problem stringlengths 10 5.15k | answer dict |
|---|---|
A sphere intersects the $xy$-plane in a circle centered at $(2, 3, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 3, -8),$ with radius $r.$ Find $r.$ | {
"answer": "2\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of tetrahedron PQRS with edge lengths PQ = 4, PR = 5, PS = 6, QR = 3, QS = √37, and RS = 7. | {
"answer": "10.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $8^x - 8^{x-1} = 60$, calculate the value of $(3x)^x$. | {
"answer": "58.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(a+b)(\sin A - \sin B) = (c-b)\sin C.$
(1) Determine the measure of angle $A$.
(2) If $2c=3b$ and the area of $\triangle ABC$ is $6 \sqrt{3}$, find the value of $a$. | {
"answer": "\\frac{2\\sqrt{21}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest four-digit number that satisfies the following system of congruences:
\begin{align*}
3x &\equiv 6 \pmod{12} \\
5x + 20 &\equiv 25 \pmod{15} \\
3x - 2 &\equiv 2x \pmod{35}
\end{align*} | {
"answer": "1274",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the value of $$64^{-\frac{1}{3}}+lg0.001$$. | {
"answer": "- \\frac{11}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The total number of workers in Workshop A and Workshop C is $x + y$. If a sample of 45 people is drawn from the factory with 20 people from Workshop A and 10 people from Workshop C, determine the relationship between the number of workers in Workshop A, Workshop B, and Workshop C. | {
"answer": "900",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Squares of integers that are palindromes (i.e., they read the same left-to-right and right-to-left) are an interesting subject of study. For example, the squares of $1, 11, 111,$ and $1111$ are $1, 121, 12321,$ and $1234321$ respectively, and all these numbers are palindromes. This rule applies to any number of ones up to 9. However, there are irregular cases, such as $264^2 = 69696$ and $2285^2 = 5221225$.
In all the above examples, the number of digits is odd. Could the reader provide examples with an even number of digits? | {
"answer": "698896",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A numismatist has 100 identical-looking coins, knowing that among them are 30 genuine coins and 70 counterfeit coins. The masses of all genuine coins are the same, while the masses of all counterfeit coins are different, with each counterfeit coin being heavier than a genuine coin. The exact masses of the coins are unknown. There is a two-pan balance scale available, which can be used to compare the masses of two groups of coins that have the same number of coins. What is the minimum number of weighings needed for the numismatist to reliably find at least one genuine coin? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of n terms of an arithmetic progression is 180, and the common difference is 3. If the first term must be a positive integer, and n > 1, then find the number of possible values for n. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To investigate the growth inhibitory effect of a certain drug on mice, $40$ mice were divided into two groups, a control group (without the drug) and an experimental group (with the drug).<br/>$(1)$ Suppose the number of mice in the control group among two mice is $X$, find the probability distribution and mathematical expectation of $X$;<br/>$(2)$ The weights of the $40$ mice are as follows (unit: $g) (already arranged in ascending order):<br/>Control group: $17.3, 18.4, 20.1, 20.4, 21.5, 23.2, 24.6, 24.8, 25.0, 25.4$<br/>$26.1, 26.3, 26.4, 26.5, 26.8, 27.0, 27.4, 27.5, 27.6, 28.3$<br/>Experimental group: $5.4, 6.6, 6.8, 6.9, 7.8, 8.2, 9.4, 10.0, 10.4, 11.2$<br/>$14.4, 17.3, 19.2, 20.2, 23.6, 23.8, 24.5, 25.1, 25.2, 26.0$<br/>$(i)$ Find the median $m$ of the weights of the $40$ mice and complete the following $2\times 2$ contingency table:<br/><table><tbody><tr><td align="center"></td><td align="center">$ \lt m$</td><td align="center">$\geq m$</td></tr><tr><td align="center">Control Group</td><td align="center">_____</td><td align="center">_____</td></tr><tr><td align="center">Experimental Group</td><td align="center">_____</td><td align="center">_____</td></tr></tbody></table>$($ii)$ Based on the $2\times 2$ contingency table, can we be $95\%$ confident that the drug inhibits the growth of mice.<br/>Reference data:<br/><table><tbody><tr><td align="center">$k_{0}$</td><td align="center">$0.10$</td><td align="center">$0.05$</td><td align="center">$0.010$</td></tr><tr><td align="center">$P(k^{2}\ge k_{0})$</td><td align="center">$2.706$</td><td align="center">$3.841$</td><td align="center">$6.635$</td></tr></tbody></table> | {
"answer": "95\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos ( \frac {π}{6}+α) \cdot \cos ( \frac {π}{3}-α)=- \frac {1}{4}$, where $α \in ( \frac {π}{3}, \frac {π}{2})$, find the value of $\sin 2α$ and the value of $\tan α - \frac {1}{\tan α}$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The escalator of the department store, which at any given time can be seen at $75$ steps section, moves up one step in $2$ seconds. At time $0$ , Juku is standing on an escalator step equidistant from each end, facing the direction of travel. He goes by a certain rule: one step forward, two steps back, then again one step forward, two back, etc., taking one every second in increments of one step. Which end will Juku finally get out and at what point will it happen? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest positive integer $N $ for which one can choose $N $ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$ .
Proposed by Mikhail Evdokimov | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \) and \( y \) be positive integers, with \( x < y \). The leading digit of \( \lg x \) is \( a \), and the trailing digit is \( \alpha \); the leading digit of \( \lg y \) is \( b \), and the trailing digit is \( \beta \). They satisfy the conditions \( a^{2} + b^{2} = 5 \) and \( \alpha + \beta = 1 \). What is the maximum value of \( x \)? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit '3'? | {
"answer": "3125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $7, n,$ and $n+2$ cents, $120$ cents is the greatest postage that cannot be formed. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express the sum of $0.\overline{123}+0.\overline{0123}+0.\overline{000123}$ as a common fraction. | {
"answer": "\\frac{123 \\times 1000900}{999 \\times 9999 \\times 100001}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory implements a time-based wage system, where each worker is paid 6 yuan for every hour worked, for a total of 8 hours per day. However, the clock used for timing is inaccurate: it takes 69 minutes for the minute hand to coincide with the hour hand once. Calculate the amount of wages underpaid to each worker per day. | {
"answer": "2.60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px + 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | {
"answer": "2028",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a, b, c$ are the sides opposite to angles $A, B, C$ respectively in $\triangle ABC$, and $\cos A \sin B \sin C + \cos B \sin A \sin C = 2 \cos C \sin A \sin B$, find the maximum value of $C$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right rectangular prism $Q$ has integral side lengths $a, b, c$ with $a \le b \le c$. A plane parallel to one of the faces of $Q$ cuts $Q$ into two prisms, one of which is similar to $Q$, with both having nonzero volumes. The middle side length $b = 3969$. Determine the number of ordered triples $(a, b, c)$ that allow such a plane to exist. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A solid cube of side length 4 cm is cut into two pieces by a plane that passed through the midpoints of six edges. Find the surface area of each half cube created. | {
"answer": "69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The increasing sequence of positive integers $b_1, b_2, b_3, \ldots$ follows the rule:
\[ b_{n + 2} = b_{n + 1} + b_n \]
for all $n \geq 1$. If $b_5 = 55$, find $b_6$. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A conical flask has a base radius of $15$ cm and a height of $30$ cm. The flask is filled with a liquid to a height of $10$ cm from the base, and a sphere is inscribed such that it just touches the liquid surface. What is the radius of the sphere? Express your answer in cm. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\phi = \tfrac{1+\sqrt 5}2$ be the positive root of $x^2=x+1$ . Define a function $f:\mathbb N\to\mathbb N$ by
\begin{align*}
f(0) &= 1
f(2x) &= \lfloor\phi f(x)\rfloor
f(2x+1) &= f(2x) + f(x).
\end{align*}
Find the remainder when $f(2007)$ is divided by $2008$ . | {
"answer": "2007",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 1 = 0,$ find $7 \alpha^5 + 8 \beta^4.$ | {
"answer": "1448",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $M$ is the midpoint of $\overline{BC}$, $AB = 15$, and $AC = 24$. Let $E$ be a point on $\overline{AC}$, and $H$ be a point on $\overline{AB}$, and let $G$ be the intersection of $\overline{EH}$ and $\overline{AM}$. If $AE = 3AH$, find $\frac{EG}{GH}$. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $P(2,2)$, and circle $C$: $x^{2}+y^{2}-8y=0$. A moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$, and $O$ being the origin.
$(1)$ Find the equation of the trajectory of point $M$;
$(2)$ When $|OP|=|OM|$, find the equation of line $l$ and the area of $\Delta POM$. | {
"answer": "\\frac{16}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the triangle (see the diagram) on graph paper. (Each side of a square is 1 unit.) | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of three positive integers $a$, $b$, and $c$ equals 1176. What is the minimum possible value of the sum $a + b + c$? | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find real numbers \( x, y, z \) greater than 1 that satisfy the equation
\[ x + y + z + \frac{3}{x - 1} + \frac{3}{y - 1} + \frac{3}{z - 1} = 2(\sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}). \] | {
"answer": "\\frac{3 + \\sqrt{13}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
5 points in a plane are situated so that no two of the lines joining a pair of points are coincident, parallel, or perpendicular. Through each point, lines are drawn perpendicular to each of the lines through two of the other 4 points. Determine the maximum number of intersections these perpendiculars can have. | {
"answer": "315",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a dark room drawer, there are 100 red socks, 80 green socks, 60 blue socks, and 40 black socks. A young person picks out one sock at a time without seeing its color. To ensure that at least 10 pairs of socks are obtained, what is the minimum number of socks they must pick out?
(Assume that two socks of the same color make a pair, and a single sock cannot be used in more than one pair)
(37th American High School Mathematics Examination, 1986) | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the function $y=f(x)$, if there exists $x_{0} \in D$ such that $f(-x_{0})+f(x_{0})=0$, then the function $f(x)$ is called a "sub-odd function" and $x_{0}$ is called a "sub-odd point" of the function. Consider the following propositions:
$(1)$ Odd functions are necessarily "sub-odd functions";
$(2)$ There exists an even function that is a "sub-odd function";
$(3)$ If the function $f(x)=\sin (x+ \frac {\pi}{5})$ is a "sub-odd function", then all "sub-odd points" of this function are $\frac {k\pi}{2} (k\in \mathbb{Z})$;
$(4)$ If the function $f(x)=\lg \frac {a+x}{1-x}$ is a "sub-odd function", then $a=\pm1$;
$(5)$ If the function $f(x)=4^{x}-m\cdot 2^{x+1}$ is a "sub-odd function", then $m\geqslant \frac {1}{2}$. Among these, the correct propositions are ______. (Write down the numbers of all propositions you think are correct) | {
"answer": "(1)(2)(4)(5)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $g(x) = \frac{6x^2 + 11x + 17}{7(2 + x)}$, find the minimum value of $g(x)$ for $x \ge 0$. | {
"answer": "\\frac{127}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Jessica uses 150 grams of lemon juice and 100 grams of sugar, and there are 30 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar, and water contains no calories, compute the total number of calories in 300 grams of her lemonade. | {
"answer": "152.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number 123456789 is written on the board. Two adjacent digits are selected from the number, if neither of them is 0, 1 is subtracted from each digit, and the selected digits are swapped (for example, from 123456789, one operation can result in 123436789). What is the smallest number that can be obtained as a result of these operations? | {
"answer": "101010101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The region shown is bounded by the arcs of circles having radius 5 units, each with a central angle measure of 45 degrees, intersecting at points of tangency. The area of the region can be expressed in the form $a\sqrt{b} + c\pi$ square units, where $\sqrt{b}$ is a radical in simplest form. Calculate the value of $a + b + c$. | {
"answer": "-12.625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the palindromes between 1000 and 2000 contain at least one 7? | {
"answer": "12\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest four-digit number that is divisible by $35$? | {
"answer": "1200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function $g$ from the integers to the integers is defined as follows:
\[g(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $m$ is odd and $g(g(g(m))) = 39.$ Find $m.$ | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory estimates that the total demand for a particular product in the first $x$ months starting from the beginning of 2016, denoted as $f(x)$ (in units of 'tai'), is approximately related to the month $x$ as follows: $f(x)=x(x+1)(35-2x)$, where $x \in \mathbb{N}^*$ and $x \leqslant 12$.
(1) Write the relationship expression between the demand $g(x)$ in the $x$-th month of 2016 and the month $x$;
(2) If the factory produces $a$ 'tai' of this product per month, what is the minimum value of $a$ to ensure that the monthly demand is met? | {
"answer": "171",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression \[ \frac{a^2 + 2a}{a^2 + a} \cdot \frac{b^2 - 4}{b^2 - 6b + 8} \cdot \frac{c^2 + 16c + 64}{c^2 + 12c + 36} \]
given that \(c = b - 20\), \(b = a + 4\), \(a = 2\), and ensuring none of the denominators are zero. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest negative root of the equation
$$
4 \sin (3 x) + 13 \cos (3 x) = 8 \sin (x) + 11 \cos (x)
$$ | {
"answer": "-0.1651",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the integer \[M = 8 + 88 + 888 + 8888 + \cdots + \underbrace{88\ldots 88}_\text{150 digits}.\] Find the sum of the digits of $M$. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( p, q, r, s, t, u, v, \) and \( w \) be real numbers such that \( pqrs = 16 \) and \( tuvw = 25 \). Find the minimum value of
\[
(pt)^2 + (qu)^2 + (rv)^2 + (sw)^2.
\] | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of the sum $\frac{3}{4}+\frac{3^2}{4^2}+\frac{3^3}{4^3}+ \ldots +\frac{3^{15}}{4^{15}}$? Express your answer as a common fraction. | {
"answer": "\\frac{3177884751}{1073741824}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=( \sqrt {3}\sin x,m+\cos x)$ and $\overrightarrow{b}=(\cos x,-m+\cos x)$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$
(1) Find the analytical expression for the function $f(x)$;
(2) When $x\in\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$, the minimum value of $f(x)$ is $-4$. Find the maximum value of the function $f(x)$ and the corresponding value of $x$ in this interval. | {
"answer": "-\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the following sequence of sets of natural numbers. The first set \( I_{0} \) consists of two ones, 1,1. Then, between these numbers, we insert their sum \( 1+1=2 \); we obtain the set \( I_{1}: 1,2,1 \). Next, between each pair of numbers in \( I_{1} \) we insert their sum; we obtain the set \( I_{2}: 1,3,2,3,1 \). Proceeding in the same way with the set \( I_{2} \), we obtain the set \( I_{3}: 1,4,3,5,2,5,3,4,1 \), and so on. How many times will the number 1973 appear in the set \( I_{1000000} \)? | {
"answer": "1972",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(a, b, c, a+b-c, a+c-b, b+c-a, a+b+c\) are 7 distinct prime numbers, and the sum of any two of \(a, b, c\) is 800. Let \(d\) be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of \(d\). | {
"answer": "1594",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 10 by 10 table \(A\), some numbers are written. Let \(S_1\) be the sum of all numbers in the first row, \(S_2\) in the second row, and so on. Similarly, let \(t_1\) be the sum of all numbers in the first column, \(-t_2\) in the second column, and so on. A new table \(B\) of size 10 by 10 is created with numbers written as follows: in the first cell of the first row, the smaller of \(S_1\) and \(t_1\) is written, in the third cell of the fifth row, the smaller of \(S_5\) and \(t_3\) is written, and similarly the entire table is filled. It turns out that it is possible to number the cells of table \(B\) from 1 to 100 such that in the cell with number \(k\), the number will be less than or equal to \(k\). What is the maximum value that the sum of all numbers in table \(A\) can take under these conditions? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The arithmetic mean of a set of $60$ numbers is $42$. If three numbers from the set, $48$, $58$, and $52$, are removed, find the arithmetic mean of the remaining set of numbers. | {
"answer": "41.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $ABC$ with the sides opposite to angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, let vectors $\overrightarrow{m}=(1-\cos(A+B), \cos \frac{A-B}{2})$ and $\overrightarrow{n}=(\frac{5}{8}, \cos \frac{A-B}{2})$, and it's known that $\overrightarrow{m} \cdot \overrightarrow{n} = \frac{9}{8}$.
1. Find the value of $\tan A \cdot \tan B$.
2. Find the maximum value of $\frac{a b \sin C}{a^2 + b^2 - c^2}$. | {
"answer": "-\\frac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A larger equilateral triangle ABC with side length 5 has a triangular corner DEF removed from one corner, where DEF is an isosceles triangle with DE = EF = 2, and DF = 2\sqrt{2}. Calculate the perimeter of the remaining quadrilateral. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the probability of rolling eight standard, six-sided dice and getting exactly three pairs of identical numbers, while the other two numbers are distinct from each other and from those in the pairs? Express your answer as a common fraction. | {
"answer": "\\frac{525}{972}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of $y=g(x)$, defined on a limited domain shown, is conceptualized through the function $g(x) = \frac{(x-6)(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{945} - 2.5$. If each horizontal grid line represents a unit interval, determine the sum of all integers $d$ for which the equation $g(x) = d$ has exactly six solutions. | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 8 Olympic volunteers, among them volunteers $A_{1}$, $A_{2}$, $A_{3}$ are proficient in Japanese, $B_{1}$, $B_{2}$, $B_{3}$ are proficient in Russian, and $C_{1}$, $C_{2}$ are proficient in Korean. One volunteer proficient in Japanese, Russian, and Korean is to be selected from them to form a group.
(Ⅰ) Calculate the probability of $A_{1}$ being selected;
(Ⅱ) Calculate the probability that neither $B_{1}$ nor $C_{1}$ is selected. | {
"answer": "\\dfrac {5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $q(x) = x^5 - 4x^3 + 5$, then find the coefficient of the $x^3$ term in the polynomial $(q(x))^2$. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $n$ mathematicians attending a conference. Each mathematician has exactly 3 friends (friendship is mutual). If they are seated around a circular table such that each person has their friends sitting next to them on both sides, the number of people at the table is at least 7. Find the minimum possible value of $n$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A train is scheduled to arrive at a station randomly between 1:00 PM and 3:00 PM, and it waits for 15 minutes before leaving. If Alex arrives at the station randomly between 1:00 PM and 3:00 PM as well, what is the probability that he will find the train still at the station when he arrives? | {
"answer": "\\frac{105}{1920}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix},$ determine the scalar $s$ such that
\[\begin{pmatrix} 5 \\ -4 \\ 1 \end{pmatrix} = s(\mathbf{a} \times \mathbf{b}) + p\mathbf{a} + q\mathbf{b},\]
where $p$ and $q$ are scalars. | {
"answer": "-\\frac{1}{45}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $\frac{17!}{7!10!}$. | {
"answer": "408408",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polynomial \( x^{103} + Cx + D \) is divisible by \( x^2 + 2x + 1 \) for some real numbers \( C \) and \( D \). Find \( C + D \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest three-digit number in a format $abc$ (where $a, b, c$ are digits, $a \neq 0$) such that when multiplied by 111, the result is not a palindrome. | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases}x=a\cos \varphi \\ y=b\sin \varphi\end{cases} (a > b > 0,$ $\varphi$ is the parameter), and the point $M(2, \sqrt{3})$ on curve $C1$ corresponds to the parameter $\varphi= \frac{\pi}{3}$. Using $O$ as the pole and the positive half-axis of $x$ as the polar axis to establish a polar coordinate system, curve $C2$ is a circle with its center on the polar axis and passing through the pole. The ray $\theta= \frac{\pi}{4}$ intersects curve $C2$ at point $D(\sqrt{2}, \frac{\pi}{4})$.
$(1)$ Find the standard equation of curve $C1$ and the polar equation of curve $C2$;
$(2)$ If $A(\rho_{1},\theta),B(\rho_{2},\theta+ \frac{\pi}{2})$ are two points on curve $C1$, find the value of $\frac{1}{\rho_{1}^{2}+\rho_{2}^{2}}$. | {
"answer": "\\frac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression: $2\log_{2}\;\sqrt {2}-\lg 2-\lg 5+ \frac{1}{ 3(\frac{27}{8})^{2} }$. | {
"answer": "\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the linear function $y=(7-m)x-9$ does not pass through the second quadrant, and the fractional equation about $y$ $\frac{{2y+3}}{{y-1}}+\frac{{m+1}}{{1-y}}=m$ has a non-negative solution, calculate the sum of all integer values of $m$ that satisfy the conditions. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, given $a \geqslant \frac{b+c}{3}$, it holds that
$$
a c + b c - c^{2} \leqslant \lambda\left(a^{2} + b^{2} + 3 c^{2} + 2 a b - 4 b c\right).
$$ | {
"answer": "\\frac{2\\sqrt{2} + 1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the center C of a moving circle is on the parabola $x^2=2py$ ($p>0$), the circle passes through point A $(0, p)$, and intersects the x-axis at two points M and N. The maximum value of $\sin\angle MCN$ is. | {
"answer": "\\frac{1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $s(\theta) = \frac{1}{2 - \theta}$. What is $s(s(s(s(s(s(s(s(s(\frac{1}{2})))))))))$ (where $s$ is applied 9 times)? | {
"answer": "\\frac{13}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube of mass $m$ slides down the felt end of a ramp semicircular of radius $h$ , reaching a height $h/2$ at the opposite extreme.
Find the numerical coefficient of friction $\mu_k$ bretween the cube and the surface.
*Proposed by Danilo Tejeda, Atlantida* | {
"answer": "\\frac{1}{\\sqrt{1 - \\left(\\frac{1}{2\\pi}\\right)^2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The circumcenter of a regular tetrahedron \( ABCD \) is \( O \). If \( E \) is the midpoint of \( BC \), what is the measure of the dihedral angle between \( A-BO-E \)? | {
"answer": "\\frac{2}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=x^{2}+px+q$ and $g(x)=x+\frac{1}{x^{2}}$ on the interval $[1,2]$, determine the maximum value of $f(x)$. | {
"answer": "4 - \\frac{5}{2} \\sqrt[3]{2} + \\sqrt[3]{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the first 4 terms of the geometric sequence $\{a_n\}$. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f : \mathbb{C} \to \mathbb{C} \) be defined by \( f(z) = z^2 - 2iz + 2 \). Determine the number of complex numbers \( z \) such that \( \text{Im}(z) > 0 \) and both the real and the imaginary parts of \( f(z) \) are integers within the range \(-5\) to \(5\). | {
"answer": "143",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence \(a_1\), \(a_2\), \(\ldots\) of non-negative integers is defined by the rule \(a_{n+2}=|a_{n+1}-a_n|\) for \(n\geq1\). If \(a_1=1010\), \(a_2<1010\), and \(a_{2023}=0\), how many different values of \(a_2\) are possible? | {
"answer": "399",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is a multiple of $150$. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$, $Q$, and $R$ be points on a circle of radius $12$. If $\angle PRQ = 110^\circ,$ find the circumference of the minor arc $PQ$. Express your answer in terms of $\pi$. | {
"answer": "\\frac{22}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A merchant acquires goods at a discount of $30\%$ of the list price and intends to sell them with a $25\%$ profit margin after a $25\%$ discount on the marked price. Determine the required percentage of the original list price that the goods should be marked. | {
"answer": "124\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$ ? | {
"answer": "3267",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers \( n \) are there such that \( n \) is a multiple of 4, and the least common multiple of \( 4! \) and \( n \) equals 4 times the greatest common divisor of \( 8! \) and \( n \)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the function $g(x) = x^{-2} + \frac{x^{-2}}{1+x^{-2}}$. Determine $g(g(3))$. | {
"answer": "\\frac{72596100}{3034921}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $n\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest $n$ for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are the sides opposite the angles $A$, $B$, and $C$ in $\triangle ABC$ respectively, and the equation $\sqrt{3}b\sin A - a\cos B - 2a = 0$ holds, then the measure of $\angle B$ is ______. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a cone with a height of 4 and a slant height of 8, three spheres of radius \( r \) are placed. They touch each other (externally), the lateral surface of the cone, and the first two spheres touch the base of the cone. Find the maximum value of \( r \). | {
"answer": "\\frac{12}{5+2\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $n$ is a positive integer, and $4^7 + 4^n + 4^{1998}$ is a perfect square, then one value of $n$ is. | {
"answer": "3988",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $995 + 997 + 999 + 1001 + 1003 = 5100 - N$, determine $N$. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right square prism $M-ABCD$, the base $ABCD$ is a rectangle, $MD \perp$ face $ABCD$ with $MD$ being an integer, and the lengths of $MA$, $MC$, and $MB$ are three consecutive even numbers. What is the volume of the right square prism $M-ABCD$? | {
"answer": "$24 \\sqrt{5}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|\overrightarrow {a}|=4$, $|\overrightarrow {b}|=2$, and the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$ is $120^{\circ}$, find:
1. $\left(\overrightarrow {a}-2\overrightarrow {b}\right)\cdot \left(\overrightarrow {a}+\overrightarrow {b}\right)$;
2. The projection of $\overrightarrow {a}$ onto $\overrightarrow {b}$;
3. The angle between $\overrightarrow {a}$ and $\overrightarrow {a}+\overrightarrow {b}$. | {
"answer": "\\dfrac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Estimate the probability of a rifle student hitting the target using a random simulation method by analyzing the ratio of the number of sets of three random numbers where exactly one represents a hit to the total number of sets of three random numbers. | {
"answer": "\\frac{9}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim. | {
"answer": "1004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $g(x)$ satisfies
\[g(x) - 2 g \left( \frac{1}{x} \right) = 3^x\] for all \( x \neq 0 \). Find $g(2)$. | {
"answer": "-\\frac{29}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x > 0$, $y > 0$, and the inequality $2\log_{\frac{1}{2}}[(a-1)x+ay] \leq 1 + \log_{\frac{1}{2}}(xy)$ always holds, find the minimum value of $4a$. | {
"answer": "\\sqrt{6}+\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We call a natural number \( b \) lucky if for any natural \( a \) such that \( a^{5} \) is divisible by \( b^{2} \), the number \( a^{2} \) is divisible by \( b \).
Find the number of lucky natural numbers less than 2010. | {
"answer": "1961",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta ABC$, it is known that $\overrightarrow{AB} \cdot \overrightarrow{AC} + 2\overrightarrow{BA} \cdot \overrightarrow{BC} = 3\overrightarrow{CA} \cdot \overrightarrow{CB}$. The minimum value of $\cos C$ is ______. | {
"answer": "\\dfrac{ \\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If two distinct numbers from the set $\{ 5, 15, 21, 35, 45, 63, 70, 90 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 105? Express your answer as a common fraction. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start with three toggle switches with one of them *on* and two of them *off*. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be *on* and two of them will be *off*. Find $m+n$ . | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers $n \le 1500$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n\] for some real number $x.$ | {
"answer": "854",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the coordinate plane, consider points $A = (0, 0)$, $B = (8, 0)$, and $C = (15, 0)$. Line $\ell_A$ has slope 1 and passes through $A$. Line $\ell_B$ is vertical and passes through $B$. Line $\ell_C$ has slope $-1$ and passes through $C$. Additionally, introduce point $D = (20, 0)$ with line $\ell_D$ having a slope of 2 and passing through $D$. All four lines $\ell_A$, $\ell_B$, $\ell_C$, and $\ell_D$ begin rotating clockwise about their respective points, $A$, $B$, $C$, and $D$, at the same angular rate. At any given time, the four lines form a quadrilateral. Determine the largest possible area of such a quadrilateral. | {
"answer": "110.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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