problem stringlengths 10 5.15k | answer dict |
|---|---|
If line $l_1: ax+2y+6=0$ is parallel to line $l_2: x+(a-1)y+(a^2-1)=0$, then the real number $a=$ . | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the first four terms of an arithmetic progression, as well as the sum of the first seven terms, are natural numbers. Furthermore, its first term \(a_1\) satisfies the inequality \(a_1 \leq \frac{2}{3}\). What is the greatest value that \(a_1\) can take? | {
"answer": "9/14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the isosceles trapezoid $ABCD$, $AD \parallel BC$, $\angle B = 45^\circ$. Point $P$ is on the side $BC$. The area of $\triangle PAD$ is $\frac{1}{2}$, and $\angle APD = 90^\circ$. Find the minimum value of $AD$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given circle $C$: $x^{2}+y^{2}+8x+ay-5=0$ passes through the focus of parabola $E$: $x^{2}=4y$. The length of the chord formed by the intersection of the directrix of parabola $E$ and circle $C$ is $\_\_\_\_\_\_$. | {
"answer": "4 \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters? | {
"answer": "89",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
$$
\lim_{x \rightarrow 1} (2-x)^{\sin \left(\frac{\pi x}{2}\right) / \ln (2-x)}
$$ | {
"answer": "e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many five-digit natural numbers are divisible by 9, where the last digit is greater than the second last digit by 2? | {
"answer": "800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), with a variance of $\frac{5}{3}$ (unit: rings$^{2}$). The shooting scores of the $2$ new members are $3$ rings and $5$ rings, respectively. What is the variance of the shooting scores of these $8$ members? | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p> \sqrt{n}$ and $p<20$ . What is the value of $N$ ? | {
"answer": "69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a $5 \times 18$ rectangle, the numbers from 1 to 90 are placed. This results in five rows and eighteen columns. In each column, the median value is chosen, and among the medians, the largest one is selected. What is the minimum possible value that this largest median can take?
Recall that among 99 numbers, the median is such a number that is greater than 49 others and less than 49 others. | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A truncated pyramid has a square base with a side length of 4 units, and every lateral edge is also 4 units. The side length of the top face is 2 units. What is the greatest possible distance between any two vertices of the truncated pyramid? | {
"answer": "\\sqrt{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $[x]$ represent the greatest integer less than or equal to the real number $x$. Define the sets
$$
\begin{array}{l}
A=\{y \mid y=[x]+[2x]+[4x], x \in \mathbf{R}\}, \\
B=\{1,2, \cdots, 2019\}.
\end{array}
$$
Find the number of elements in the intersection $A \cap B$. | {
"answer": "1154",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of a triangle are 5, 6, and 7. Find the area of the orthogonal projection of the triangle onto a plane that forms an angle equal to the smallest angle of the triangle with the plane of the triangle. | {
"answer": "\\frac{30 \\sqrt{6}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can you select two letters from the word "УЧЕБНИК" such that one of the letters is a consonant and the other is a vowel? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is ( )
A 36 B 48 C 52 D 54 | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set $\{[x] + [2x] + [3x] \mid x \in \mathbb{R}\} \mid \{x \mid 1 \leq x \leq 100, x \in \mathbb{Z}\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$. | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many perfect squares less than 20,000 can be represented as the difference of squares of two integers that differ by 2? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction. | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a ship travels in one direction and Emily walks parallel to the riverbank in the opposite direction, counting 210 steps from back to front and 42 steps from front to back, determine the length of the ship in terms of Emily's equal steps. | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let function $G(n)$ denote the number of solutions to the equation $\cos x = \sin nx$ on the interval $[0, 2\pi]$. For each integer $n$ greater than 2, what is the sum $\sum_{n=3}^{100} G(n)$? | {
"answer": "10094",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $k \in [-2, 2]$, find the probability that for the value of $k$, there can be two tangents drawn from the point A(1, 1) to the circle $x^2 + y^2 + kx - 2y - \frac{5}{4}k = 0$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ , with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$ . Determine the length of the line segment $AP$ such that for any position of the point $M\in BC$ , the triangle $MNP$ is right angled at $N$ . | {
"answer": "27/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin (α- \frac{π}{3})= \frac{3}{5} $, where $a∈(\frac{π}{4}, \frac{π}{2})$, find $\tan α=\_\_\_\_\_\_\_\_\_$. | {
"answer": "- \\frac{48+25 \\sqrt{3}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
$$
\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}}
$$ | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are nine parts in a bag, including five different genuine parts and four different defective ones. These parts are being drawn and inspected one by one. If the last defective part is found exactly on the fifth draw, calculate the total number of different sequences of draws. | {
"answer": "480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya and Vasya competed in a 60-meter race. When Petya finished, Vasya was 9 meters behind him. In the second race, Petya stood exactly 9 meters behind Vasya at the starting point. Who finished first in the second race and by how many meters did he lead his opponent? (Assume that each boy ran both times at his own constant speed). | {
"answer": "1.35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price of Margit néni's favorite chocolate was increased by 30%, and at the same time her pension increased by 15%. By what percentage does Margit néni's chocolate consumption decrease if she can spend only 15% more on chocolate? | {
"answer": "11.54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( D \) and \( E \) are located on the diagonals \( A B_{1} \) and \( C A_{1} \) of the lateral faces of the prism \( A B C A_{1} B_{1} C_{1} \) such that the lines \( D E \) and \( B C_{1} \) are parallel. Find the ratio of the segments \( D E \) and \( B C_{1} \). | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of \(\sin \left(-\frac{5 \pi}{3}\right) + \cos \left(-\frac{5 \pi}{4}\right) + \tan \left(-\frac{11 \pi}{6}\right) + \cot \left(-\frac{4 \pi}{3}\right)\). | {
"answer": "\\frac{\\sqrt{3} - \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive value of \( x \) that satisfies the equation \( \sqrt{3x} = 5x - 1 \). | {
"answer": "\\frac{13 - \\sqrt{69}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with a radius of 2 is inscribed in triangle \(ABC\) and touches side \(BC\) at point \(D\). Another circle with a radius of 4 touches the extensions of sides \(AB\) and \(AC\), as well as side \(BC\) at point \(E\). Find the area of triangle \(ABC\) if the measure of angle \(\angle ACB\) is \(120^{\circ}\). | {
"answer": "\\frac{56}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sequence \\(\{a_n\}\) have a sum of the first \\(n\\) terms denoted by \\(S_n\\), and it is known that \\(S_n = 2a_n - 2^{n+1} (n \in \mathbb{N}^*)\).
\\((1)\\) Find the general formula for the sequence \\(\{a_n\}\).
\\((2)\\) Let \\(b_n = \log_{\frac{a_n}{n+1}} 2\), and the sum of the first \\(n\\) terms of the sequence \\(\{b_n\}\) be \\(B_n\). If there exists an integer \\(m\\) such that for any \\(n \in \mathbb{N}^*\) and \\(n \geqslant 2\), \\(B_{3n} - B_n > \frac{m}{20}\) holds, find the maximum value of \\(m\\). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( A B C \), the base of the height \( C D \) lies on side \( A B \), and the median \( A E \) is equal to 5. The height \( C D \) is equal to 6.
Find the area of triangle \( A B C \), given that the area of triangle \( A D C \) is three times the area of triangle \( B C D \). | {
"answer": "96/7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \) and \( B \) be two moving points on the ellipse \( x^{2}+3 y^{2}=1 \), and \( OA \) is perpendicular to \( OB \) (where \( O \) is the origin). Then, the product of the maximum and minimum values of \( |AB| \) is ______. | {
"answer": "\\frac{2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 26 bricks, and two brothers are vying to take them. The younger brother arrives first and arranges the bricks. The elder brother arrives and thinks the younger brother has taken too many, so he takes half of the bricks from the younger brother. The younger brother, feeling confident, takes half of the bricks from the elder brother. The elder brother doesn't allow this, so the younger brother has to give 5 bricks to the elder brother. In the end, the elder brother ends up with 2 more bricks than the younger brother. How many bricks did the younger brother originally plan to take? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of children required in a school to be sure that at least 3 of them have their birthday on the same day? (Keep in mind that some people are born on February 29.) | {
"answer": "733",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(\log_{5} x + \log_{25} x = \log_{1/5} \sqrt{3}\). | {
"answer": "\\frac{1}{\\sqrt[3]{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \(A B C D\), the base \(A D\) is four times larger than \(B C\). A line passing through the midpoint of diagonal \(B D\) and parallel to \(A B\) intersects side \(C D\) at point \(K\). Find the ratio \(D K : K C\). | {
"answer": "2:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a\in \mathbb{R}$, $b\in \mathbb{R}$, if the set $\{a,\frac{b}{a},1\}=\{a^{2},a+b,0\}$, determine the value of $a^{2023}+b^{2024}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, the parametric equations of the curve C₁ are $$\begin{cases} x=2+2\cos\alpha \\ y=2\sin\alpha \end{cases}$$ (where α is the parameter). Establish a polar coordinate system with the origin of the coordinate system as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of the curve C₂ is ρ=2sinθ.
(I) Write the polar equation of the curve C₁ and the rectangular equation of the curve C₂.
(II) Suppose point P is on C₁, point Q is on C₂, and ∠POQ=$$\frac {π}{3}$$, find the maximum area of △POQ. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y=\log_a(x+3)-1$ ($a > 0$ and $a \neq 1$) always passes through a fixed point $A$. If the point $A$ lies on the line $mx+ny+1=0$, where $mn > 0$, then the minimum value of $\frac{1}{m}+\frac{1}{n}$ is | {
"answer": "3+2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number $m$ for which there exists a natural number $n$ such that the sets of the last 2014 digits in the decimal representation of the numbers $a=2015^{3 m+1}$ and $b=2015^{6 n+2}$ are identical, with the condition that $a<b$. | {
"answer": "671",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sample with a sample size of $7$, an average of $5$, and a variance of $2$. If a new data point of $5$ is added to the sample, what will be the variance of the sample? | {
"answer": "\\frac{7}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive number such as 3.27, 3 is called the integer part of the number, and 0.27 is called the decimal part of the number. Find a positive number whose decimal part, integer part, and the number itself form three consecutive terms of a geometric sequence.
(The 7th Canadian Mathematical Olympiad, 1975) | {
"answer": "\\frac{1 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively. It is given that $c \cos B = (2a - b) \cos C$.
1. Find the magnitude of angle $C$.
2. If $AB = 4$, find the maximum value of the area $S$ of $\triangle ABC$. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers between $1500$ and $2500$ have the property that their units digit is the sum of the other digits?
**A)** $76$
**B)** $81$
**C)** $85$
**D)** $91$
**E)** $96$ | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$ and satisfies $f(x+2)=- \frac{1}{f(x)}$, and $f(x)=x$ when $1 \leq x \leq 2$, find $f(- \frac{11}{2})$. | {
"answer": "- \\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On one side of the acute angle \(A\), points \(P\) and \(Q\) are marked such that \(AP = 4\), \(AQ = 12\). On the other side, points \(M\) and \(N\) are marked at distances of 6 and 10 from the vertex. Find the ratio of the areas of triangles \(MNO\) and \(PQO\), where \(O\) is the intersection point of the lines \(MQ\) and \(NP\). | {
"answer": "1:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. The length of $Q R$ is | {
"answer": "$\\sqrt{6}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the exact value of the expression $\left|3 - |9 - \pi^2| \right|$. Write your answer using only integers, $\pi$, and necessary mathematical operations, without any absolute value signs. | {
"answer": "12 - \\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\theta = \arctan \frac{5}{12}\), find the principal value of the argument of the complex number \(z = \frac{\cos 2\theta + i \sin 2\theta}{239 + i}\). | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\left(\frac{\pi}{3} - \alpha\right) = \frac{1}{3}$, calculate $\sin\left(\frac{\pi}{6} - 2\alpha\right)$. | {
"answer": "- \\frac{7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\frac{\cos \alpha + \sin \alpha}{\cos \alpha - \sin \alpha} = 2$, find the value of $\frac{1 + \sin 4\alpha - \cos 4\alpha}{1 + \sin 4\alpha + \cos 4\alpha}$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangular prism $S-ABC$, where the base is an isosceles right triangle with $AB$ as the hypotenuse, $SA = SB = SC = 2$, and $AB = 2$, and points $S$, $A$, $B$, and $C$ all lie on a sphere centered at point $O$, find the distance from point $O$ to the plane $ABC$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an integer coefficient polynomial
$$
f(x)=x^{5}+a_{1} x^{4}+a_{2} x^{3}+a_{3} x^{2}+a_{4} x+a_{5}
$$
If $f(\sqrt{3}+\sqrt{2})=0$ and $f(1)+f(3)=0$, then $f(-1)=$ $\qquad$ | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory's annual fixed cost for producing a certain product is 2.5 million yuan. For every $x$ thousand units produced, an additional cost $C(x)$ is incurred. When the annual output is less than 80 thousand units, $C(x)=\frac{1}{3}x^2+10x$ (in million yuan). When the annual output is not less than 80 thousand units, $C(x)=51x+\frac{10000}{x}-1450$ (in million yuan). The selling price per thousand units of the product is 50 million yuan. Market analysis shows that the factory can sell all the products it produces.
(Ⅰ) Write the function expression for the annual profit $L(x)$ (in million yuan) in terms of the annual output, $x$ (in thousand units);
(Ⅱ) What is the annual output in thousand units for which the factory achieves the maximum profit from this product, and what is the maximum value? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In isosceles $\triangle ABC$, $\angle BAC=120^{\circ}$, $AB=AC=2$, $\overrightarrow{BC}=2 \overrightarrow{BD}$, $\overrightarrow{AC}=3 \overrightarrow{AE}$. Calculate the value of $\overrightarrow{AD}\cdot \overrightarrow{BE}$. | {
"answer": "-\\dfrac {2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular tetrahedron has an edge length of $2$. Calculate the surface area of the sphere circumscribed around this tetrahedron. | {
"answer": "6\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the complex number \( z = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13} \). Find the value of \( \left(z^{-12} + z^{-11} + z^{-10}\right)\left(z^{3} + 1\right)\left(z^{6} + 1\right) \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From 51 consecutive odd numbers $1, 3, 5, \cdots, 101$, select $\mathrm{k}$ numbers such that their sum is 1949. What is the maximum value of $\mathrm{k}$? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the island of Misfortune with a population of 96 people, the government decided to implement five reforms. Each reform is disliked by exactly half of the citizens. A citizen protests if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the government can expect at the protest? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For $i = 1, 2, \cdots, n$, if $\left|x_{i}\right| < 1$ and $\left|x_{1}\right| + \left|x_{2}\right| + \cdots + \left|x_{n}\right| = 2005 + \left|x_{1} + x_{2} + \cdots + x_{n} \right|$, find the minimum value of the positive integer $n$. | {
"answer": "2006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a podcast series that lasts for 837 minutes needs to be stored on CDs and each CD can hold up to 75 minutes of audio, determine the number of minutes of audio that each CD will contain. | {
"answer": "69.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the region \(D\) bounded by the curves
\[ x^{2} + y^{2} = 12, \quad x \sqrt{6} = y^{2} \quad (x \geq 0) \] | {
"answer": "3\\pi + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum (when written as an irreducible fraction)? | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The tangents of the angles of a triangle are in the ratio $1: 2: 3$. How do the sides of the triangle compare to each other in length? | {
"answer": "\\sqrt{5} : 2\\sqrt{2} : 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following four propositions:
(1) If line $a$ is parallel to line $b$, and line $a$ is parallel to plane $\alpha$, then line $b$ is parallel to plane $\alpha$.
(2) If line $a$ is parallel to plane $\alpha$, and line $b$ is contained in plane $\alpha$, then plane $\alpha$ is parallel to line $b$.
(3) If line $a$ is parallel to plane $\alpha$, then line $a$ is parallel to all lines within plane $\alpha$.
(4) If line $a$ is parallel to plane $\alpha$, and line $a$ is parallel to line $b$, and line $b$ is not contained in plane $\alpha$, then line $b$ is parallel to plane $\alpha$.
The correct proposition(s) is/are __________. | {
"answer": "(4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(1,\sin x)$, $\overrightarrow{b}=(\sin x,-1)$, $\overrightarrow{c}=(1,\cos x)$, where $x\in(0,\pi)$.
(Ⅰ) If $(\overrightarrow{a}+ \overrightarrow{b})\nparallel \overrightarrow{c}$, find $x$;
(Ⅱ) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $B$ is the $x$ found in (Ⅰ), $2\sin^2B+2\sin^2C-2\sin^2A=\sin B\sin C$, find the value of $\sin \left(C- \frac{\pi}{3}\right)$. | {
"answer": "\\frac{1-3\\sqrt{5}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Please write an irrational number that is greater than -3 and less than -2. | {
"answer": "-\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least five-digit positive integer which is congruent to 6 (mod 17)? | {
"answer": "10,017",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the circle $\odot C: x^{2}+y^{2}-2 x-2 y=-1$, points $A(2 a, 0)$ and B(0,2 b), where $a > 1$ and $b > 1$, determine the minimum area of triangle $\triangle A O B$ (where $O$ is the origin) when the circle $\odot C$ is tangent to the line $AB$. | {
"answer": "3 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A meal at a diner includes a burger weighing 150 grams, of which 40 grams are filler. What percent of the burger is not filler? | {
"answer": "73.33\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1 \ (b > 0)$, respectively. A line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $5$, determine the eccentricity of the ellipse. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{m} = (\sqrt{3}\sin x - \cos x, 1)$ and $\overrightarrow{n} = (\cos x, \frac{1}{2})$, let the function $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$.
(1) Find the interval where the function $f(x)$ is monotonically increasing.
(2) If $a$, $b$, $c$ are the lengths of the sides opposite to angles A, B, C in triangle $\triangle ABC$, with $a=2\sqrt{3}$, $c=4$, and $f(A) = 1$, find the area of triangle $\triangle ABC$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Hooligan Vasya loves running on the metro escalator, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving down, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down on an escalator that is moving up? (The escalator always moves at a constant speed.) | {
"answer": "324",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ellipse \(\frac{x^{2}}{2002^{2}}+\frac{y^{2}}{1949^{2}}=1\) intersects with two lines passing through the origin at points \(A, B, C,\) and \(D\). The slope of line \(AB\) is \(k\), and the slope of line \(CD\) is \(-\frac{1949^{2}}{2002^{2} \cdot k}\). Given that there exists a triangle \(\triangle PQR\) such that \(PQ=OA\), \(PR=OC\), and \(\angle QPR=\left|\angle AOC-\frac{\pi}{2}\right|\), find \(QR-\qquad\). | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. How many girls could have answered affirmatively? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Below is pictured a regular seven-pointed star. Find the measure of angle \(a\) in radians. | {
"answer": "\\frac{5\\pi}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the standard products of a certain factory, on average 15% are of the second grade. What is the probability that the percentage of second-grade products among 1000 standard products of this factory differs from 15% by less than 2% in absolute value? | {
"answer": "0.9232",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( g(x) \) satisfies
\[ g(x) - 2 g \left( \frac{1}{x} \right) = 3^x + x \]
for all \( x \neq 0 \). Find \( g(2) \). | {
"answer": "-4 - \\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{4^{5 x}-9^{-2 x}}{\sin x-\operatorname{tg}(x^{3})}
$$ | {
"answer": "\\ln (1024 \\cdot 81)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sqrt{3}\cos (\frac{\pi }{2}+x)\bullet \cos x+\sin^{2}x$, where $x\in R$.
(I) Find the interval where $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $B=\frac{\pi }{4}$, $a=2$ and angle $A$ satisfies $f(A)=0$, find the area of $\triangle ABC$. | {
"answer": "\\frac{3+\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin ^{4} 3 x \cos ^{4} 3 x \, d x
$$ | {
"answer": "\\frac{3\\pi}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A total of 120 is divided into four parts proportional to 3, 2, 4, and 5. What is the second largest part? | {
"answer": "\\frac{240}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evgeny is laying tiles on the floor of his living room, which measures 12 by 16 meters. He plans to place square tiles of size 1 m × 1 m along the border of the room and to lay the rest of the floor with square tiles of size 2 m × 2 m. How many tiles will he need in total? | {
"answer": "87",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The segment \( PQ \) is divided into several smaller segments. On each of them, a square is constructed (see illustration).
What is the length of the path along the arrows if the length of the segment \( PQ \) is 73? If necessary, round your answer to 0.01 or write the answer as a common fraction. | {
"answer": "219",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A biased coin has a probability of $\frac{3}{4}$ of landing heads and $\frac{1}{4}$ of landing tails on each toss. The outcomes of the tosses are independent. The probability of winning Game C, where the player tosses the coin four times and wins if either all four outcomes are heads or all four are tails, can be compared to the probability of winning Game D, where the player tosses the coin five times and wins if the first two tosses are the same, the third toss is different from the first two, and the last two tosses are the same as the first two. Determine the difference in the probabilities of winning Game C and Game D. | {
"answer": "\\frac{61}{256}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polynomial $f(x) = x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64$, calculate the value of $v_4$ when $x = 2$ using Horner's method. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\sin x + \sin y = 0.6\) and \(\cos x + \cos y = 0.8\), find \(\cos x \cdot \cos y\). | {
"answer": "-\\frac{11}{100}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( A, B, C, \) and \( D \) lie on a straight line in that order. For a point \( E \) outside the line,
\[ \angle AEB = \angle BEC = \angle CED = 45^\circ. \]
Let \( F \) be the midpoint of segment \( AC \), and \( G \) be the midpoint of segment \( BD \). What is the measure of angle \( FEG \)? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$25$ checkers are placed on $25$ leftmost squares of $1 \times N$ board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal $N$ such that all the checkers could be placed in the row of $25$ successive squares but in the reverse order. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the value $-2\frac{1}{2}$, calculate its absolute value. | {
"answer": "2\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In front of you is a clock face. Divide it into three parts using two straight lines so that the sum of the numbers in each part is equal. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangular pyramid $S-ABC$ with the base being an isosceles right triangle with $AB$ as the hypotenuse, and $SA = SB = SC = 2$, $AB = 2$, let points $S$, $A$, $B$, and $C$ all lie on the surface of a sphere centered at $O$. What is the distance from point $O$ to the plane $ABC$?
| {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three musicians, Janek, Mikeš, and Vávra usually divide their shared fee in the ratio $4: 5: 6$, with Janek receiving the least and Vávra the most. This time, Vávra did not perform well, so he gave up his portion. Janek suggested that Vávra's share should be divided equally between him and Mikeš. However, Mikeš insisted that they should still divide this share unevenly as usual, in the ratio $4: 5$ because, under Janek's proposal, he would receive 40 CZK less than under his own proposal.
Determine the total amount of the shared fee.
(L. Šimünek) | {
"answer": "1800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^\circ$. Determine the largest possible value of $A$ such that, in any such division, each of the seven arcs contains at least $A^\circ$. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$ . Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$ . (See diagram). Given $EC=1$ , find the radius of the circle.
[asy]
size(6cm);
pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle);
draw(unitcircle);
draw(O--C);
draw(B--D);
dot(O);
dot(B);
dot(C);
dot(D);
dot(E);
label(" $B$ ", B, B);
label(" $C$ ", C, C);
label(" $D$ ", D, D);
label(" $E$ ", E, dir(280));
label(" $O$ ", O, dir(270));
[/asy] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?
A) $\frac{273}{288}$
B) $\frac{275}{288}$
C) $\frac{277}{288}$
D) $\frac{279}{288}$ | {
"answer": "\\frac{277}{288}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The midpoints of the sides of a regular octagon \( ABCDEFGH \) are joined to form a smaller octagon. What fraction of the area of \( ABCDEFGH \) is enclosed by the smaller octagon? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a magical tree with 123 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number of fruits that should fall on a given day, the falling process restarts from 1 fruit on that day and a new cycle begins. Following this pattern, on which day will all the fruits have fallen from the tree? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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