problem stringlengths 10 5.15k | answer dict |
|---|---|
A circle has a radius of 15 units. Suppose a chord in this circle bisects a radius perpendicular to the chord and the distance from the center of the circle to the chord is 9 units. What is the area of the smaller segment cut off by the chord?
A) $100 \pi$
B) $117.29$
C) $120 \pi$
D) $180$ | {
"answer": "117.29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a television program, five children (Tian Tian, Shi Tou, Kimi, Cindy, Angela) need to switch fathers (each child can choose any one of the other four fathers except their own). Calculate the total number of different combinations of choices for the five children. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the USA, standard letter-size paper is 8.5 inches wide and 11 inches long. What is the largest integer that cannot be written as a sum of a whole number (possibly zero) of 8.5's and a whole number (possibly zero) of 11's? | {
"answer": "159",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\pi$. | {
"answer": "300\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ is an angle in the third quadrant, and $f(\alpha)=\frac{{\sin({\frac{\pi}{2}-\alpha})\cos(-\alpha)\tan(\pi+\alpha)}}{{\cos(\pi-\alpha)}}$.
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $f(\alpha)=\frac{{2\sqrt{5}}}{5}$, find the value of $\cos \alpha$. | {
"answer": "-\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The solution to the equation \(\arcsin x + \arcsin 2x = \arccos x + \arccos 2x\) is | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If five points are given on a plane, then by considering all possible triples of these points, 30 angles can be formed. Denote the smallest of these angles by $\alpha$. Find the maximum value of $\alpha$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ is an obtuse angle, if $\sin (α- \frac{3π}{4})= \frac{3}{5}$, find $\cos (α+ \frac{π}{4})=\_\_\_\_\_\_\_\_\_\_\_\_\_.$ | {
"answer": "- \\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $A.\sqrt{21}$, $2\sin 2A\cos A-\sin 3A+\sqrt{3}\cos A=\sqrt{3}$
$(1)$ Find the size of angle $A$;
$(2)$ Given that $a,b,c$ are the sides opposite to angles $A,B,C$ respectively, and $a=1$ and $\sin A+\sin (B-C)=2\sin 2C$, find the area of triangle $A.\sqrt{21}$. | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
19) A puck is kicked up a ramp, which makes an angle of $30^{\circ}$ with the horizontal. The graph below depicts the speed of the puck versus time. What is the coefficient of friction between the puck and the ramp?
A) 0.07
B) 0.15
C) 0.22
D) 0.29
E) 0.37 | {
"answer": "0.29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Students from three different schools worked on a summer project. Six students from Atlas school worked for $4$ days, five students from Beacon school worked for $6$ days, and three students from Cedar school worked for $10$ days. The total payment for the students' work was $774. Given that each student received the same amount for a day's work, determine how much the students from Cedar school earned altogether. | {
"answer": "276.43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Anchuria, a standardized state exam is conducted. The probability of guessing the correct answer to each exam question is 0.25. In 2011, in order to obtain a certificate, it was necessary to answer correctly three out of 20 questions. In 2012, the School Management of Anchuria decided that three questions were too few. Now it is required to correctly answer six out of 40 questions. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of obtaining an Anchurian certificate higher - in 2011 or in 2012? | {
"answer": "2012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journey, and halved its speed again for the next eighth part of the journey, and so on. After the eighth decrease in speed, the car did not change its speed until the end of the trip. By how many times did the second car take more time to complete the entire journey than the first car? | {
"answer": "5/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two concentric circles have radii of $12$ meters and $24$ meters respectively. A rabbit starts at point X and runs along a sixth of the circumference of the larger circle, followed by a straight line to the smaller circle, then along a third of the circumference of the smaller circle, and finally across the diameter of the smaller circle back to the closest point on the larger circle's circumference. Calculate the total distance the rabbit runs. | {
"answer": "16\\pi + 36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( z_{1} \) and \( z_{2} \) be complex numbers such that \( \left|z_{1}\right|=3 \), \( \left|z_{2}\right|=5 \), and \( \left|z_{1} + z_{2}\right|=7 \). Find the value of \( \arg \left(\left( \frac{z_{2}}{z_{1}} \right)^{3}\right) \). | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A = \{1, 2, \cdots, 10\} \). If the equation \( x^2 - bx - c = 0 \) satisfies \( b, c \in A \) and the equation has at least one root \( a \in A \), then the equation is called a "beautiful equation". Find the number of "beautiful equations". | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=e^{ax}-x-1$, where $a\neq 0$. If $f(x)\geqslant 0$ holds true for all $x\in R$, then the set of possible values for $a$ is \_\_\_\_\_\_. | {
"answer": "\\{1\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a pot can hold 2 cakes at a time and it takes 5 minutes to cook both sides of a cake, calculate the minimum time it will take to cook 3 cakes thoroughly. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equations of curve $C_1$ are $$\begin{cases} x=2\cos\theta \\ y=\sin\theta\end{cases}(\theta \text{ is the parameter}),$$ and the parametric equations of curve $C_2$ are $$\begin{cases} x=-3+t \\ y= \frac {3+3t}{4}\end{cases}(t \text{ is the parameter}).$$
(1) Convert the parametric equations of curves $C_1$ and $C_2$ into standard equations;
(2) Find the maximum and minimum distances from a point on curve $C_1$ to curve $C_2$. | {
"answer": "\\frac {12-2 \\sqrt {13}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ostap Bender organized an elephant distribution for the residents in the city of Fuks. 28 members of a union and 37 non-union members came for the distribution. Ostap distributed the elephants equally among all union members and also equally among all non-union members.
It turned out that there was only one possible way to distribute the elephants (such that all elephants were distributed). What is the largest number of elephants that Ostap Bender could have? (It is assumed that each attendee received at least one elephant.) | {
"answer": "1036",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A flock of geese was flying over several lakes. On each lake, half of the geese and an additional half goose landed, while the rest continued flying. All the geese landed after seven lakes. How many geese were in the flock? | {
"answer": "127",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 white tiles, 4 yellow tiles, and 8 green tiles, all of which are square tiles with a side length of 1. These tiles are used to create square patterns with a side length of 4. How many different patterns can be created? | {
"answer": "900900",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To monitor the skier's movements, the coach divided the track into three sections of equal length. It was found that the skier's average speeds on these three separate sections were different. The time required for the skier to cover the first and second sections together was 40.5 minutes, and for the second and third sections - 37.5 minutes. Additionally, it was determined that the skier's average speed on the second section was the same as the average speed for the first and third sections combined. How long did it take the skier to reach the finish? | {
"answer": "58.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the quadrilateral $ABCD$, it is known that $\angle BAC = \angle CAD = 60^{\circ}$, and $AB + AD = AC$. Additionally, it is known that $\angle ACD = 23^{\circ}$. How many degrees is the angle $\angle ABC$? | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$ , $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$ . What is the most probable sum he can get? | {
"answer": "127",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $l$: $2mx - y - 8m - 3 = 0$ and the circle $C$: $x^2 + y^2 - 6x + 12y + 20 = 0$, find the shortest length of the chord that line $l$ cuts on circle $C$. | {
"answer": "2\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the New-Vasyuki currency exchange, 11 tugriks are traded for 14 dinars, 22 rupees for 21 dinars, 10 rupees for 3 thalers, and 5 crowns for 2 thalers. How many tugriks can be exchanged for 13 crowns? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ,$ $\angle Y = 45^\circ,$ $\angle Z = 90^\circ,$ and $XZ = 6.$ Find $YZ.$ | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( 9210 - 9124 = 210 - \square \), the value represented by the \( \square \) is: | {
"answer": "124",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elective 4-5: Selected Topics on Inequalities. Given the function $f(x) = |2x-1| + |2x+3|$.
$(1)$ Solve the inequality $f(x) \geqslant 6$;
$(2)$ Let the minimum value of $f(x)$ be $m$, and let the positive real numbers $a, b$ satisfy $2ab + a + 2b = m$. Find the minimum value of $a + 2b$. | {
"answer": "2\\sqrt{5} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\left\{a_{n}\right\}$, where $a_{1}=a_{2}=1$, $a_{3}=-1$, and $a_{n}=a_{n-1} a_{n-3}$, find $a_{1964}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB = AC = 17$, $BC = 16$, and point $G$ is the centroid of the triangle. Points $A'$, $B'$, and $C'$ are the images of $A$, $B$, and $C$, respectively, after a $90^\circ$ clockwise rotation about $G$. Determine the area of the union of the two regions enclosed by triangles $ABC$ and $A'B'C'$. | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right-angled triangle has an area of \( 36 \mathrm{~m}^2 \). A square is placed inside the triangle such that two sides of the square are on two sides of the triangle, and one vertex of the square is at one-third of the longest side.
Determine the area of this square. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\sin (x + \sin x) = \cos (x - \cos x)\), where \(x \in [0, \pi]\). Find \(x\). | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a square of side length 13. Let \(E\) and \(F\) be points on rays \(AB\) and \(AD\), respectively, so that the area of square \(ABCD\) equals the area of triangle \(AEF\). If \(EF\) intersects \(BC\) at \(X\) and \(BX=6\), determine \(DF\). | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the lengths of the arcs of curves defined by the equations in polar coordinates.
$$
\rho=5(1-\cos \varphi),-\frac{\pi}{3} \leq \varphi \leq 0
$$ | {
"answer": "20 \\left(1 - \\frac{\\sqrt{3}}{2}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the dividend to the divisor is 9:2, and the ratio of the divisor to the quotient is ____. | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y= \sqrt {x^{2}-ax+4}$, find the set of all possible values of $a$ such that the function is monotonically decreasing on the interval $[1,2]$. | {
"answer": "\\{4\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( g(x) = \log_{\frac{1}{3}}\left(\log_9\left(\log_{\frac{1}{9}}\left(\log_{81}\left(\log_{\frac{1}{81}}x\right)\right)\right)\right) \). Determine the length of the interval that forms the domain of \( g(x) \), and express it in the form \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers. What is \( p+q \)?
A) 80
B) 82
C) 84
D) 86
E) 88 | {
"answer": "82",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose you have two bank cards for making purchases: a debit card and a credit card. Today you decided to buy airline tickets worth 20,000 rubles. If you pay for the purchase with the credit card (the credit limit allows it), you will have to repay the bank within $\mathrm{N}$ days to stay within the grace period in which the credit can be repaid without extra charges. Additionally, in this case, the bank will pay cashback of $0.5 \%$ of the purchase amount after 1 month. If you pay for the purchase with the debit card (with sufficient funds available), you will receive a cashback of $1 \%$ of the purchase amount after 1 month. It is known that the annual interest rate on the average monthly balance of funds on the debit card is $6 \%$ per year (Assume for simplicity that each month has 30 days, the interest is credited to the card at the end of each month, and the interest accrued on the balance is not compounded). Determine the minimum number of days $\mathrm{N}$, under which all other conditions being equal, it is more profitable to pay for the airline tickets with the credit card. (15 points) | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the length of the chord cut by the ray $θ= \dfrac {π}{4}$ on the circle $ρ=4\sin θ$ is __________. | {
"answer": "2\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of a triangle with side lengths 13, 14, and 14. | {
"answer": "6.5\\sqrt{153.75}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate or simplify:
\\((1)\\dfrac{\\sqrt{1-2\\sin {15}^{\\circ}\\cos {15}^{\\circ}}}{\\cos {15}^{\\circ}-\\sqrt{1-\\cos^2 {165}^{\\circ}}}\\)；
\\((2)\\)Given \\(| \\vec{a} |=4\\), \\(| \\vec{b} |=2\\), and the angle between \\(\\vec{a}\\) and \\(\\vec{b}\\) is \\(\\dfrac{2\\pi }{3}\\), find the value of \\(| \\vec{a} + \\vec{b} |\\). | {
"answer": "2\\\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chemistry student conducted an experiment: starting with a bottle filled with syrup solution, the student poured out one liter of liquid, refilled the bottle with water, then poured out one liter of liquid again, and refilled the bottle with water once more. As a result, the syrup concentration decreased from 36% to 1%. Determine the volume of the bottle in liters. | {
"answer": "1.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \(ABC\) with sides opposite to the angles \(A\), \(B\), and \(C\) being \(a\), \(b\), and \(c\) respectively, and it is known that \( \sqrt {3}\sin A-\cos (B+C)=1\) and \( \sin B+\sin C= \dfrac {8}{7}\sin A\) with \(a=7\):
(Ⅰ) Find the value of angle \(A\);
(Ⅱ) Calculate the area of \( \triangle ABC\). | {
"answer": "\\dfrac {15 \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a,\) \(b,\) \(c\) be distinct real numbers such that
\[\frac{a}{1 + b} = \frac{b}{1 + c} = \frac{c}{1 + a} = k.\] Find the product of all possible values of \(k.\) | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( f(x)=a \sin ((x+1) \pi)+b \sqrt[3]{x-1}+2 \), where \( a \) and \( b \) are real numbers and \( f(\lg 5) = 5 \), find \( f(\lg 20) \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the parallelepiped $ABCD A_1B_1C_1D_1$, the face $ABCD$ is a square with side length 5, the edge $AA_1$ is also equal to 5, and this edge forms angles of $60^\circ$ with the edges $AB$ and $AD$. Find the length of the diagonal $BD_1$. | {
"answer": "5\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 4. The point $K$ is defined such that $\overrightarrow{L K}=3 \overrightarrow{F A}-\overrightarrow{F B}$. Determine whether $K$ lies inside, on the boundary, or outside of $A B C D E F$, and find the length of the segment $K A$. | {
"answer": "\\frac{4 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the infinite geometric series: $$\frac{4}{3} - \frac{5}{12} + \frac{25}{144} - \frac{125}{1728} + \dots$$ | {
"answer": "\\frac{64}{63}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a square and a regular decagon that share an edge. One side of the square is extended to meet an extended edge of the decagon. What is the value of \( x \)?
A) 15
B) 18
C) 21
D) 24
E) 27 | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin x+\lambda\cos x (\lambda\in\mathbb{R})$ is symmetric about $x=-\frac{\pi}{4}$, find the equation of one of the axes of symmetry of function $g(x)$ obtained by expanding the horizontal coordinate of each point of the graph of $f(x)$ by a factor of $2$ and then shifting it right by $\frac{\pi}{3}$. | {
"answer": "\\frac{11\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $2.29^{\star \star} N$ is an integer. Its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a perfect fourth power. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A balloon that inflates into the shape of a perfect cube is being blown up at a rate such that at time \( t \) (in fortnights), it has a surface area of \( 6t \) square furlongs. At what rate, in cubic furlongs per fortnight, is the air being pumped in when the surface area is 144 square furlongs? | {
"answer": "3\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a company needs to select 8 engineering and technical personnel from its 6 subsidiaries to form a task force, with each subsidiary contributing at least one person, calculate the total number of ways to allocate these 8 positions. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the two-variable function
$$
f(a, b)=\max _{x \in[-1,1]}\left\{\left|x^{2}-a x-b\right|\right\},
$$
find the minimum value of \( f(a, b) \). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
30 beads (blue and green) were arranged in a circle. 26 beads had a neighboring blue bead, and 20 beads had a neighboring green bead. How many blue beads were there? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Peter's most listened-to CD contains eleven tracks. His favorite is the eighth track. When he inserts the CD into the player and presses one button, the first track starts, and by pressing the button seven more times, he reaches his favorite song. If the device is in "random" mode, he can listen to the 11 tracks in a randomly shuffled order. What are the chances that he will reach his favorite track with fewer button presses in this way? | {
"answer": "7/11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of the expression \(\cos ^{4} \frac{7 \pi}{24}+\sin ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{17 \pi}{24}+\cos ^{4} \frac{13 \pi}{24}\). | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point P in the plane satisfying $|PM| - |PN| = 2\sqrt{2}$, with $M(-2,0)$, $N(2,0)$, and $O(0,0)$,
(1) Find the locus S of point P;
(2) A straight line passing through the point $(2,0)$ intersects with S at points A and B. Find the minimum value of the area of triangle $\triangle OAB$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a cabinet, there are 3 pairs of different shoes. If 2 shoes are randomly taken out, let event A denote "the taken out shoes do not form a pair"; event B denote "both taken out shoes are for the same foot"; event C denote "one shoe is for the left foot and the other is for the right foot, but they do not form a pair".
(Ⅰ) Please list all the basic events;
(Ⅱ) Calculate the probabilities of events A, B, and C respectively. | {
"answer": "\\dfrac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A,B,C,D$ , be four different points on a line $\ell$ , so that $AB=BC=CD$ . In one of the semiplanes determined by the line $\ell$ , the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$ . | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the midsegment (median) of an isosceles trapezoid, if its diagonal is 25 and its height is 15. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x_{1}, x_{2}, \cdots, x_{1993} \) satisfy:
\[
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993,
\]
and
\[
y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k} \quad (k=1,2,\cdots,1993),
\]
what is the maximum possible value of \( \left|y_{1}-y_{2}\right|+\left|y_{2}-y_{3}\right|+\cdots+\left|y_{1992}-y_{1993}\right| \)? | {
"answer": "1992",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $8x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$? | {
"answer": "356",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$. | {
"answer": "680",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An infinite arithmetic progression of positive integers contains the terms 7, 11, 15, 71, 75, and 79. The first term in the progression is 7. Kim writes down all the possible values of the one-hundredth term in the progression. What is the sum of the numbers Kim writes down? | {
"answer": "714",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the set $\{10, 11, 12, \ldots, 19\}$, 5 different numbers were chosen, and from the set $\{90, 91, 92, \ldots, 99\}$, 5 different numbers were also chosen. It turned out that the difference of any two numbers from the ten chosen numbers is not divisible by 10. Find the sum of all 10 chosen numbers. | {
"answer": "545",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\sin \theta, 2)$ and $\overrightarrow{b}=(\cos \theta, 1)$, which are collinear, where $\theta \in (0, \frac{\pi}{2})$.
1. Find the value of $\tan (\theta + \frac{\pi}{4})$.
2. If $5\cos (\theta - \phi)=3 \sqrt{5}\cos \phi, 0 < \phi < \frac{\pi}{2}$, find the value of $\phi$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a convex quadrilateral \( ABCD \) with \( X \) being the midpoint of the diagonal \( AC \). It is found that \( CD \parallel BX \). Find \( AD \) given that \( BX = 3 \), \( BC = 7 \), and \( CD = 6 \). | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer $n$ such that $$\frac 1{\sin 30^\circ\sin 31^\circ}+\frac 1{\sin 32^\circ\sin 33^\circ}+\cdots+\frac 1{\sin 88^\circ\sin 89^\circ}+\cos 89^\circ=\frac 1{\sin n^\circ}.$$ | {
"answer": "n = 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_0=29$ , $b_0=1$ and $$ a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1} $$ for $n\geq 1$ . Determine the smallest positive integer $k$ for which $29$ divides $\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$ . | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the conditions $a+acosC=\sqrt{3}csinA$, $\left(a+b+c\right)\left(a+b-c\right)=3ab$, $\left(a-b\right)\sin \left(B+C\right)+b\sin B=c\sin C$. Choose any one of these three conditions and complete the following question, then solve it. In triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, _____. Find the value of angle $C$; If the angle bisector of angle $C$ intersects $AB$ at point $D$ and $CD=2\sqrt{3}$, find the minimum value of $2a+b$. | {
"answer": "6 + 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \) and \( y \) be real numbers with \( x > y \) such that \( x^{2} y^{2} + x^{2} + y^{2} + 2xy = 40 \) and \( xy + x + y = 8 \). Find the value of \( x \). | {
"answer": "3 + \\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadrilateral pyramid $S-ABCD$ with a right trapezoid as its base, where $\angle ABC = 90^\circ$, $SA \perp$ plane $ABCD$, $SA = AB = BC = 1$, and $AD = \frac{1}{2}$, find the tangent of the angle between plane $SCD$ and plane $SBA$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average of \( p, q, r \) is 12. The average of \( p, q, r, t, 2t \) is 15. Find \( t \).
\( k \) is a real number such that \( k^{4} + \frac{1}{k^{4}} = t + 1 \), and \( s = k^{2} + \frac{1}{k^{2}} \). Find \( s \).
\( M \) and \( N \) are the points \( (1, 2) \) and \( (11, 7) \) respectively. \( P(a, b) \) is a point on \( MN \) such that \( MP:PN = 1:s \). Find \( a \).
If the curve \( y = ax^2 + 12x + c \) touches the \( x \)-axis, find \( c \). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers 407 and 370 equal the sum of the cubes of their digits. For example, \( 4^3 = 64 \), \( 0^3 = 0 \), and \( 7^3 = 343 \). Adding 64, 0, and 343 gives you 407. Similarly, the cube of 3 (27), added to the cube of 7 (343), gives 370.
Could you find a number, not containing zero and having the same property? Of course, we exclude the trivial case of the number 1. | {
"answer": "153",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among 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"
} |
A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were initially written on the board. | {
"answer": "998",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Lucas's odometer showed 27372 miles, which is a palindrome, and 3 hours later it showed another palindrome, calculate Lucas's average speed, in miles per hour, during this 3-hour period. | {
"answer": "33.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The "Tiao Ri Method", invented by mathematician He Chengtian during the Southern and Northern Dynasties of China, is an algorithm for finding a more accurate fraction to represent a numerical value. Its theoretical basis is as follows: If the deficient approximate value and the excess approximate value of a real number $x$ are $\frac{b}{a}$ and $\frac{d}{c}$ ($a, b, c, d \in \mathbb{N}^*$) respectively, then $\frac{b+d}{a+c}$ is a more accurate deficient approximate value or excess approximate value of $x$. We know that $\pi = 3.14159...$, and if we let $\frac{31}{10} < \pi < \frac{49}{15}$, then after using the "Tiao Ri Method" once, we get $\frac{16}{5}$ as a more accurate excess approximate value of $\pi$, i.e., $\frac{31}{10} < \pi < \frac{16}{5}$. If we always choose the simplest fraction, then what approximate fraction can we get for $\pi$ after using the "Tiao Ri Method" four times? | {
"answer": "\\frac{22}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=1+\frac{1}{2}t}\\{y=\sqrt{3}+\frac{{\sqrt{3}}}{2}t}\end{array}}\right.$ (where $t$ is a parameter), establish a polar coordinate system with the origin as the pole and the non-negative x-axis as the polar axis. The polar coordinate equation of curve $C$ is $\rho =4\sin \theta$.
$(1)$ Find the rectangular coordinate equation of curve $C$ and the polar coordinate equation of line $l$.
$(2)$ If $M(1,\sqrt{3})$, and the line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $\frac{{|MB|}}{{|MA|}}+\frac{{|MA|}}{{|MB|}}$. | {
"answer": "\\frac{3\\sqrt{3} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four cars $A, B, C,$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ travel clockwise, while $C$ and $D$ travel counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at that same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. After how much time from the start of the race will $C$ and $D$ meet for the first time? | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\{a_{n}\}$ is an arithmetic sequence, with the sum of its first $n$ terms denoted as $S_{n}$, and $a_{4}=-3$, choose one of the following conditions as known: <br/>$(Ⅰ)$ The arithmetic sequence $\{a_{n}\}$'s general formula; <br/>$(Ⅱ)$ The minimum value of $S_{n}$ and the value of $n$ when $S_{n}$ reaches its minimum value. <br/>Condition 1: $S_{4}=-24$; <br/>Condition 2: $a_{1}=2a_{3}$. | {
"answer": "-30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Yangyang leaves home for school. If she walks 60 meters per minute, she arrives at school at 6:53. If she walks 75 meters per minute, she arrives at school at 6:45. What time does Yangyang leave home? | {
"answer": "6:13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Person A, Person B, and Person C start from point $A$ to point $B$. Person A starts at 8:00, Person B starts at 8:20, and Person C starts at 8:30. They all travel at the same speed. After Person C has been traveling for 10 minutes, Person A is exactly halfway between Person B and point $B$, and at that moment, Person C is 2015 meters away from point $B$. Find the distance between points $A$ and $B$. | {
"answer": "2418",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving circle that passes through the fixed point $F(1,0)$ and is tangent to the fixed line $l$: $x=-1$.
(1) Find the equation of the trajectory $C$ of the circle's center;
(2) The midpoint of the chord $AB$ formed by the intersection of line $l$ and $C$ is $(2,1}$. $O$ is the coordinate origin. Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$ and $| \overrightarrow{AB}|$. | {
"answer": "\\sqrt{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( M \) lies on the parabola \( y = 2x^2 - 3x + 4 \), and point \( F \) lies on the line \( y = 3x - 4 \).
Find the minimum value of \( MF \). | {
"answer": "\\frac{7 \\sqrt{10}}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the function \( f(x) = \operatorname{tg}^{2} x - 4 \operatorname{tg} x - 12 \operatorname{ctg} x + 9 \operatorname{ctg}^{2} x - 3 \) on the interval \(\left( -\frac{\pi}{2}, 0 \right)\). | {
"answer": "3 + 8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square \( ABCD \) has center \( O \). Points \( P \) and \( Q \) are on \( AB \), \( R \) and \( S \) are on \( BC \), \( T \) and \( U \) are on \( CD \), and \( V \) and \( W \) are on \( AD \), so that \( \triangle APW \), \( \triangle BRQ \), \( \triangle CTS \), and \( \triangle DVU \) are isosceles and \( \triangle POW \), \( \triangle ROQ \), \( \triangle TOS \), and \( \triangle VOU \) are equilateral. What is the ratio of the area of \( \triangle PQO \) to that of \( \triangle BRQ \)? | {
"answer": "1:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that line $l$ passes through point $P(-1,2)$ with a slope angle of $\frac{2\pi}{3}$, and the circle's equation is $\rho=2\cos (\theta+\frac{\pi}{3})$:
(1) Find the parametric equation of line $l$;
(2) Let line $l$ intersect the circle at points $M$ and $N$, find the value of $|PM|\cdot|PN|$. | {
"answer": "6+2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown onto the segment [6, 11], and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2k-24\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the distances from one vertex of a rectangle with length 3 and width 4 to the midpoints of each of its sides.
A) $6 + \sqrt{5}$
B) $7 + \sqrt{12}$
C) $7.77 + \sqrt{13}$
D) $9 + \sqrt{15}$ | {
"answer": "7.77 + \\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$ | {
"answer": "\\sqrt[4]{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Please choose one of the following two sub-questions to answer. If multiple choices are made, the score will be based on the first chosen question.
$(①)$ The sum of the internal angles of a regular hexagon is $ $ degrees.
$(②)$ Xiaohua saw a building with a height of $(137)$ meters at its signboard. From the same horizontal plane at point $B$, he measured the angle of elevation to the top of the building $A$ to be $30^{\circ}$. The distance from point $B$ to the building is $ $ meters (rounded to the nearest whole number, and ignore the measuring instrument error, $\sqrt{3} \approx 1.732$). | {
"answer": "237",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. Find \(\angle A\) if \(\angle MAN = 50^\circ\). | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion? | {
"answer": "999000000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$, it is known that $\frac{a_{11}}{a_{10}} + 1 < 0$. Determine the maximum value of $n$ for which $S_n > 0$ holds. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the three cutlets needs to be fried on a pan on both sides for five minutes each side. The pan can hold only two cutlets at a time. Is it possible to fry all three cutlets in less than 20 minutes (ignoring the time for flipping and transferring the cutlets)? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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