problem stringlengths 10 5.15k | answer dict |
|---|---|
Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to definitely get inside?
b) On average, how much time will Petya need?
c) What is the probability that Petya will get inside in less than a minute? | {
"answer": "\\frac{29}{120}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$ ?
*Proposed by Giacomo Rizzo* | {
"answer": "444",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( t = \frac{1}{1 - \sqrt[4]{2}} \), simplify the expression for \( t \). | {
"answer": "-(1 + \\sqrt[4]{2})(1 + \\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $3a + 5b = 47$ and $7a + 2b = 52$, what is the value of $a + b$? | {
"answer": "\\frac{35}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the first $500$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]
where \( x \) is a real number? | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \) be a positive integer. Given a real number \( x \), let \( \lfloor x \rfloor \) be the greatest integer less than or equal to \( x \). For example, \( \lfloor 2.4 \rfloor = 2 \), \( \lfloor 3 \rfloor = 3 \), and \( \lfloor \pi \rfloor = 3 \). Define a sequence \( a_1, a_2, a_3, \ldots \) where \( a_1 = n \) and
\[
a_m = \left\lfloor \frac{a_{m-1}}{3} \right\rfloor,
\]
for all integers \( m \geq 2 \). The sequence stops when it reaches zero. The number \( n \) is said to be lucky if 0 is the only number in the sequence that is divisible by 3. For example, 7 is lucky, since \( a_1 = 7, a_2 = 2, a_3 = 0 \), and none of 7, 2 are divisible by 3. But 10 is not lucky, since \( a_1 = 10, a_2 = 3, a_3 = 1, a_4 = 0 \), and \( a_2 = 3 \) is divisible by 3. Determine the number of lucky positive integers less than or equal to 1000. | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of eight occurrences of the term $5^5$:
\[ 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 \]
A) $5^6$
B) $5^7$
C) $5^8$
D) $5^{6.29248125}$
E) $40^5$ | {
"answer": "5^{6.29248125}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest common divisor of $8!$ and $(6!)^2.$ | {
"answer": "7200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, find the minimum value of the sum of the distance between points $P$ and $Q$ and the distance between point $P$ and the axis of symmetry of the parabola. | {
"answer": "\\sqrt{17}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that \(4 \operatorname{tg}^{2} Y + 4 \operatorname{ctg}^{2} Y - \frac{1}{\sin ^{2} \gamma} - \frac{1}{\cos ^{2} \gamma} = 17\). Find the value of the expression \(\cos ^{2} Y - \cos ^{4} \gamma\). | {
"answer": "\\frac{3}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has one side of length 5 and the other side less than 4. When the rectangle is folded so that two opposite corners coincide, the length of the crease is \(\sqrt{6}\). Calculate the length of the other side. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the closed shape formed by the graph of the function $y=2\sin x$, where $x \in \left[\frac{\pi}{2}, \frac{5\pi}{2}\right]$, and the lines $y=\pm2$ is what? | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distances between the points on a line are given as $2, 4, 5, 7, 8, k, 13, 15, 17, 19$. Determine the value of $k$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three vectors such that $\mathbf{a}$ is a unit vector, $\mathbf{b}$ and $\mathbf{c}$ have magnitudes of 2, and $\mathbf{a} \cdot \mathbf{b} = 0$, $\mathbf{b} \cdot \mathbf{c} = 4$. Given that
\[
\mathbf{a} = p(\mathbf{a} \times \mathbf{b}) + q(\mathbf{b} \times \mathbf{c}) + r(\mathbf{c} \times \mathbf{a}),
\]
and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2$, find the value of $p + q + r$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square $R_1$ with an area of 25, where each side is trisected to form a smaller square $R_2$, and this process is repeated to form $R_3$ using the same trisection points strategy, calculate the area of $R_3$. | {
"answer": "\\frac{400}{81}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existing stack (if there are already bags there). If given a choice to carry a bag from the truck or from the gate, the worker randomly chooses each option with a probability of 0.5. Eventually, all the bags end up in the shed.
a) (7th grade level, 1 point). What is the probability that the bags end up in the shed in the reverse order compared to how they were placed in the truck?
b) (7th grade level, 1 point). What is the probability that the bag that was second from the bottom in the truck ends up as the bottom bag in the shed? | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( |z| = 2 \) and \( u = \left|z^{2} - z + 1\right| \), find the minimum value of \( u \) (where \( z \in \mathbf{C} \)). | {
"answer": "\\frac{3}{2} \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kolya's parents give him pocket money once a month based on the following criteria: for each A in math, he gets 100 rubles; for each B, he gets 50 rubles; for each C, they subtract 50 rubles; for each D, they subtract 200 rubles. If the total amount is negative, Kolya gets nothing. The math teacher assigns the quarterly grade by calculating the average grade and rounding according to standard rounding rules. How much money could Kolya have gotten at most if it is known that his quarterly grade was two, the quarter lasted exactly 2 months, each month had 14 math lessons, and Kolya gets no more than one grade per lesson? | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A clock has a hour hand $OA$ and a minute hand $OB$ with lengths of $3$ and $4$ respectively. If $0$ hour is represented as $0$ time, then the analytical expression of the area $S$ of $\triangle OAB$ with respect to time $t$ (unit: hours) is ______, and the number of times $S$ reaches its maximum value within a day (i.e., $t\in \left[0,24\right]$ hours) is ______. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can a bamboo trunk (a non-uniform natural material) of length 4 meters be cut into three parts, the lengths of which are multiples of 1 decimeter, and from which a triangle can be formed? | {
"answer": "171",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven students are standing in a line for a graduation photo. Among them, student A must be in the middle, and students B and C must stand together. How many different arrangements are possible? | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coin collector has 100 identical-looking coins. Among them, there are 30 genuine coins and 70 counterfeit coins. The collector knows that all genuine coins have the same weight, all counterfeit coins have different weights, and all counterfeit coins are heavier than the genuine coins. The collector has a balance scale that can be used to compare the weights of two equal-sized groups of coins. What is the minimum number of weighings needed to be sure of finding at least one genuine coin? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A soccer team has 16 members. We need to select a starting lineup including a goalkeeper, a defender, a midfielder, and two forwards. However, only 3 players can play as a goalkeeper, 5 can play as a defender, 8 can play as a midfielder, and 4 players can play as forwards. In how many ways can the team select a starting lineup with these specific constraints? | {
"answer": "1440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x$, $y$, and $a \in R^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Then, calculate the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$. | {
"answer": "6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\sqrt{3}\cos^2{\omega}x+\sin{\omega}x\cos{\omega}x (\omega>0)$, if there exists a real number $x_{0}$ such that for any real number $x$, $f(x_{0})\leq f(x)\leq f(x_{0}+2022\pi)$ holds, then the minimum value of $\omega$ is ____. | {
"answer": "\\frac{1}{4044}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of six-digit palindromes. | {
"answer": "9000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $C: x^2+y^2-2x+4y-4=0$, and a line $l$ with a slope of 1 intersects the circle $C$ at points $A$ and $B$.
(1) Express the equation of the circle in standard form, and identify the center and radius of the circle;
(2) Does there exist a line $l$ such that the circle with diameter $AB$ passes through the origin? If so, find the equation of line $l$; if not, explain why;
(3) When the line $l$ moves parallel to itself, find the maximum area of triangle $CAB$. | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ is a multiple of $18720$, what is the greatest common divisor of $f(x)=(5x+3)(8x+2)(12x+7)(3x+11)$ and $x$? | {
"answer": "462",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)$ is an even function with a period of $2$, and when $x \in (0,1)$, $f(x) = 2^x - 1$, find the value of $f(\log_{2}{12})$. | {
"answer": "-\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer with exactly 12 positive integer divisors? | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A decagon is inscribed in a rectangle such that the vertices of the decagon divide each side of the rectangle into five equal segments. The perimeter of the rectangle is 160 centimeters, and the ratio of the length to the width of the rectangle is 3:2. What is the number of square centimeters in the area of the decagon? | {
"answer": "1413.12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ordered pair $(a, c)$ of integers, each of which has an absolute value less than or equal to 6, is chosen at random. What is the probability that the equation $ax^2 - 3ax + c = 0$ will not have distinct real roots both greater than 2?
A) $\frac{157}{169}$ B) $\frac{167}{169}$ C) $\frac{147}{169}$ D) $\frac{160}{169}$ | {
"answer": "\\frac{167}{169}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $n$ is an integer between $1$ and $60$, inclusive, determine for how many values of $n$ the expression $\frac{((n+1)^2 - 1)!}{(n!)^{n+1}}$ is an integer. | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all $x \in (0, +\infty)$, the inequality $(2x - 2a + \ln \frac{x}{a})(-2x^{2} + ax + 5) \leq 0$ always holds. Determine the range of values for the real number $a$. | {
"answer": "\\left\\{ \\sqrt{5} \\right\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The "2023 MSI" Mid-Season Invitational of "League of Legends" is held in London, England. The Chinese teams "$JDG$" and "$BLG$" have entered the finals. The finals are played in a best-of-five format, where the first team to win three games wins the championship. Each game must have a winner, and the outcome of each game is not affected by the results of previous games. Assuming that the probability of team "$JDG$" winning a game is $p (0 \leq p \leq 1)$, let the expected number of games be denoted as $f(p)$. Find the maximum value of $f(p)$. | {
"answer": "\\frac{33}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the same Rotokas alphabet, how many license plates of five letters are possible that begin with G, K, or P, end with T, cannot contain R, and have no letters that repeat? | {
"answer": "630",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2023 \left( \frac{2}{x} + \frac{2}{y} + 2022 \right)\] for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s.$ | {
"answer": "2023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leaves them symmetrical? | {
"answer": "8100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that exactly $5$ of his answers are True. What is the probability he answers at least $5$ questions correctly? | {
"answer": "9/14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, points $A$, $B$, $C$, $D$, $E$, and $F$ lie on a straight line with $AB=BC=CD=DE=EF=3$. Semicircles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ create a shape as depicted. What is the area of the shaded region underneath the largest semicircle that exceeds the areas of the other semicircles combined, given that $AF$ is after the diameters were tripled compared to the original configuration?
[asy]
size(5cm); defaultpen(fontsize(9));
pair one = (0.6, 0);
pair a = (0, 0); pair b = a + one; pair c = b + one; pair d = c + one; pair e = d + one; pair f = e + one;
path region = a{up}..{down}f..{up}e..{down}d..{up}c..{down}b..{up}a--cycle;
filldraw(region, gray(0.75), linewidth(0.75));
draw(a--f, dashed + linewidth(0.75));
// labels
label("$A$", a, W); label("$F$", f, E);
label("$B$", b, 0.8 * SE); label("$D$", d, 0.8 * SE);
label("$C$", c, 0.8 * SW); label("$E$", e, 0.8 * SW);
[/asy] | {
"answer": "\\frac{45}{2}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in acute triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively, and bsinA = acos(B - $ \frac{\pi}{6}$).
(1) Find the value of angle B.
(2) If b = $\sqrt{13}$, a = 4, and D is a point on AC such that S<sub>△ABD</sub> = 2$\sqrt{3}$, find the length of AD. | {
"answer": "\\frac{2 \\sqrt{13}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangles $\triangle DEF$ and $\triangle D'E'F'$ are in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,15)$, $E'(30,15)$, $F'(20,5)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\triangle DEF$ to $\triangle D'E'F'$. Find $n+u+v$. | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is inscribed in an equilateral triangle such that each vertex of the square touches the perimeter of the triangle. One side of the square intersects and forms a smaller equilateral triangle within which we inscribe another square in the same manner, and this process continues infinitely. What fraction of the equilateral triangle's area is covered by the infinite series of squares? | {
"answer": "\\frac{3 - \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Let \( x_i \in \{0,1\} \) (for \( i=1,2,\cdots,n \)). If the value of the function \( f=f(x_1, x_2, \cdots, x_n) \) only takes 0 or 1, then \( f \) is called an n-ary Boolean function. We denote
\[
D_{n}(f)=\left\{(x_1, x_2, \cdots, x_n) \mid f(x_1, x_2, \cdots, x_n)=0\right\}.
\]
(1) Find the number of n-ary Boolean functions.
(2) Let \( g \) be a 10-ary Boolean function such that
\[
g(x_1, x_2, \cdots, x_{10}) \equiv 1+\sum_{i=1}^{10} \prod_{j=1}^{i} x_{j} \ (\bmod \ 2),
\]
find the number of elements in the set \( D_{10}(g) \), and evaluate
\[
\sum_{\left(x_1, x_2, \cdots, x_{10}\right) \in D_{10}(g)}\left(x_1+x_2+\cdots+x_{10}\right).
\] | {
"answer": "565",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle with integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<100$, determine the number of possible integer values for $b$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The local junior football team is deciding on their new uniforms. The team's ninth-graders will choose the color of the socks (options: red, green, or blue), and the tenth-graders will pick the color for the t-shirts (options: red, yellow, green, blue, or white). Neither group will discuss their choices with the other group. If each color option is equally likely to be selected, what is the probability that both the socks and the t-shirt are either both white or different colors? | {
"answer": "\\frac{13}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the one-variable quadratic equation $x^{2}+2x+m+1=0$ has two distinct real roots with respect to $x$, determine the value of $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many triangles can be formed using the vertices of a regular pentadecagon (a 15-sided polygon), if no side of the triangle can be a side of the pentadecagon? | {
"answer": "440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a point outside a circle $\Gamma$ centered at point $O$ , and let $PA$ and $PB$ be tangent lines to circle $\Gamma$ . Let segment $PO$ intersect circle $\Gamma$ at $C$ . A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$ , respectively. Given that $EF=8$ and $\angle{APB}=60^\circ$ , compute the area of $\triangle{AOC}$ .
*2020 CCA Math Bonanza Individual Round #6* | {
"answer": "12 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During the 2011 Universiade in Shenzhen, a 12-person tour group initially stood in two rows with 4 people in the front row and 8 people in the back row. The photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that these two people are not adjacent in the front row. Calculate the number of different ways to adjust their positions. | {
"answer": "560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In cube \(ABCDA_1B_1C_1D_1\) with side length 1, a sphere is inscribed. Point \(E\) is located on edge \(CC_1\) such that \(C_1E = \frac{1}{8}\). From point \(E\), a tangent to the sphere intersects the face \(AA_1D_1D\) at point \(K\), with \(\angle KEC = \arccos \frac{1}{7}\). Find \(KE\). | {
"answer": "\\frac{7}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set of vectors $\mathbf{u}$ such that
\[\mathbf{u} \cdot \mathbf{u} = \mathbf{u} \cdot \begin{pmatrix} 6 \\ -28 \\ 12 \end{pmatrix}\] forms a solid in space. Find the volume of this solid. | {
"answer": "\\frac{4}{3} \\pi \\cdot 241^{3/2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is calculated. | {
"answer": "18185",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles trapezoid \(ABCD\), where \(AD \parallel BC\), \(BC = 2AD = 4\), \(\angle ABC = 60^\circ\), and \(\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CD}\), find \(\overrightarrow{CA} \cdot \overrightarrow{BE}\). | {
"answer": "-10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the decimal representation of an even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) are used, and the digits may repeat. It is known that the sum of the digits of the number \( 2M \) equals 39, and the sum of the digits of the number \( M / 2 \) equals 30. What values can the sum of the digits of the number \( M \) take? List all possible answers. | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x^3 - 6x + 5, x \in \mathbb{R}$.
(1) Find the equation of the tangent line to the function $f(x)$ at $x = 1$;
(2) Find the extreme values of $f(x)$ in the interval $[-2, 2]$. | {
"answer": "5 - 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The digits from 1 to 9 are to be written in the nine cells of a $3 \times 3$ grid, one digit in each cell.
- The product of the three digits in the first row is 12.
- The product of the three digits in the second row is 112.
- The product of the three digits in the first column is 216.
- The product of the three digits in the second column is 12.
What is the product of the digits in the shaded cells? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers does not exceed $5000$ yuan (commonly known as the "threshold") is not taxable, and the part exceeding $5000$ yuan is the taxable income for the whole month. The new tax rate table is as follows:
2019年1月1日后个人所得税税率表
| 全月应纳税所得额 | 税率$(\%)$ |
|------------------|------------|
| 不超过$3000$元的部分 | $3$ |
| 超过$3000$元至$12000$元的部分 | $10$ |
| 超过$12000$元至$25000$元的部分 | $20$ |
| 超过$25000$元至$35000$元的部分 | $25$ |
Individual income tax special additional deductions refer to the six special additional deductions specified in the individual income tax law, including child education, continuing education, serious illness medical treatment, housing loan interest, housing rent, and supporting the elderly. Among them, supporting the elderly refers to the support expenses for parents and other legally supported persons aged $60$ and above paid by taxpayers. It can be deducted at the following standards: for taxpayers who are only children, a standard deduction of $2000$ yuan per month is allowed; for taxpayers with siblings, the deduction amount of $2000$ yuan per month is shared among them, and the amount shared by each person cannot exceed $1000$ yuan per month.
A taxpayer has only one older sister, and both of them meet the conditions for supporting the elderly as specified. If the taxpayer's personal income tax payable in May 2020 is $180$ yuan, then the taxpayer's monthly salary after tax in that month is ____ yuan. | {
"answer": "9720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\triangle ABC$ has side lengths $AB=20$ , $BC=15$ , and $CA=7$ . Let the altitudes of $\triangle ABC$ be $AD$ , $BE$ , and $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$ ? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y=g(x)$ is given. For all $x > 5$, it is observed that $g(x) > 0.1$. If $g(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A, B, C$ are integers, determine $A+B+C$ knowing that the vertical asymptotes occur at $x = -3$ and $x = 4$. | {
"answer": "-108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(\triangle ABC\) and \(\triangle CDE\) are equilateral triangles. Given that \(\angle EBD = 62^\circ\) and \(\angle AEB = x^\circ\), what is the value of \(x\)? | {
"answer": "122",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is chosen at random on the number line between 0 and 1, and this point is colored red. Another point is then chosen at random on the number line between 0 and 2, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point but less than three times the number of the red point? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $z$ be a nonreal complex number such that $|z| = 1$. Find the real part of $\frac{1}{z - i}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \(\lim _{x \rightarrow -1} \frac{3 x^{4} + 2 x^{3} - x^{2} + 5 x + 5}{x^{3} + 1}\). | {
"answer": "-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a cube where each pair of opposite faces sums to 8 instead of the usual 7. If one face shows 1, the opposite face will show 7; if one face shows 2, the opposite face will show 6; if one face shows 3, the opposite face will show 5. Calculate the largest sum of three numbers whose faces meet at one corner of the cube. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose the probability distribution of the random variable $X$ is given by $P\left(X=\frac{k}{5}\right)=ak$, where $k=1,2,3,4,5$.
(1) Find the value of $a$.
(2) Calculate $P\left(X \geq \frac{3}{5}\right)$.
(3) Find $P\left(\frac{1}{10} < X \leq \frac{7}{10}\right)$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $n$ between 1 and 500 inclusive does the decimal representation of $\frac{n}{2520}$ terminate? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 65 cents worth of coins come up heads? | {
"answer": "\\dfrac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the remainder when $123456789012$ is divided by $240$. | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$ . Find the measure of the angle $\angle PBC$ . | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Under the threat of severe punishment, he decreed that each family should have no more than one girl. As a result, in this country, each woman's last - and only last - child was a girl because no woman dared to have more children after giving birth to a girl. What proportion of boys comprised the total number of children in this country, assuming the chances of giving birth to a boy or a girl are equal? | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the height of an external tangent cone of a sphere is three times the radius of the sphere, determine the ratio of the lateral surface area of the cone to the surface area of the sphere. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real numbers \(x\), \(y\), and \(z\), consider the matrix
\[
\begin{pmatrix} x+y & x & y \\ x & y+z & y \\ y & x & x+z \end{pmatrix}
\]
Determine whether this matrix is invertible. If not, list all possible values of
\[
\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.
\] | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a recent test, $15\%$ of the students scored $60$ points, $20\%$ got $75$ points, $30\%$ scored $85$ points, $10\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and the median score on this test. | {
"answer": "-1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of positive integers $n$ satisfying:
- $n<10^6$
- $n$ is divisible by 7
- $n$ does not contain any of the digits 2,3,4,5,6,7,8.
| {
"answer": "104",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.
*Proposed by Lewis Chen* | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, a curve $C$ has the polar equation $ρ^2 - 4ρ\sinθ + 3 = 0$. Points $A$ and $B$ have polar coordinates $(1,π)$ and $(1,0)$, respectively.
(1) Find the parametric equation of curve $C$;
(2) Take a point $P$ on curve $C$ and find the maximum and minimum values of $|AP|^2 + |BP|^2$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let p, q, r, s, and t be distinct integers such that (9-p)(9-q)(9-r)(9-s)(9-t) = -120. Calculate the value of p+q+r+s+t. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the sum of the digits of a natural number \( n \) is subtracted from \( n \), the result is 2016. Find the sum of all such natural numbers \( n \). | {
"answer": "20245",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near future. If the robot works for $\mathrm{L}$ hours a day on producing blocks, it will be fully out of order in $8-\sqrt{L}$ days. It is known that the hourly average productivity of the robot during a shift is determined by the function $\frac{1}{\sqrt{L}}$, and the value of $\mathrm{L}$ can only be set once and cannot be changed thereafter.
(a) What value of $\mathrm{L}$ should be chosen if the main objective is to maximize the number of construction blocks produced by the robot in one day?
(b) What value of $\mathrm{L}$ should be chosen if the main objective is to maximize the total number of construction blocks produced by the robot over its entire lifespan?
(c) The Japanese company that owns the robot decided to maximize its revenue from the robot. Therefore, in addition to producing goods, it was decided to send the robot to a 24-hour exhibition, where it will be displayed as an exhibit and generate 4 monetary units per hour during non-production hours. If the robot goes out of order, it cannot be used as an exhibit. What value of $\mathrm{L}$ should be chosen to ensure maximum revenue from owning the robot? What will be the company's revenue?
(d) What is the minimum hourly revenue from displaying the robot at the exhibition, at which the company would decide to abandon the production of construction blocks? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1 \le a < b < c \le 20$ . | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a checkerboard with 31 rows and 29 columns, where each corner square is black and the squares alternate between red and black, determine the number of black squares on this checkerboard. | {
"answer": "465",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and vectors $\overrightarrow{m}=(1-\cos (A+B),\cos \frac {A-B}{2})$ and $\overrightarrow{n}=( \frac {5}{8},\cos \frac {A-B}{2})$ with $\overrightarrow{m}\cdot \overrightarrow{n}= \frac {9}{8}$,
(1) Find the value of $\tan A\cdot\tan B$;
(2) Find the maximum value of $\frac {ab\sin C}{a^{2}+b^{2}-c^{2}}$. | {
"answer": "-\\frac {3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an extended hexagonal lattice, each point is still one unit from its nearest neighbor. The lattice is now composed of two concentric hexagons where the outer hexagon has sides twice the length of the inner hexagon. All vertices are connected to their nearest neighbors. How many equilateral triangles have all three vertices in this extended lattice? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\] | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**A person rolls seven standard, six-sided dice. What is the probability that there is at least one pair but no three dice show the same value?** | {
"answer": "\\frac{315}{972}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set S contains triangles whose sides have integer lengths less than 7, and no two elements of S are congruent or similar. Calculate the largest number of elements that S can have. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic-geometric sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $\frac{S_6}{S_3} = -\frac{19}{8}$ and $a_4 - a_2 = -\frac{15}{8}$. Find the value of $a_3$. | {
"answer": "\\frac{9}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 11 ones, 22 twos, 33 threes, and 44 fours on a blackboard. The following operation is performed: each time, erase 3 different numbers and add 2 more of the fourth number that was not erased. For example, if 1 one, 1 two, and 1 three are erased in one operation, write 2 more fours. After performing this operation several times, only 3 numbers remain on the blackboard, and it's no longer possible to perform the operation. What is the product of the remaining three numbers?
| {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Use Horner's method to find the value of the polynomial $f(x) = 5x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$ when $x=1$, and find the value of $v_3$. | {
"answer": "8.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\leqslant 1-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______. | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a $4 \times 4$ grid of squares with 25 grid points. Determine the number of different lines passing through at least 3 of these grid points. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\left(\frac{a^2}{a+1}-a+1\right) \div \frac{a^2-1}{a^2+2a+1}$, then choose a suitable integer from the inequality $-2 \lt a \lt 3$ to substitute and evaluate. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Equilateral triangle $ABC$ has a side length of $12$. There are three distinct triangles $AD_1E_1$, $AD_2E_2$, and $AD_3E_3$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = BD_3 = 6$. Find $\sum_{k=1}^3(CE_k)^2$. | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | {
"answer": "63_{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Right triangle $PQR$ has one leg of length 9 cm, one leg of length 12 cm and a right angle at $P$. A square has one side on the hypotenuse of triangle $PQR$ and a vertex on each of the two legs of triangle $PQR$. What is the length of one side of the square, in cm? Express your answer as a common fraction. | {
"answer": "\\frac{45}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A survey conducted at a conference found that 70% of the 150 male attendees and 75% of the 850 female attendees support a proposal for new environmental legislation. What percentage of all attendees support the proposal? | {
"answer": "74.2\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ant has one sock and one shoe for each of its six legs, and on one specific leg, both the sock and shoe must be put on last. Find the number of different orders in which the ant can put on its socks and shoes. | {
"answer": "10!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = e^{-x}(ax^2 + bx + 1)$ (where $e$ is a constant, $a > 0$, $b \in \mathbb{R}$), the derivative of the function $f(x)$ is denoted as $f'(x)$, and $f'(-1) = 0$.
1. If $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f(0))$.
2. When $a > \frac{1}{5}$, if the maximum value of the function $f(x)$ in the interval $[-1, 1]$ is $4e$, try to find the values of $a$ and $b$. | {
"answer": "\\frac{12e^2 - 2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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