problem stringlengths 10 5.15k | answer dict |
|---|---|
Two circles centered at \( O_{1} \) and \( O_{2} \) have radii 2 and 3 and are externally tangent at \( P \). The common external tangent of the two circles intersects the line \( O_{1} O_{2} \) at \( Q \). What is the length of \( PQ \)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten children were given 100 pieces of macaroni each on their plates. Some children didn't want to eat and started playing. With one move, one child transfers one piece of macaroni from their plate to each of the other children's plates. What is the minimum number of moves needed such that all the children end up with a different number of pieces of macaroni on their plates? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divide the sequence successively into groups with the first parenthesis containing one number, the second parenthesis two numbers, the third parenthesis three numbers, the fourth parenthesis four numbers, the fifth parenthesis one number, and so on in a cycle: $(3)$, $(5,7)$, $(9,11,13)$, $(15,17,19,21)$, $(23)$, $(25,27)$, $(29,31,33)$, $(35,37,39,41)$, $(43)$, $…$, then calculate the sum of the numbers in the 104th parenthesis. | {
"answer": "2072",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A three-digit number \( \mathrm{abc} \) divided by the sum of its digits leaves a remainder of 1. The three-digit number \( \mathrm{cba} \) divided by the sum of its digits also leaves a remainder of 1. If different letters represent different digits and \( a > c \), then \( \overline{\mathrm{abc}} = \) ____. | {
"answer": "452",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are four identical balls numbered $1$, $2$, $3$, $4$, and four boxes also numbered $1$, $2$, $3$, $4$.
$(1)$ If each box contains one ball, find the number of ways such that exactly one box has the same number as the ball inside it.
$(2)$ Find the number of ways such that exactly one box is empty. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $θ$ is an angle in the third quadrant and $\sin (θ- \frac {π}{4})= \frac {3}{5}$, find $\tan (θ+ \frac {π}{4})=$____. | {
"answer": "\\frac {4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2c=a+\cos A\frac{b}{\cos B}$.
$(1)$ Find the measure of angle $B$.
$(2)$ If $b=4$ and $a+c=3\sqrt{2}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x, y \in (-1, 1)$, find the minimum value of the expression $$\sqrt {(x+1)^{2}+(y-1)^{2}}+\sqrt {(x+1)^{2}+(y+1)^{2}}+\sqrt {(x-1)^{2}+(y+1)^{2}}+\sqrt {(x-1)^{2}+(y-1)^{2}}.$$ | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $P$ is $\sqrt3$ units away from plane $A$ . Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$ ? | {
"answer": "8\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the letters $A$, $B$, $C$, $D$, and $E$, calculate the total number of different arrangements of these letters in a row with the constraints that $A$ and $E$ are not placed at the two ends. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \cdots, 2005\} \). Find the minimum value of \( n \) such that any set of \( n \) pairwise coprime elements from \( S \) contains at least one prime number. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, $\sin A = \frac{4}{5}$, $\cos B = \frac{5}{13}$, and $c=56$. Find $\sin C$ and the circumference of the circumscribed circle of triangle ABC. | {
"answer": "65\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the power of _n_ of a natural number _m_ greater than or equal to 2, the following decomposition formula is given:
2<sup>2</sup> = 1 + 3, 3<sup>2</sup> = 1 + 3 + 5, 4<sup>2</sup> = 1 + 3 + 5 + 7…
2<sup>3</sup> = 3 + 5, 3<sup>3</sup> = 7 + 9 + 11…
2<sup>4</sup> = 7 + 9…
According to this pattern, the third number in the decomposition of 5<sup>4</sup> is ______. | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When $x=$____, the expressions $\frac{x-1}{2}$ and $\frac{x-2}{3}$ are opposite in sign. | {
"answer": "\\frac{7}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of line $l$:
$$
\begin{cases}
x=t+1 \\
y= \sqrt {3}t
\end{cases}
$$
(where $t$ is the parameter), and the polar equation of curve $C$ is $\rho=2\cos\theta$, then the polar radius (taking the positive value) of the intersection point of line $l$ and curve $C$ is ______. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \(AB = 1\), and the slanted segments form an angle of \(45^\circ\) with \(AB\). There are \(n\) vertices above \(AB\).
What is the length of the broken line? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the share of the Chinese yuan in the currency structure of the National Wealth Fund (NWF) as of 01.07.2021 by one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Chinese yuan as of 01.07.2021:
\[ CNY_{22} = 1213.76 - 3.36 - 38.4 - 4.25 - 600.3 - 340.56 - 0.29 = 226.6 \text{ (billion rubles)} \]
b) Determine the share of the Chinese yuan in the currency structure of the NWF funds as of 01.07.2021:
\[ \alpha_{07}^{CNY} = \frac{226.6}{1213.76} \approx 18.67\% \]
c) Calculate by how many percentage points and in which direction the share of the Chinese yuan in the currency structure of the NWF funds changed during the period considered in the table:
\[ \Delta \alpha^{CNY} = \alpha_{07}^{CNY} - \alpha_{06}^{CNY} = 18.67 - 17.49 = 1.18 \approx 1.2 \text{ (percentage points)} \]
Second method:
a) Determine the share of the euro in the currency structure of the NWF funds as of 01.07.2021:
\[ \alpha_{07}^{EUR} = \frac{38.4}{1213.76} \approx 3.16\% \]
b) Determine the share of the Chinese yuan in the currency structure of the NWF funds as of 01.07.2021:
\[ \alpha_{07}^{CNY} = 100 - 0.28 - 3.16 - 0.35 - 49.46 - 28.06 - 0.02 = 18.67\% \]
c) Calculate by how many percentage points and in which direction the share of the Chinese yuan in the currency structure of the NWF funds changed during the period considered in the table:
\[ \Delta \alpha^{CNY} = \alpha_{07}^{CNY} - \alpha_{06}^{CNY} = 18.67 - 17.49 = 1.18 \approx 1.2 \text{ (percentage points)} \] | {
"answer": "1.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the central cell of a $21 \times 21$ board, there is a piece. In one move, the piece can be moved to an adjacent cell sharing a side. Alina made 10 moves. How many different cells can the piece end up in? | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select 5 different numbers from $0,1,2,3,4,5,6,7,8,9$ to form a five-digit number such that this five-digit number is divisible by $3$, $5$, $7$, and $13$. What is the largest such five-digit number? | {
"answer": "94185",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From points A and B, a motorcyclist and a cyclist respectively set off towards each other simultaneously and met at a distance of 4 km from B. At the moment the motorcyclist arrived in B, the cyclist was at a distance of 15 km from A. Find the distance AB. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Billy Bones has two coins — one gold and one silver. One of these coins is fair, while the other is biased. It is unknown which coin is biased but it is known that the biased coin lands heads with a probability of $p=0.6$.
Billy Bones tossed the gold coin and it landed heads immediately. Then, he started tossing the silver coin, and it landed heads only on the second toss. Find the probability that the gold coin is the biased one. | {
"answer": "5/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( M \) and \( N \) are taken on the diagonals \( AB_1 \) and \( BC_1 \) of the faces of the parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), and the segments \( MN \) and \( A_1 C \) are parallel. Find the ratio of these segments. | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose an amoeba is placed in a container one day, and on that day it splits into three amoebas. Each subsequent day, every surviving amoeba splits into three new amoebas. However, at the end of every second day starting from day two, only half of the amoebas survive. How many amoebas are in the container at the end of one week (after seven days)? | {
"answer": "243",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(x_1, x_2, \ldots, x_n\) be real numbers in arithmetic sequence, which satisfy \(|x_i| < 1\) for \(i = 1, 2, \dots, n,\) and \[|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|.\] What is the smallest possible value of \(n\)? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a set of circles in the upper half-plane, all tangent to the $x$-axis. Begin with two circles of radii $50^2$ and $53^2$, externally tangent to each other, defined as Layer $L_0$. For each pair of consecutive circles in $\bigcup_{j=0}^{k-1}L_j$, a new circle in Layer $L_k$ is constructed externally tangent to each circle in the pair. Construct this way up to Layer $L_3$. Let $S=\bigcup_{j=0}^{3}L_j$. For each circle $C$ denote by $r(C)$ its radius. Find
\[
\sum_{C \in S} \frac{1}{{r(C)}}.
\]
A) $\frac{1}{1450}$
B) $\frac{1}{1517}$
C) $\frac{1}{1600}$
D) $\frac{1}{1700}$ | {
"answer": "\\frac{1}{1517}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in the decimal system 2004 equates to $4 \times 10^0 + 0 \times 10^1 + 0 \times 10^2 + 2 \times 10^3$, by analogy, calculate the equivalent value of $2004$ in base 5 in decimal form. | {
"answer": "254",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of $$\frac {1}{\tan 20^\circ} - \frac {1}{\cos 10^\circ}$$ is equal to \_\_\_\_\_\_. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grandma's garden has three types of apples: Antonovka, Grushovka, and White Naliv. If the amount of Antonovka apples were tripled, the total number of apples would increase by 70%. If the amount of Grushovka apples were tripled, the total number of apples would increase by 50%. By what percentage would the total number of apples change if the amount of White Naliv apples were tripled? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bag containing 7 apples and 1 orange, the probability of randomly picking an apple is ______, and the probability of picking an orange is ______. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the sequence $\{a_n\}$, $a_n+a_{n+1}+a_{n+2}=(\sqrt{2})^{n}$. Find the sum of the first $9$ terms of the sequence $\{a_n\}$ (express the answer as a numerical value). | {
"answer": "4+9\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bowling ball must have a diameter of 9 inches. Calculate both the surface area and the volume of the bowling ball before the finger holes are drilled. Express your answers as common fractions in terms of \(\pi\). | {
"answer": "\\frac{729\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\clubsuit$, the second card is a $\heartsuit$, and the third card is a king? | {
"answer": "\\frac{13}{2550}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three boxes \(A\), \(B\), and \(C\) containing 100, 50, and 80 balls respectively. Each box has a certain number of black balls. It is known that box \(A\) contains 15 black balls. If a box is chosen randomly and then a ball is randomly drawn from that box, the probability of drawing a black ball is \(\frac{101}{600}\). Determine the maximum number of black balls that can be in box \(C\). | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
31 cars simultaneously started from the same point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on up to the 31st car at 91 km/h. The track is narrow, and if one car overtakes another, they collide and both crash out of the race. Eventually, one car remains. What is its speed? | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest integer divisible by all positive integers less than its cube root? | {
"answer": "420",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many pairs of consecutive integers in $\{3000,3001,3002,\ldots,4000\}$ is no borrowing required when the first integer is subtracted from the second? | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the radius of the smallest circle into which any system of points with a diameter of 1 can be enclosed? | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets. | {
"answer": "409510",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a\_n\}$, for any $k \in \mathbb{N}^*$, when $n = 3k$, $a\_n = a\_{\frac{n}{3}}$; when $n \neq 3k$, $a\_n = n$. The 10th occurrence of 2 in this sequence is the \_\_\_\_\_\_th term. | {
"answer": "2 \\cdot 3^{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Person A and person B start walking towards each other from locations A and B simultaneously. The speed of person B is $\frac{3}{2}$ times the speed of person A. After meeting for the first time, they continue to their respective destinations, and then immediately return. Given that the second meeting point is 20 kilometers away from the first meeting point, what is the distance between locations A and B? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles trapezoid with a perimeter of 8 and an area of 2, a circle can be inscribed. Find the distance from the point of intersection of the diagonals of the trapezoid to its shorter base. | {
"answer": "\\frac{2 - \\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of rectangle \(ABCD\) is 48, and the diagonal is 10. On the plane where the rectangle is located, a point \(O\) is chosen such that \(OB = OD = 13\). Find the distance from point \(O\) to the vertex of the rectangle farthest from it. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A trapezoid \(ABCD\) is inscribed in a circle, with bases \(AB = 1\) and \(DC = 2\). Let \(F\) denote the intersection point of the diagonals of this trapezoid. Find the ratio of the sum of the areas of triangles \(ABF\) and \(CDF\) to the sum of the areas of triangles \(AFD\) and \(BCF\). | {
"answer": "5/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(PQR\) with side \(PQ = 3\), a median \(PM = \sqrt{14}\) and an altitude \(PH = \sqrt{5}\) are drawn from vertex \(P\) to side \(QR\).
Find side \(PR\), given that \(\angle QPR + \angle PRQ < 90^\circ\). | {
"answer": "\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You have 5 identical buckets, each with a maximum capacity of some integer number of liters, and a 30-liter barrel containing an integer number of liters of water. All the water from the barrel was poured into the buckets, with the first bucket being half full, the second one-third full, the third one-quarter full, the fourth one-fifth full, and the fifth one-sixth full. How many liters of water were in the barrel? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Teacher Li and three students (Xiao Ma, Xiao Lu, and Xiao Zhou) depart from school one after another and walk the same route to a cinema. The walking speed of the three students is equal, and Teacher Li's walking speed is 1.5 times that of the students. Currently, Teacher Li is 235 meters from the school, Xiao Ma is 87 meters from the school, Xiao Lu is 59 meters from the school, and Xiao Zhou is 26 meters from the school. After they walk a certain number of meters, the distance of Teacher Li from the school will be equal to the sum of the distances of the three students from the school. Find the number of meters they need to walk for this condition to be true. | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Most countries in the world use the Celsius temperature scale to forecast the weather, but countries like the United States and the United Kingdom still use the Fahrenheit temperature scale. A student looked up data and found the following information in the table:
| Celsius Temperature $x/^{\circ}\mathrm{C}$ | $0$ | $10$ | $20$ | $30$ | $40$ | $50$ |
|-------------------------------------------|-----|------|------|------|------|------|
| Fahrenheit Temperature $y/^{\circ}F$ | $32$| $50$ | $68$ | $86$ | $104$| $122$|
$(1)$ Analyze whether the correspondence between the two temperature scales is a linear function. Fill in the blank with "yes" or "no".
$(2)$ Based on the data, calculate the Celsius temperature at $0^{\circ}F$. | {
"answer": "-\\frac{160}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triline is a line with the property that three times its slope is equal to the sum of its \(x\)-intercept and its \(y\)-intercept. For how many integers \(q\) with \(1 \leq q \leq 10000\) is there at least one positive integer \(p\) so that there is exactly one triline through \((p, q)\)? | {
"answer": "57",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $E$ with eccentricity $\frac{\sqrt{2}}{2}$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ that passes through point $A(1, \frac{\sqrt{2}}{2})$.
$(1)$ Find the equation of ellipse $E$;
$(2)$ If a line $l$: $y= \frac{\sqrt{2}}{2}x+m$ that does not pass through point $A$ intersects ellipse $E$ at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers \( x_{1}, x_{2}, \cdots, x_{1991} \) that satisfy the condition
\[
\sum_{i=1}^{1990} \left| x_{i} - x_{i+1} \right| = 1991,
\]
and \( y_{k} = \frac{1}{k} \sum_{i=1}^{k} x_{i} \) for \( k = 1, 2, \cdots, 1991 \), determine the maximum value of \( \sum_{i=1}^{1990} \left| y_{i} - y_{i+1} \right| \). | {
"answer": "1990",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square $WXYZ$ with side length 8 units is divided into four smaller squares by drawing lines from the midpoints of one side to the midpoints of the opposite sides. The top right square of each iteration is shaded. If this dividing and shading process is done 100 times, what is the total area of the shaded squares?
A) 18
B) 21
C) $\frac{64}{3}$
D) 25
E) 28 | {
"answer": "\\frac{64}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $$\sqrt {0.1587}$$≈0.3984, $$\sqrt {1.587}$$≈1.260, $$\sqrt[3]{0.1587}$$≈0.5414, $$\sqrt[3]{1.587}$$≈1.166, can you find without using a calculator:
(1) $$\sqrt {15.87}$$≈\_\_\_\_\_\_;
(2) - $$\sqrt {0.001587}$$≈\_\_\_\_\_\_;
(3) $$\sqrt[3]{1.587\times 10^{-4}}$$≈\_\_\_\_\_\_. | {
"answer": "0.05414",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $2\cos(2\alpha)=\sin\left(\alpha-\frac{\pi}{4}\right)$ and $\alpha\in\left(\frac{\pi}{2},\pi\right)$, calculate the value of $\cos 2\alpha$. | {
"answer": "\\frac{\\sqrt{15}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $F_{1}$, $F_{2}$ are the foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), and point $M(x_{0},y_{0})$ ($x_{0} > 0$, $y_{0} > 0$) lies on the asymptote of the hyperbola such that $MF_{1} \perp MF_{2}$. If there is a parabola with focus $F_{2}$ described by $y^2 = 2px$ ($p > 0$) passing through point $M$, determine the eccentricity of this hyperbola. | {
"answer": "2 + \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the positive numbers $x$ and $y$ satisfy the equation $$3x+y+ \frac {1}{x}+ \frac {2}{y}= \frac {13}{2}$$, find the minimum value of $$x- \frac {1}{y}$$. | {
"answer": "- \\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the dihedral angle $\alpha-l-\beta$ is $60^{\circ}$, points $P$ and $Q$ are on planes $\alpha$ and $\beta$ respectively. The distance from $P$ to plane $\beta$ is $\sqrt{3}$, and the distance from $Q$ to plane $\alpha$ is $2 \sqrt{3}$. What is the minimum distance between points $P$ and $Q$? | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many five-digit numbers divisible by 3 are there that include the digit 6? | {
"answer": "12504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $B$ is an acute angle. If $\frac{\sin A}{\sin B} = \frac{5c}{2b}$, $\sin B = \frac{\sqrt{7}}{4}$, and $S_{\triangle ABC} = \frac{5\sqrt{7}}{4}$, find the value of $b$. | {
"answer": "\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\sin x,\frac{3}{2})$ and $\overrightarrow{b}=(\cos x,-1)$.
$(1)$ When $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $\sin 2x$.
$(2)$ Find the minimum value of $f(x)=(\overrightarrow{a}+\overrightarrow{b}) \cdot \overrightarrow{b}$ for $x \in [-\frac{\pi}{2},0]$. | {
"answer": "-\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the times between $8$ and $9$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$. | {
"answer": "8:22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A phone fully charges in 1 hour 20 minutes on fast charge and in 4 hours on regular charge. Fedya initially put the completely discharged phone on regular charge, then switched to fast charge once he found the proper adapter, completing the charging process. Determine the total charging time of the phone, given that it spent one-third of the total charging time on the fast charge. Assume that both fast and regular charging happen uniformly. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square \(ABCD\) on the plane, find the minimum of the ratio \(\frac{OA+OC}{OB+OD}\), where \(O\) is an arbitrary point on the plane. | {
"answer": "\\frac{1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$15\times 36$ -checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$
Tiles are supposed to cover whole squares of the board and be non-overlapping.
What is the maximum number of squares to be covered? | {
"answer": "540",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A three-digit number \( X \) was composed of three different digits, \( A, B, \) and \( C \). Four students made the following statements:
- Petya: "The largest digit in the number \( X \) is \( B \)."
- Vasya: "\( C = 8 \)."
- Tolya: "The largest digit is \( C \)."
- Dima: "\( C \) is the arithmetic mean of the digits \( A \) and \( B \)."
Find the number \( X \), given that exactly one of the students was mistaken. | {
"answer": "798",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\angle B=60^{\circ}$, $AC=2\sqrt{3}$, $BC=4$, then the area of $\triangle ABC$ is $\_\_\_\_\_\_$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Yesterday, Sasha cooked soup and added too little salt, requiring additional seasoning. Today, he added twice as much salt as yesterday, but still had to season the soup additionally, though with half the amount of salt he used for additional seasoning yesterday. By what factor does Sasha need to increase today's portion of salt so that tomorrow he does not have to add any additional seasoning? (Each day Sasha cooks the same portion of soup.) | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( a, b, c, d \) are prime numbers (they can be the same), and \( abcd \) is the sum of 35 consecutive positive integers, find the minimum value of \( a + b + c + d \). | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Through the vertices \(A\), \(C\), and \(D_1\) of a rectangular parallelepiped \(ABCD A_1 B_1 C_1 D_1\), a plane is drawn forming a dihedral angle of \(60^\circ\) with the base plane. The sides of the base are 4 cm and 3 cm. Find the volume of the parallelepiped. | {
"answer": "\\frac{144 \\sqrt{3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The route from $A$ to $B$ is traveled by a passenger train 3 hours and 12 minutes faster than a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train travels 288 km more. If the speed of each train is increased by $10 \mathrm{km} / h$, the passenger train would travel from $A$ to $B$ 2 hours and 24 minutes faster than the freight train. Determine the distance from $A$ to $B$. | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An old clock's minute and hour hands overlap every 66 minutes of standard time. Calculate how much the old clock's 24 hours differ from the standard 24 hours. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$ , inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime positive integers $m,n$ . Find $100m+n$ *Proposed by Yannick Yao* | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $a\sin B=-b\sin \left(A+ \frac {\pi}{3}\right)$.
$(1)$ Find $A$;
$(2)$ If the area of $\triangle ABC$, $S= \frac { \sqrt {3}}{4}c^{2}$, find the value of $\sin C$. | {
"answer": "\\frac { \\sqrt {7}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a store where all items cost an integer number of rubles, there are two special offers:
1) A customer who buys at least three items simultaneously can choose one item for free, whose cost does not exceed the minimum of the prices of the paid items.
2) A customer who buys exactly one item costing at least $N$ rubles receives a 20% discount on their next purchase (regardless of the number of items).
A customer, visiting this store for the first time, wants to purchase exactly four items with a total cost of 1000 rubles, where the cheapest item costs at least 99 rubles. Determine the maximum $N$ for which the second offer is more advantageous than the first. | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $\lfloor 3.998 \rfloor + \lceil 7.002 \rceil$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \frac{1}{2}x^2 - 2ax + b\ln(x) + 2a^2$ achieves an extremum of $\frac{1}{2}$ at $x = 1$, find the value of $a+b$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors \(\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right)\) and \(\overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}- x\right), \cos 2 x\right)\), and the function \(f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}\). If \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively in an acute triangle \( \triangle ABC \), and it is given that \( f(A) = 1 \), \( b + c = 5 + 3 \sqrt{2} \), and \( a = \sqrt{13} \), find the area \( S \) of \( \triangle ABC \). | {
"answer": "15/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, point $M$ lies inside the triangle such that $\angle MBA = 30^\circ$ and $\angle MAB = 10^\circ$. Given that $\angle ACB = 80^\circ$ and $AC = BC$, find $\angle AMC$. | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\left(2x-a\right)^{7}=a_{0}+a_{1}(x+1)+a_{2}(x+1)^{2}+a_{3}(x+1)^{3}+\ldots +a_{7}(x+1)^{7}$, and $a_{4}=-560$.<br/>$(1)$ Find the value of the real number $a$;<br/>$(2)$ Find the value of $|a_{1}|+|a_{2}|+|a_{3}|+\ldots +|a_{6}|+|a_{7}|$. | {
"answer": "2186",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the eccentricity of the ellipse $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $\frac{\sqrt{2}}{2}$, and it passes through the point $P(\sqrt{2},1)$. The line $y=\frac{\sqrt{2}}{2}x+m$ intersects the ellipse at two points $A$ and $B$.
(1) Find the equation of the ellipse $C$; (2) Find the maximum area of $\triangle PAB$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^{2}}{a-3} + \frac{y^{2}}{2-a} = 1$, with foci on the $y$-axis and a focal distance of $4$, determine the value of $a$. The options are:
A) $\frac{3}{2}$
B) $5$
C) $7$
D) $\frac{1}{2}$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median. | {
"answer": "3\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=4\cos(3x+\phi)(|\phi|<\frac{\pi}{2})$, its graph is symmetrical about the line $x=\frac{11\pi}{12}$. When $x_1,x_2\in(-\frac{7\pi}{12},-\frac{\pi}{12})$, $x_1\neq x_2$, and $f(x_1)=f(x_2)$, find $f(x_1+x_2)$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a student speech competition held at a school, there were a total of 7 judges. The final score for a student was the average score after removing the highest and the lowest scores. The scores received by a student were 9.6, 9.4, 9.6, 9.7, 9.7, 9.5, 9.6. The mode of this data set is _______, and the student's final score is _______. | {
"answer": "9.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many perfect squares are between 100 and 500? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Masha talked a lot on the phone with her friends, and the charged battery discharged exactly after a day. It is known that the charge lasts for 5 hours of talk time or 150 hours of standby time. How long did Masha talk with her friends? | {
"answer": "126/29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-6x+k=0$, and (choose one of the conditions $A$ or $B$ to answer the following questions).<br/>$A$: $x_1^2x_2^2-x_1-x_2=115$;<br/>$B$: $x_1^2+x_2^2-6x_1-6x_2+k^2+2k-121=0$.<br/>$(1)$ Find the value of $k$;<br/>$(2)$ Solve this equation. | {
"answer": "-11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence $\{a_n\}$ with a common ratio greater than $1$, $a_3a_7=72$, $a_2+a_8=27$, calculate $a_{12}$. | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ lies on the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and $F\_1$, $F\_2$ are the foci of the ellipse with $\angle F\_1 P F\_2 = 60^{\circ}$, find the area of $\triangle F\_1 P F\_2$. | {
"answer": "3 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A toy store sells a type of building block set: each starship is priced at 8 yuan, and each mech is priced at 26 yuan. A starship and a mech can be combined to form an ultimate mech, which sells for 33 yuan per set. If the store owner sold a total of 31 starships and mechs in one week, earning 370 yuan, how many starships were sold individually? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of sequences consisting of 100 R's and 2011 S's that satisfy the property that among the first \( k \) letters, the number of S's is strictly more than 20 times the number of R's for all \( 1 \leq k \leq 2111 \). | {
"answer": "\\frac{11}{2111} \\binom{2111}{100}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the sequence \(\{a_n\}\) satisfies \(a_1 = \frac{2}{3}\) and \(a_{n+1} - a_n = \sqrt{\frac{2}{3} \left(a_{n+1} + a_n\right)}\), then find the value of \(a_{2015}\). | {
"answer": "1354080",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of prime dates in a non-leap year where the day and the month are both prime numbers, and the year has one fewer prime month than usual. | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, there are two curves $C_1: x + y = 4$ and $C_2: \begin{cases} x = 1 + \cos \theta \\ y = \sin \theta \end{cases}$ (where $\theta$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.
(I) Find the polar equations of the curves $C_1$ and $C_2$.
(II) If the line $l: \theta = \alpha (\rho > 0)$ intersects $C_1$ and $C_2$ at points $A$ and $B$ respectively, find the maximum value of $\frac{|OB|}{|OA|}$. | {
"answer": "\\frac{1}{4}(\\sqrt{2} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are four different passwords, $A$, $B$, $C$, and $D$, used by an intelligence station. Each week, one of the passwords is used, and each week it is randomly chosen with equal probability from the three passwords not used in the previous week. Given that the password used in the first week is $A$, find the probability that the password used in the seventh week is also $A$ (expressed as a simplified fraction). | {
"answer": "61/243",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( m \) be the largest positive integer such that for every positive integer \( n \leqslant m \), the following inequalities hold:
\[
\frac{2n + 1}{3n + 8} < \frac{\sqrt{5} - 1}{2} < \frac{n + 7}{2n + 1}
\]
What is the value of the positive integer \( m \)? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 8. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the base of the triangular pyramid \( S-ABC \) is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = 2 \), and \( AB = 2 \), if points \( S \), \( A \), \( B \), and \( C \) are all on the surface of a sphere centered at \( O \), then the distance from point \( O \) to the plane \( ABC \) is \(\qquad\). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( A B C \) with the side ratio \( A B: A C = 5:4 \), the angle bisector of \( \angle B A C \) intersects side \( B C \) at point \( L \). Find the length of segment \( A L \), given that the length of the vector \( 4 \cdot \overrightarrow{A B} + 5 \cdot \overrightarrow{A C} \) is 2016. | {
"answer": "224",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the rational number that is the value of the expression
$$
\cos ^{6}(3 \pi / 16)+\cos ^{6}(11 \pi / 16)+3 \sqrt{2} / 16
$$ | {
"answer": "5/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the height \(BD\) is equal to 6, the median \(CE\) is equal to 5, and the distance from point \(K\) (the intersection of segments \(BD\) and \(CE\)) to side \(AC\) is 1.
Find the side \(AB\). | {
"answer": "\\frac{2 \\sqrt{145}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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