marked with a tick mark have length 2 feet. If the region has an area of 140 square feet, what is the number of feet in the perimeter of ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.