problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that $\alpha$ is an obtuse angle and $\beta$ is also an obtuse angle with $\cos\alpha = -\frac{2\sqrt{5}}{5}$ and $\sin\beta = \frac{\sqrt{10}}{10}$, find the value of $\alpha + \beta$. | {
"answer": "\\frac{7\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using toothpicks of equal length, a rectangular grid is constructed. The grid measures 25 toothpicks in height and 15 toothpicks in width. Additionally, there is an internal horizontal partition at every fifth horizontal line starting from the bottom. Calculate the total number of toothpicks used. | {
"answer": "850",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\sqrt{3}b\cos A=a\cos B$.
$(1)$ Find the angle $A$;
$(2)$ If $a= \sqrt{2}$ and $\frac{c}{a}= \frac{\sin A}{\sin B}$, find the perimeter of $\triangle ABC$. | {
"answer": "3 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many pairs of positive integers have greatest common divisor 5! and least common multiple 50! ? | {
"answer": "16384",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2016} > 0$, $S_{2017} < 0$. For any positive integer $n$, we have $|a_n| \geqslant |a_k|$. Determine the value of $k$. | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x^2 - (2t + 1)x + t \ln x$ where $t \in \mathbb{R}$,
(1) If $t = 1$, find the extreme values of $f(x)$.
(2) Let $g(x) = (1 - t)x$, and suppose there exists an $x_0 \in [1, e]$ such that $f(x_0) \geq g(x_0)$ holds. Find the maximum value of the real number $t$. | {
"answer": "\\frac{e(e - 2)}{e - 1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x, y, z$ be real numbers such that:
\begin{align*}
y+z & = 16, \\
z+x & = 18, \\
x+y & = 20.
\end{align*}
Find $\sqrt{xyz(x+y+z)}$. | {
"answer": "9\\sqrt{77}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle \( K \) goes through the vertices \( A \) and \( C \) of triangle \( ABC \). The center of circle \( K \) lies on the circumcircle of triangle \( ABC \). Circle \( K \) intersects side \( AB \) at point \( M \). Find the angle \( BAC \) if \( AM : AB = 2 : 7 \) and \( \angle B = \arcsin \frac{4}{5} \). | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A building has seven rooms numbered 1 through 7 on one floor, with various doors connecting these rooms. The doors can be either one-way or two-way. Additionally, there is a two-way door between room 1 and the outside, and there is a treasure in room 7. Design the arrangement of rooms and doors such that:
(a) It is possible to enter room 1, reach the treasure in room 7, and return outside.
(b) The minimum number of steps required to achieve this (each step involving walking through a door) is as large as possible. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a Cartesian coordinate system, the parametric equation for curve $C_1$ is
$$
\left\{ \begin{aligned}
x &= 2\cos\alpha, \\
y &= \sqrt{2}\sin\alpha
\end{aligned} \right.
$$
with $\alpha$ as the parameter. Using the origin as the pole, the positive half of the $x$-axis as the polar axis, and the same unit length as the Cartesian coordinate system, establish a polar coordinate system. The polar equation for curve $C_2$ is $\rho = \cos\theta$.
(1) Find the general equation for curve $C_1$ and the Cartesian coordinate equation for curve $C_2$;
(2) If $P$ and $Q$ are any points on the curves $C_1$ and $C_2$, respectively, find the minimum value of $|PQ|$. | {
"answer": "\\frac{\\sqrt{7} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What code will be produced for this message in the new encoding where the letter А is replaced by 21, the letter Б by 122, and the letter В by 1? | {
"answer": "211221121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers 1, 2, 3, 4, 5, two numbers are randomly selected to be the base and the true number (antilogarithm) of a logarithm, respectively. The total number of different logarithmic values that can be obtained is ___. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular tetrahedron ABCD with an edge length of 2, G is the centroid of triangle BCD, and M is the midpoint of line segment AG. The surface area of the circumscribed sphere of the tetrahedron M-BCD is __________. | {
"answer": "6\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$ .
*Proposed by Lewis Chen* | {
"answer": "2017",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the circumference of the region defined by the equation $x^2+y^2 - 10 = 3y - 6x + 3$? | {
"answer": "\\pi \\sqrt{73}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The letter T is formed by placing a $2\:\text{inch} \times 6\:\text{inch}$ rectangle vertically and a $2\:\text{inch} \times 4\:\text{inch}$ rectangle horizontally across the top center of the vertical rectangle. What is the perimeter of the T, in inches? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction. | {
"answer": "\\dfrac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape. The parallel sides of the trapezoid have lengths $18$ and $30$ meters. What fraction of the yard is occupied by the flower beds?
A) $\frac{1}{6}$
B) $\frac{1}{5}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( c_{n}=11 \ldots 1 \) be a number in which the decimal representation contains \( n \) ones. Then \( c_{n+1}=10 \cdot c_{n}+1 \). Therefore:
\[ c_{n+1}^{2}=100 \cdot c_{n}^{2} + 22 \ldots 2 \cdot 10 + 1 \]
For example,
\( c_{2}^{2}=11^{2}=(10 \cdot 1+1)^{2}=100+2 \cdot 10+1=121 \),
\( c_{3}^{2} = 111^{2} = 100 \cdot 11^{2} + 220 + 1 = 12100 + 220 + 1 = 12321 \),
\( c_{4}^{2} = 1111^{2} = 100 \cdot 111^{2} + 2220 + 1 = 1232100 + 2220 + 1 = 1234321 \), etc.
We observe that in all listed numbers \( c_{2}^{2}, c_{3}^{2}, c_{4}^{2} \), the digit with respect to which these numbers are symmetric (2 in the case of \( c_{2}^{2}, 3 \) in the case of \( c_{3}^{2}, 4 \) in the case of \( c_{4}^{2} \)) coincides with the number of ones in the number that was squared.
The given number \( c=123456787654321 \) is also symmetric with respect to the digit 8, which suggests that it might be the square of the number \( c_{8} = 11111111 \). This can be verified by performing multiplication by columns or using the recursive relation. | {
"answer": "11111111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C: \frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and the sum of the distances from a point on the ellipse to the two foci is $2\sqrt{2}$. A line $l$ with slope $k(k\neq 0)$ passes through the upper focus of the ellipse and intersects the ellipse at points $P$ and $Q$. The perpendicular bisector of segment $PQ$ intersects the $y$-axis at point $M(0,m)$.
(1) Find the standard equation of the ellipse;
(2) Find the range of values for $m$;
(3) Express the area $S$ of triangle $\Delta MPQ$ in terms of $m$, and find the maximum value of the area $S$. | {
"answer": "\\frac{3\\sqrt{6}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polar coordinate equation of curve C is given by C: ρ² = $\frac{12}{5 - \cos(2\theta)}$, and the parametric equations of line l are given by $\begin{cases} x = 1 + \frac{\sqrt{2}}{2}t \\ y = \frac{\sqrt{2}}{2}t \end{cases}$ (where t is the parameter).
1. Write the rectangular coordinate equation of C and the standard equation of l.
2. Line l intersects curve C at two points A and B. Let point M be (0, -1). Calculate the value of $\frac{|MA| + |MB|}{|MA| \cdot |MB|}$. | {
"answer": "\\frac{4\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e = \dfrac{\sqrt{2}}{2}$, the length of the minor axis is $2$.
$(1)$ Find the equation of the ellipse;
$(2)$ Point $A$ is a moving point on the ellipse (not the endpoints of the major axis), the extension line of $AF_2$ intersects the ellipse at point $B$, and the extension line of $AO$ intersects the ellipse at point $C$. Find the maximum value of the area of $\triangle ABC$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive odd integers greater than 1 and less than $200$ are square-free? | {
"answer": "79",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If four coins are tossed at the same time, what is the probability of getting exactly three heads and one tail? Follow this by rolling a six-sided die. What is the probability that the die shows a number greater than 4? | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the radius of the smallest circle inside which every planar closed polygonal line with a perimeter of 1 can be enclosed? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Half of the yellow flowers are tulips, one third of the blue flowers are daisies, and seven tenths of the flowers are yellow. Find the percentage of flowers that are daisies. | {
"answer": "45\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya invented four distinct natural numbers and wrote down all their pairwise sums on the board. Below those, he wrote all their sums taken three at a time. It turned out that the sum of the two largest pairwise sums and the two smallest sums from those taken three at a time (a total of four sums) is 2017. Find the largest possible value of the sum of the four numbers that Petya invented. | {
"answer": "1006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest possible sum of the digits in the base-nine representation of a positive integer less than $3000$? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a settlement $C$ (a point) located at the intersection of roads $A$ and $B$ (straight lines). Sasha walks along road $A$ towards $C$, taking 45 steps per minute with a step length of 60 cm. At the start, Sasha is 290 m away from $C$. Dania walks along road $B$ towards $C$ at a rate of 55 steps per minute with a step length of 65 cm, and at the start of their movement, Dania is 310 m away from $C$. Each person continues walking along their road without stopping after passing point $C$. We record the moments in time when both Dania and Sasha have taken whole numbers of steps. Find the minimum possible distance between them (along the roads) at such moments in time. Determine the number of steps each of them has taken by the time this minimum distance is achieved. | {
"answer": "57",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$ is a positive real number and $b$ is an integer between $1$ and $201$, inclusive, find the number of ordered pairs $(a,b)$ such that $(\log_b a)^{2023}=\log_b(a^{2023})$. | {
"answer": "603",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle with integral sides and an isosceles perimeter of 11, calculate the area of the triangle. | {
"answer": "\\frac{5\\sqrt{2.75}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 5 students taking a test, and each student's score ($a, b, c, d, e$) is an integer between 0 and 100 inclusive. It is known that $a \leq b \leq c \leq d \leq e$. If the average score of the 5 students is $p$, then the median score $c$ is at least $\qquad$ . | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a dog show, each dog was assigned a sequential number from 1 to 24. Due to health reasons, one of the dogs was unable to participate in the competition. It turns out that among the remaining 23 dogs, one has a number equal to the arithmetic mean of the remaining dogs' numbers. What was the number assigned to the dog that could not participate in the show? If there are multiple solutions, list these numbers in ascending order without spaces. | {
"answer": "124",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the region \(B\) in the complex plane consisting of all points \(z\) such that both \(\frac{z}{50}\) and \(\frac{50}{\overline{z}}\) have real and imaginary parts between 0 and 1, inclusive. Find the area of \(B\). | {
"answer": "2500 - 312.5 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many unique numbers can you get by multiplying two or more distinct members of the set $\{1,2,3,5,7\}$ together? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of integers has a mode of 32, a mean of 22, a smallest number of 10, and a median of \( m \). If \( m \) is replaced by \( m+10 \), the new sequence has a mean of 24 and a median of \( m+10 \). If \( m \) is replaced by \( m-8 \), the new sequence has a median of \( m-4 \). What is the value of \( m \)? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three lines are drawn parallel to the sides of a triangle through a point inside it, dividing the triangle into six parts: three triangles and three quadrilaterals. The areas of all three inner triangles are equal. Determine the range within which the ratio of the area of each inner triangle to the area of the original triangle can lie. | {
"answer": "1/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[
\sin^6 0^\circ + \sin^6 1^\circ + \sin^6 2^\circ + \dots + \sin^6 90^\circ.
\] | {
"answer": "\\frac{229}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an geometric sequence \\(\{a_n\}\) with a common ratio less than \\(1\\), the sum of the first \\(n\\) terms is \\(S_n\\), and \\(a_1 = \frac{1}{2}\\), \\(7a_2 = 2S_3\\).
\\((1)\\) Find the general formula for the sequence \\(\{a_n\}\).
\\((2)\\) Let \\(b_n = \log_2(1-S_{n+1})\\). If \\(\frac{1}{{b_1}{b_3}} + \frac{1}{{b_3}{b_5}} + \ldots + \frac{1}{{b_{2n-1}}{b_{2n+1}}} = \frac{5}{21}\\), find \\(n\\). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = |2x+1| + |3x-2|$, and the solution set of the inequality $f(x) \leq 5$ is $\left\{x \mid -\frac{4a}{5} \leq x \leq \frac{3a}{5}\right\}$, where $a, b \in \mathbb{R}$.
1. Find the values of $a$ and $b$;
2. For any real number $x$, the inequality $|x-a| + |x+b| \geq m^2 - 3m$ holds, find the maximum value of the real number $m$. | {
"answer": "\\frac{3 + \\sqrt{21}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of natural numbers $\left\{x_{n}\right\}$ is constructed according to the following rules:
$$
x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \text{ for } n \geq 1.
$$
It is known that some term in the sequence is 1000. What is the smallest possible value of $a+b$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given $a^{2}-c^{2}=b^{2}- \frac {8bc}{5}$, $a=6$, $\sin B= \frac {4}{5}$.
(I) Find the value of $\sin A$;
(II) Find the area of $\triangle ABC$. | {
"answer": "\\frac {168}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive integers \(a\), \(b\), \(c\), and \(d\) satisfy \(a > b > c > d\), \(a + b + c + d = 2200\), and \(a^2 - b^2 + c^2 - d^2 = 2200\). Find the number of possible values of \(a\). | {
"answer": "548",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) The definite integral $\int_{-1}^{1}(x^{2}+\sin x)dx=$ ______.
(2) There are 2 red balls, 1 white ball, and 1 blue ball in a box. The probability of drawing two balls with at least one red ball is ______.
(3) Given the function $f(x)=\begin{cases}1-\log_{a}(x+2), & x\geqslant 0 \\ g(x), & x < 0\end{cases}$ is an odd function, then the root of the equation $g(x)=2$ is ______.
(4) Given the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(0 < b < a < \sqrt{2}b)$, its foci are $F_{1}$ and $F_{2}$ respectively. Circle $N$ has $F_{2}$ as its center, and its minor axis length as the diameter. A tangent line to circle $N$ passing through point $F_{1}$ touches it at points $A$ and $B$. If the area of quadrilateral $F_{1}AF_{2}B$ is $S= \frac{2}{3}a^{2}$, then the eccentricity of ellipse $M$ is ______. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people, A and B, visit the "2011 Xi'an World Horticultural Expo" together. They agree to independently choose 4 attractions from numbered attractions 1 to 6 to visit, spending 1 hour at each attraction. Calculate the probability that they will be at the same attraction during their last hour. | {
"answer": "\\dfrac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular prism has dimensions of 1 by 1 by 2. Calculate the sum of the areas of all triangles whose vertices are also vertices of this rectangular prism, and express the sum in the form $m + \sqrt{n} + \sqrt{p}$, where $m, n,$ and $p$ are integers. Find $m + n + p$. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $l$: $x=my+1$ passes through the right focus $F$ of the ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the focus of the parabola $x^{2}=4\sqrt{3}y$ is the upper vertex of the ellipse $C$, and the line $l$ intersects the ellipse $C$ at points $A$ and $B$.
1. Find the equation of the ellipse $C$.
2. If the line $l$ intersects the $y$-axis at point $M$, and $\overrightarrow{MA}=\lambda_{1}\overrightarrow{AF}, \overrightarrow{MB}=\lambda_{2}\overrightarrow{BF}$, is the value of $\lambda_{1}+\lambda_{2}$ a constant as $m$ varies? If so, find this constant. If not, explain why. | {
"answer": "-\\frac{8}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of checkers that can be placed on an $8 \times 8$ board so that each one is being attacked? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many values of $k$ is $18^{18}$ the least common multiple of the positive integers $6^9$, $9^9$, and $k$? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the three-digit number \( n = abc \). If \( a, b, \) and \( c \) as the lengths of the sides can form an isosceles (including equilateral) triangle, then how many such three-digit numbers \( n \) are there? | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ordered pairs $(s, d)$ of positive integers with $4 \leq s \leq d \leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\frac{1}{2}$ ? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\angle ACB=60^{\circ}$, $BC > 1$, and $AC=AB+\frac{1}{2}$. When the perimeter of $\triangle ABC$ is at its minimum, the length of $BC$ is $\_\_\_\_\_\_\_\_\_\_$. | {
"answer": "1 + \\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:<br/>$(1)2x\left(x-1\right)=1$;<br/>$(2)x^{2}+8x+7=0$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function that satisfies
\[ f(1) = 2, \]
\[ f(2) = 1, \]
\[ f(3n) = 3f(n), \]
\[ f(3n + 1) = 3f(n) + 2, \]
\[ f(3n + 2) = 3f(n) + 1. \]
Find how many integers \( n \leq 2014 \) satisfy \( f(n) = 2n \). | {
"answer": "127",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$ ? | {
"answer": "87",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "12\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ has an asymptote that is perpendicular to the line $x + 2y + 1 = 0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola such that $|F_1A| = 2|F_2A|$. Find $\cos \angle AF_2F_1$. | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle touches the extensions of two sides \( AB \) and \( AD \) of a square \( ABCD \) with a side length of 4 cm. From point \( C \), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \( 60^{\circ} \). | {
"answer": "4 (\\sqrt{2} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The wholesale department operates a product with a wholesale price of 500 yuan per unit and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank at a monthly interest rate of 4.2‰, and the storage and operating cost is 0.30 yuan per unit per day. Determine the maximum average storage period for the product without incurring a loss. | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$ , $B$ and $C$ , respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$ . On each turn, Ming chooses a two-line intersection inside $ABC$ , and draws the straight line determined by the intersection and one of $A$ , $B$ and $C$ of his choice. Find the maximum possible number of three-line intersections within $ABC$ after 300 turns.
*Proposed by usjl* | {
"answer": "45853",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing the given edge. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When \( \frac{1}{2222} \) is expressed as a decimal, what is the sum of the first 50 digits after the decimal point? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given numbers $5, 6, 7, 8, 9, 10, 11, 12, 13$ are written in a $3\times3$ array, with the condition that two consecutive numbers must share an edge. If the sum of the numbers in the four corners is $32$, calculate the number in the center of the array. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the ray $(0,+\infty)$ of the number line, there are several (more than two) segments of length 1. For any two different segments, you can select one number from each so that these numbers differ exactly by a factor of 2. The left end of the leftmost segment is the number $a$, and the right end of the rightmost segment is the number $b$. What is the maximum value that the quantity $b-a$ can take? | {
"answer": "5.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( M \) belongs to the edge \( CD \) of the parallelepiped \( ABCDA_1B_1C_1D_1 \), where \( CM: MD = 1:2 \). Construct the section of the parallelepiped with a plane passing through point \( M \) parallel to the lines \( DB \) and \( AC_1 \). In what ratio does this plane divide the diagonal \( A_1C \) of the parallelepiped? | {
"answer": "1 : 11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=-\frac{1}{2}x^{2}+x$ with a domain that contains an interval $[m,n]$, and its range on this interval is $[3m,3n]$. Find the value of $m+n$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a meeting room, the first row has a total of 8 seats. Now 3 people are seated, and the requirement is that there should be empty seats to the left and right of each person. Calculate the number of different seating arrangements. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^3}{9^3 - 1} + \frac{3^4}{9^4 - 1} + \cdots.$$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$ . Let $N$ be the smallest positive integer such that $S(N) = 2013$ . What is the value of $S(5N + 2013)$ ? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$ . Let $\omega$ be the circle that touches the sides $AB$ , $BC$ , and $AD$ . A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$ . Determine the ratio $\frac{AK}{KD}$ .
*Proposed by Giorgi Arabidze, Georgia* | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the function \( f(x) = 3^x - 9^x \) for real numbers \( x \). | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\cos \left(x+ \frac {\pi}{3}\right)$, derive the horizontal shift of the graph of the function $y=\sin x$. | {
"answer": "\\frac {5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The axial section of a cone is an equilateral triangle with a side length of 1. Find the radius of the sphere that is tangent to the axis of the cone, its base, and its lateral surface. | {
"answer": "\\frac{\\sqrt{3} - 1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\{ 2, 3, 5, 7 \}$ . | {
"answer": "1932",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Find the minimum value of $a$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When \( N \) takes all values from 1, 2, 3, ..., to 2015, how many numbers of the form \( 3^n + n^3 \) are divisible by 7? | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin x + \cos x$.
(1) If $f(x) = 2f(-x)$, find the value of $\frac{\cos^2x - \sin x\cos x}{1 + \sin^2x}$;
(2) Find the maximum value and the intervals of monotonic increase for the function $F(x) = f(x) \cdot f(-x) + f^2(x)$. | {
"answer": "\\frac{6}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:
(a) Michael K had an average test score of $90$ , Michael M had an average test score of $91$ , and Michael R had an average test score of $92$ .
(b) Michael K took more tests than Michael M, who in turn took more tests than Michael R.
(c) Michael M got a higher total test score than Michael R, who in turn got a higher total test score than Michael K. (The total test score is the sum of the test scores over all tests)
What is the least number of tests that Michael K, Michael M, and Michael R could have taken combined?
*Proposed by James Lin* | {
"answer": "413",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The coefficient of $x^{3}$ in the expanded form of $(1+x-x^{2})^{10}$ is given by the binomial coefficient $\binom{10}{3}(-1)^{7} + \binom{10}{4}(-1)^{6}$. Calculate the value of this binomial coefficient expression. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse C: $mx^2+3my^2=1$ ($m>0$) with a major axis length of $2\sqrt{6}$, and O as the origin.
(1) Find the equation of ellipse C and its eccentricity.
(2) Let point A be (3,0), point B be on the y-axis, and point P be on ellipse C, with point P on the right side of the y-axis. If $BA=BP$, find the minimum value of the area of quadrilateral OPAB. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ball is dropped from a height of 150 feet and rebounds to three-fourths of the distance it fell on each bounce. How many feet will the ball have traveled when it hits the ground the fifth time? | {
"answer": "765.234375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle ABC$ is right-angled at $A,$ with $AB=45$ and $AC=108.$ The point $D$ is on $BC$ so that $AD$ is perpendicular to $BC.$ Determine the length of $AD$ and the ratio of the areas of triangles $ABD$ and $ADC$. | {
"answer": "5:12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(a_1,a_2,a_3,\ldots,a_{15})$ be a permutation of $(1,2,3,\ldots,15)$ for which
$a_1>a_2>a_3>a_4>a_5>a_6>a_7 \mathrm{\ and \ } a_7<a_8<a_9<a_{10}<a_{11}<a_{12}<a_{13}<a_{14}<a_{15}.$
An example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14,15).$ Find the number of such permutations. | {
"answer": "3003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function $f: \N\rightarrow\N$ is circular if for every $p\in\N$ there exists $n\in\N,\ n\leq{p}$ such that $f^n(p)=p$ ( $f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\in\N$ $f^i(p)\neq{p}$ for every $i=1,2,\dots,\lfloor{kp}\rfloor$ . Determine the maximum repulsion degree can have a circular function.**Note:** Here $\lfloor{x}\rfloor$ is the integer part of $x$ . | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the remainder when $5^{2021}$ is divided by $17$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Erin the ant starts at a given corner of a hypercube (4-dimensional cube) and crawls along exactly 15 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point, determine the number of paths that Erin can follow to meet these conditions. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability of an event happening is $\frac{1}{2}$, find the relation between this probability and the outcome of two repeated experiments. | {
"answer": "50\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As the Chinese New Year approaches, workers from a factory begin to go home to reunite with their families starting from Monday, January 17, 2011. If the number of workers leaving the factory each day is the same, and by January 31, 121 workers remain in the factory, while the total number of worker-days during this 15-day period is 2011 (one worker working for one day counts as one worker-day, and the day a worker leaves and any days after are not counted), with weekends (Saturdays and Sundays) being rest days and no one being absent, then by January 31, the total number of workers who have gone home for the New Year is ____. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest real number $p$ such that the inequality $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$ holds for all natural numbers $n$ . | {
"answer": "2\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The vertical coordinate of the intersection point of the new graph obtained by shifting the graph of the quadratic function $y=x^{2}+2x+1$ $2$ units to the left and then $3$ units up is ______. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha \cos \alpha = \frac{1}{8}$, and $\alpha$ is an angle in the third quadrant. Find $\frac{1 - \cos^2 \alpha}{\cos(\frac{3\pi}{2} - \alpha) + \cos \alpha} + \frac{\sin(\alpha - \frac{7\pi}{2}) + \sin(2017\pi - \alpha)}{\tan^2 \alpha - 1}$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{2}}{2}$, and one of its vertices is at $(0, -1)$.
(Ⅰ) Find the equation of the ellipse $C$.
(Ⅱ) If there exist two distinct points $A$ and $B$ on the ellipse $C$ that are symmetric about the line $y = -\frac{1}{m}x + \frac{1}{2}$, find the maximum value of the area of $\triangle OAB$ ($O$ is the origin). | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain school randomly selected several students to investigate the daily physical exercise time of students in the school. They obtained data on the daily physical exercise time (unit: minutes) and organized and described the data. Some information is as follows:
- $a$. Distribution of daily physical exercise time:
| Daily Exercise Time $x$ (minutes) | Frequency (people) | Percentage |
|-----------------------------------|--------------------|------------|
| $60\leqslant x \lt 70$ | $14$ | $14\%$ |
| $70\leqslant x \lt 80$ | $40$ | $m$ |
| $80\leqslant x \lt 90$ | $35$ | $35\%$ |
| $x\geqslant 90$ | $n$ | $11\%$ |
- $b$. The daily physical exercise time in the group $80\leqslant x \lt 90$ is: $80$, $81$, $81$, $81$, $82$, $82$, $83$, $83$, $84$, $84$, $84$, $84$, $84$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $86$, $87$, $87$, $87$, $87$, $87$, $88$, $88$, $88$, $89$, $89$, $89$, $89$, $89$.
Based on the above information, answer the following questions:
$(1)$ In the table, $m=$______, $n=$______.
$(2)$ If the school has a total of $1000$ students, estimate the number of students in the school who exercise for at least $80$ minutes per day.
$(3)$ The school is planning to set a time standard $p$ (unit: minutes) to commend students who exercise for at least $p$ minutes per day. If $25\%$ of the students are to be commended, what value can $p$ be? | {
"answer": "86",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a $4 \times 4$ square grid partitioned into $16$ unit squares, each of which is painted white or black with a probability of $\frac{1}{2}$, determine the probability that the grid is entirely black after a $90^{\circ}$ clockwise rotation and any white square landing in a position previously occupied by a black square is repainted black. | {
"answer": "\\frac{1}{65536}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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