problem stringlengths 10 5.15k | answer dict |
|---|---|
One interior angle in a triangle measures $50^{\circ}$. What is the angle between the bisectors of the remaining two interior angles? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the argument $\theta$ of the complex number
\[
e^{11\pi i/60} + e^{31\pi i/60} + e^{51 \pi i/60} + e^{71\pi i /60} + e^{91 \pi i /60}
\]
expressed in the form $r e^{i \theta}$ with $0 \leq \theta < 2\pi$. | {
"answer": "\\frac{17\\pi}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chessboard’s squares are labeled with numbers as follows:
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(9 - i + j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]
Eight of the squares are chosen such that each row and each column has exactly one selected square. Find the maximum sum of the labels of these eight chosen squares. | {
"answer": "\\frac{8}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $2\tan\alpha=3\tan \frac{\pi}{8}$, then $\tan\left(\alpha- \frac{\pi}{8}\right)=$ ______. | {
"answer": "\\frac{5\\sqrt{2}+1}{49}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
After reading the following solution, answer the question: Xiaofang found in the simplification of $\sqrt{7+4\sqrt{3}}$ that first, $\sqrt{7+4\sqrt{3}}$ can be simplified to $\sqrt{7+2\sqrt{12}}$. Since $4+3=7$ and $4\times 3=12$, that is, ${(\sqrt{4})^2}+{(\sqrt{3})^2}=7$, $\sqrt{4}×\sqrt{3}=\sqrt{12}$, so $\sqrt{7+4\sqrt{3}}=\sqrt{7+2\sqrt{12}}=\sqrt{{{(\sqrt{4})}^2}+2\sqrt{4×3}+{{(\sqrt{3})}^2}}=\sqrt{{{(\sqrt{4}+\sqrt{3})}^2}}=2+\sqrt{3}$. The question is:<br/>$(1)$ Fill in the blanks: $\sqrt{4+2\sqrt{3}}=$______, $\sqrt{5-2\sqrt{6}}=$______; <br/>$(2)$ Further research reveals that the simplification of expressions in the form of $\sqrt{m±2\sqrt{n}}$ can be done by finding two positive numbers $a$ and $b\left(a \gt b\right)$ such that $a+b=m$, $ab=n$, that is, ${(\sqrt{a})^2}+{(\sqrt{b})^2}=m$, $\sqrt{a}×\sqrt{b}=\sqrt{n}$, then we have: $\sqrt{m±2\sqrt{n}}=\_\_\_\_\_\_.$<br/>$(3)$ Simplify: $\sqrt{4-\sqrt{15}$ (Please write down the simplification process). | {
"answer": "\\frac{\\sqrt{10}}{2}-\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\tan \alpha= \sqrt {2}$, then $2\sin ^{2}\alpha-\sin \alpha\cos \alpha+\cos ^{2}\alpha=$ \_\_\_\_\_\_ . | {
"answer": "\\frac {5- \\sqrt {2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the following sum:
\[
\frac{1}{2^{2024}} \sum_{n = 0}^{1011} (-3)^n \binom{2024}{2n}.
\] | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a race, all runners must start at point $A$, touch any part of a 1500-meter wall, and then stop at point $B$. Given that the distance from $A$ directly to the wall is 400 meters and from the wall directly to $B$ is 600 meters, calculate the minimum distance a participant must run to complete this. Express your answer to the nearest meter. | {
"answer": "1803",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kelvin the Frog is playing the game of Survival. He starts with two fair coins. Every minute, he flips all his coins one by one, and throws a coin away if it shows tails. The game ends when he has no coins left, and Kelvin's score is the *square* of the number of minutes elapsed. What is the expected value of Kelvin's score? For example, if Kelvin flips two tails in the first minute, the game ends and his score is 1. | {
"answer": "\\frac{64}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$. | {
"answer": "\\frac{3-\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, it is known that $\cos C + (\cos A - \sqrt{3} \sin A) \cos B = 0$.
(1) Find the measure of angle $B$.
(2) If $\sin (A - \frac{\pi}{3}) = \frac{3}{5}$, find $\sin 2C$. | {
"answer": "\\frac{24 + 7\\sqrt{3}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
James has 6 ounces of tea in a ten-ounce mug and 6 ounces of milk in a separate ten-ounce mug. He first pours one-third of the tea from the first mug into the second mug and stirs well. Then he pours one-fourth of the mixture from the second mug back into the first. What fraction of the liquid in the first mug is now milk?
A) $\frac{1}{6}$
B) $\frac{1}{5}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number is composed of 6 millions, 3 tens of thousands, and 4 thousands. This number is written as ____, and when rewritten in terms of "ten thousands" as the unit, it becomes ____ ten thousands. | {
"answer": "603.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery? | {
"answer": "1024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the value of $\frac{1}{4}\cdot\frac{8}{1}\cdot\frac{1}{32}\cdot\frac{64}{1} \dotsm \frac{1}{1024}\cdot\frac{2048}{1}$. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$, where $a_n \in \mathbb{N}^*$, and $S_n = \frac{1}{8}(a_n+2)^2$. If $b_n = \frac{1}{2}a_n - 30$, find the minimum value of the sum of the first \_\_\_\_\_\_ terms of the sequence $\{b_n\}$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2g(a^2 + b^2) = [g(a)]^2 + [g(b)]^2\] for all nonnegative integers $a$ and $b.$
Let $n$ be the number of possible values of $g(16),$ and let $s$ be the sum of the possible values of $g(16).$ Find $n \times s.$ | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonal lengths of a rhombus are 18 units and 26 units. Calculate both the area and the perimeter of the rhombus. | {
"answer": "20\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram has side lengths of 10, $12x-2$, $5y+5$, and $4$. What is the value of $x+y$?
[asy]draw((0,0)--(24,0)--(30,20)--(6,20)--cycle);
label("$12x-2$",(15,0),S);
label("10",(3,10),W);
label("$5y+5$",(18,20),N);
label("4",(27,10),E);
[/asy] | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The difference between two perfect squares is 221. What is the smallest possible sum of the two perfect squares? | {
"answer": "24421",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a line segment $\overline{AB}=10$ cm, a point $C$ is placed on $\overline{AB}$ such that $\overline{AC} = 6$ cm and $\overline{CB} = 4$ cm. Three semi-circles are drawn with diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, external to the segment. If a line $\overline{CD}$ is drawn perpendicular to $\overline{AB}$ from point $C$ to the boundary of the smallest semi-circle, find the ratio of the shaded area to the area of a circle taking $\overline{CD}$ as its radius. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sarah subscribes to a virtual fitness class platform that charges a monthly membership fee plus a per-class fee. If Sarah paid a total of $30.72 in February for 4 classes, and $54.72 in March for 8 classes, with the monthly membership fee increasing by 10% from February to March, calculate the fixed monthly membership fee. | {
"answer": "7.47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volumes of solids formed by rotating the region bounded by the function graphs about the \( O y \) (y-axis).
$$
y = x^{2} + 1, \quad y = x, \quad x = 0, \quad x = 1
$$ | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a modified toothpick pattern, the first stage is constructed using 5 toothpicks. If each subsequent stage is formed by adding three more toothpicks than the previous stage, what is the total number of toothpicks needed for the $15^{th}$ stage? | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using systematic sampling to select a sample of size 20 from 180 students, the students are randomly numbered from 1 to 180. They are then divided into 20 groups in order of their number (group 1: numbers 1-9, group 2: numbers 10-18, ..., group 20: numbers 172-180). If the number drawn from group 20 is 176, what is the number drawn from group 3? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If each side of a regular hexagon consists of 6 toothpicks, and there are 6 sides, calculate the total number of toothpicks used to build the hexagonal grid. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^2 = 4x$, a line passing through point $P(4, 0)$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. Find the minimum value of $y_1^2 + y_2^2$. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\cos({2x+\frac{π}{3}})$, determine the horizontal shift of the graph of the function $y=\sin 2x$. | {
"answer": "\\frac{5\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A monthly cell phone plan costs $30 per month, plus $10 cents per text message, plus $15 cents for each minute used over 25 hours, and an additional $5 for every gigabyte of data used over 15GB. In February, Emily sent 150 text messages, talked for 26 hours, and used 16GB of data. How much did she have to pay?
**A)** $40.00
**B)** $45.50
**C)** $50.00
**D)** $59.00
**E)** $70.00 | {
"answer": "59.00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A self-employed individual plans to distribute two types of products, A and B. According to a survey, when the investment amount is $x$ (where $x \geq 0$) in ten thousand yuan, the profits obtained from distributing products A and B are $f(x)$ and $g(x)$ in ten thousand yuan, respectively, where $f(x) = a(x - 1) + 2$ ($a > 0$); $g(x) = 6\ln(x + b)$, ($b > 0$). It is known that when the investment amount is zero, the profit is also zero.
(1) Determine the values of $a$ and $b$;
(2) If the self-employed individual is ready to invest 5 ten thousand yuan in these two products, please help him devise an investment plan to maximize his profit, and calculate the maximum value of his income. (Round to 0.1, reference data: $\ln 3 \approx 1.10$). | {
"answer": "12.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If the area of $\triangle ABC$ is $S$, and $2S=(a+b)^{2}-c^{2}$, then $\tan C$ equals \_\_\_\_\_\_. | {
"answer": "- \\frac {4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of line $l$ as $$\begin{cases} \left.\begin{matrix}x= \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{matrix}\right.\end{cases}$$ (where $t$ is the parameter), and the polar equation of curve $C$ as $\rho=2 \sqrt {2}\sin(\theta+ \frac {\pi}{4})$, line $l$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $P$.
(1) Find the standard equation of line $l$ and the Cartesian coordinate equation of curve $C$;
(2) Calculate the value of $\frac {1}{|PA|} + \frac {1}{|PB|}$. | {
"answer": "\\sqrt {5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following statements, the correct one is:
(1) The probability of event A or B happening is definitely greater than the probability of exactly one of A or B happening;
(2) The probability of events A and B happening simultaneously is definitely less than the probability of exactly one of A or B happening;
(3) Mutually exclusive events are definitely complementary events, and complementary events are also mutually exclusive events;
(4) Complementary events are definitely mutually exclusive events, but mutually exclusive events are not necessarily complementary events;
(5) If A and B are complementary events, then A+B cannot be a certain event. | {
"answer": "(4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the radius of the circle with the equation $x^2 - 8x + y^2 - 10y + 34 = 0$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of the triangle is S, and it is given that 2$\sqrt {3}$S - $\overrightarrow {AB}$•$\overrightarrow {AC}$ = 0, and c = 2.
(I) Find the measure of angle A.
(II) If a² + b² - c² = $\frac {6}{5}$ab, find the value of b. | {
"answer": "\\frac{3 + 4\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the first six terms of the geometric series $3 + \left(\frac{1}{3}\right) + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^3 + \left(\frac{1}{3}\right)^4 + \left(\frac{1}{3}\right)^5$. Express your answer as a simplified fraction. | {
"answer": "\\frac{364}{81}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=kx^{2}+2kx+1$ defined on the interval $[-3,2]$, the maximum value of the function is $4$. Determine the value of the real number $k$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin 2x - \cos \left(2x+\dfrac{\pi}{6}\right)$.
(1) Find the value of $f\left(\dfrac{\pi}{6}\right)$.
(2) Find the minimum positive period and the interval of monotonic increase of the function $f(x)$.
(3) Find the maximum and minimum values of $f(x)$ on the interval $\left[0,\dfrac{7\pi}{12}\right]$. | {
"answer": "-\\dfrac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Convert the parametric equations of the conic curve $C$:
$$
\begin{cases}
x=t^{2}+ \frac {1}{t^{2}}-2 \\
y=t- \frac {1}{t}
\end{cases}
$$
($t$ is the parameter) into a Cartesian coordinate equation.
2. If the polar equations of two curves are $\rho=1$ and $\rho=2\cos\left(\theta+ \frac {\pi}{3}\right)$ respectively, and they intersect at points $A$ and $B$, find the length of segment $AB$. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In my office, there are two digital 24-hour clocks. One clock gains one minute every hour and the other loses two minutes every hour. Yesterday, I set both of them to the same time, but when I looked at them today, I saw that the time shown on one was 11:00 and the time on the other was 12:00. What time was it when I set the two clocks?
A) 23:00
B) 19:40
C) 15:40
D) 14:00
E) 11:20 | {
"answer": "15:40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers. | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an increasing sequence $\{a_n\}$ with $2017$ terms, and all terms are non-zero, $a_{2017}=1$. If two terms $a_i$, $a_j$ are arbitrarily chosen from $\{a_n\}$, when $i < j$, $a_j-a_i$ is still a term in the sequence $\{a_n\}$. Then, the sum of all terms in the sequence $S_{2017}=$ ______. | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDEFGH$ be a cube with each edge of length $s$. A right square pyramid is placed on top of the cube such that its base aligns perfectly with the top face $EFGH$ of the cube, and its apex $P$ is directly above $E$ at a height $s$. Calculate $\sin \angle FAP$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $AB = 10$ be a diameter of circle $P$ . Pick point $C$ on the circle such that $AC = 8$ . Let the circle with center $O$ be the incircle of $\vartriangle ABC$ . Extend line $AO$ to intersect circle $P$ again at $D$ . Find the length of $BD$ . | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S = \{1, 2,..., 8\}$ . How many ways are there to select two disjoint subsets of $S$ ? | {
"answer": "6561",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rotate a square with a side length of 1 around a line that contains one of its sides. The lateral surface area of the resulting solid is \_\_\_\_\_\_. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, and identical letters correspond to identical digits). The word "GUATEMALA" was the result. How many different numbers could Egor have originally written, if his number was divisible by 25? | {
"answer": "20160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
While waiting at the post office, Lena moved 40 feet closer to the counter over a period of 20 minutes. At this rate, how many minutes will it take her to move the remaining 100 meters to the counter? | {
"answer": "164.042",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the asymptotes of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and the axis of the parabola $x^{2} = 4y$ form a triangle with an area of $2$, calculate the eccentricity of the hyperbola. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin(2x + \frac{\pi}{3}) - \sqrt{3}\sin(2x - \frac{\pi}{6})$,
(1) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic increase;
(2) When $x \in \left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$, find the maximum and minimum values of $f(x)$, and the corresponding values of $x$ at which these extreme values are attained. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola $F$ with the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, let $F$ be its left focus. Draw a line perpendicular to one asymptote of the hyperbola passing through $F$, and denote the foot of the perpendicular as $A$ and the intersection with the other asymptote as $B$. If $3\overrightarrow{FA} = \overrightarrow{FB}$, find the eccentricity of this hyperbola.
A) $2$
B) $3$
C) $\sqrt{2}$
D) $\sqrt{3}$ | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Any five points are taken inside or on a rectangle with dimensions 2 by 1. Let b be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than b. What is b? | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $102^{4} - 4 \cdot 102^{3} + 6 \cdot 102^2 - 4 \cdot 102 + 1$? | {
"answer": "100406401",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \cos x \cdot \sin\left(\frac{\pi}{6} - x\right)$,
(1) Find the interval where $f(x)$ is monotonically decreasing;
(2) In $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. If $f(C) = -\frac{1}{4}$, $a=2$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the length of side $c$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin x= \frac {3}{5}$, and $x\in( \frac {\pi}{2},\pi)$, find the values of $\cos 2x$ and $\tan (x+ \frac {\pi}{4})$. | {
"answer": "\\frac {1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers are in the list $-50, -44, -38, \ldots, 68, 74$? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function f(x) = |lnx|, and real numbers m and n that satisfy 0 < m < n and f(m) = f(n). If the maximum value of f(x) in the interval [m^2, n] is 2, find the value of $\frac{n}{m}$. | {
"answer": "e^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $DEF$, $DE=130$, $DF=110$, and $EF=140$. The angle bisector of angle $D$ intersects $\overline{EF}$ at point $T$, and the angle bisector of angle $E$ intersects $\overline{DF}$ at point $S$. Let $R$ and $U$ be the feet of the perpendiculars from $F$ to $\overline{ES}$ and $\overline{DT}$, respectively. Find $RU$. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that vertex E of right triangle ABE, where AE=BE, is in the interior of unit square ABCD, let R be the region consisting of all points inside ABCD and outside triangle ABE whose distance from AD is between 1/4 and 1/2. Calculate the area of R. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A=\{m-1,-3\}$, $B=\{2m-1,m-3\}$. If $A\cap B=\{-3\}$, then determine the value of the real number $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ with pairwise angles of $60^\circ$, and $|\overrightarrow{a}|=|\overrightarrow{b}|=|\overrightarrow{c}|=1$, find $|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}|$. | {
"answer": "\\sqrt{2}",
"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 $4\cos C \cdot \sin^2 \frac{C}{2} + \cos 2C = 0$.
(1) If $\tan A = 2\tan B$, find the value of $\sin(A-B)$;
(2) If $3ab = 25 - c^2$, find the maximum area of $\triangle ABC$. | {
"answer": "\\frac{25\\sqrt{3}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\{a_{n}\}$ is a geometric sequence, $a_{2}a_{4}a_{5}=a_{3}a_{6}$, $a_{9}a_{10}=-8$, then $a_{7}=\_\_\_\_\_\_$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a\_n\}$, if $a\_1=0$ and $a\_i=k^2$ ($i \in \mathbb{N}^*, 2^k \leqslant i < 2^{k+1}, k=1,2,3,...$), find the smallest value of $i$ that satisfies $a\_i + a_{2i} \geq 100$. | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$. | {
"answer": "40\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady's $2010$th step, considering the procedure implies hexadecimal (base 16) operations rather than quinary (base 5)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $(x,y)$ is a distance of 15 units from the $x$-axis. It is a distance of 13 units from the point $(2,7)$. It is a distance $n$ from the origin. Given that $x>2$, what is $n$? | {
"answer": "\\sqrt{334 + 4\\sqrt{105}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If there are $1, $2, and $3 bills in the board game "Silly Bills" and let x be the number of $1 bills, then x+11, x-18, and x+11+(x-18) = 2x-7 are the respective number of $2 and $3 bills, determine the value of x when the total amount of money is $100. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the share of the Japanese yen in the currency structure of the NWF funds as of 01.07.2021 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Japanese yen as of 01.07.2021:
\[ JPY_{22} = 1213.76 - 3.36 - 38.4 - 4.25 - 226.6 - 340.56 - 0.29 = 600.3 \text{ (billion rubles)} \]
b) Determine the share of Japanese yen in the currency structure of NWF funds as of 01.07.2021:
\[ \alpha_{07}^{JPY} = \frac{600.3}{1213.76} \approx 49.46\% \]
c) Calculate by how many percentage points and in what direction the share of Japanese yen in the currency structure of NWF funds has changed over the period considered in the table:
\[ \Delta \alpha^{JPY} = \alpha_{07}^{JPY} - \alpha_{06}^{JPY} = 49.46 - 72.98 = -23.52 \approx -23.5 \text{ (p.p.)} \]
Second method:
a) Determine the share of euros in the currency structure of NWF funds as of 01.07.2021:
\[ \alpha_{07}^{\text{EUR}} = \frac{38.4}{1213.76} \approx 3.16\% \]
b) Determine the share of Japanese yen in the currency structure of NWF funds as of 01.07.2021:
\[ \alpha_{07}^{JPY} = 100 - 0.28 - 3.16 - 0.35 - 18.67 - 28.06 - 0.02 = 49.46\% \]
c) Calculate by how many percentage points and in what direction the share of Japanese yen in the currency structure of NWF funds has changed over the period considered in the table:
\[ \Delta \alpha^{JPY} = \alpha_{07}^{JPY} - \alpha_{06}^{JPY} = 49.46 - 72.98 = -23.52 \approx -23.5 \text{ (p.p.)} \] | {
"answer": "-23.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Farmer Pythagoras has now expanded his field, which remains a right triangle. The lengths of the legs of this field are $5$ units and $12$ units, respectively. He leaves an unplanted rectangular area $R$ in the corner where the two legs meet at a right angle. This rectangle has dimensions such that its shorter side runs along the leg of length $5$ units. The shortest distance from the rectangle $R$ to the hypotenuse is $3$ units. Find the fraction of the field that is planted.
A) $\frac{151}{200}$
B) $\frac{148}{200}$
C) $\frac{155}{200}$
D) $\frac{160}{200}$ | {
"answer": "\\frac{151}{200}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an acute triangle $ABC$ , the points $H$ , $G$ , and $M$ are located on $BC$ in such a way that $AH$ , $AG$ , and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$ , $AB=10$ , and $AC=14$ . Find the area of triangle $ABC$ . | {
"answer": "12\\sqrt{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given circle $O: x^2+y^2=r^2(r>0)$, $A(x_1, y_1)$, $B(x_2, y_2)$ are two points on circle $O$, satisfying $x_1+y_1=x_2+y_2=3$, $x_1x_2+y_1y_2=-\frac{1}{2}r^2$, calculate the value of $r$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse E: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with a focal length of $2\sqrt{3}$, and the ellipse passes through the point $(\sqrt{3}, \frac{1}{2})$.
(Ⅰ) Find the equation of ellipse E;
(Ⅱ) Through point P$(-2, 0)$, draw two lines with slopes $k_1$ and $k_2$ respectively. These two lines intersect ellipse E at points M and N. When line MN is perpendicular to the y-axis, find the value of $k_1 \cdot k_2$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | {
"answer": "2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2017,0),(2017,2018),$ and $(0,2018)$. What is the probability that $x > 9y$? Express your answer as a common fraction. | {
"answer": "\\frac{2017}{36324}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $α \in \left( \frac{π}{2}, π \right)$, and $\sin α = \frac{1}{3}$.
$(1)$ Find the value of $\sin 2α$;
$(2)$ If $\sin (α+β) = -\frac{3}{5}$, and $β \in (0, \frac{π}{2})$, find the value of $\sin β$. | {
"answer": "\\frac{6\\sqrt{2}+4}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of a sphere is increased to $72\pi$ cubic inches. What is the new surface area of the sphere? Express your answer in terms of $\pi$. | {
"answer": "36\\pi \\cdot 2^{2/3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and the altitude on side $BC$ is $\frac{a}{2}$. Determine the maximum value of $\frac{c}{b}$. | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)=2\sin x( \sqrt {3}\cos x-\sin x)+1$$, if $f(x-\varphi)$ is an even function, determine the value of $\varphi$. | {
"answer": "\\frac {\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many rectangles can be formed when the vertices are chosen from points on a 4x4 grid (having 16 points)? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the height of a tower from a 20-meter distant building, given that the angle of elevation to the top of the tower is 30° and the angle of depression to the base of the tower is 45°. | {
"answer": "20 \\left(1 + \\frac {\\sqrt {3}}{3}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, respectively, and it satisfies $(2b-a) \cdot \cos C = c \cdot \cos A$.
$(1)$ Find the size of angle $C$;
$(2)$ Let $y = -4\sqrt{3}\sin^2\frac{A}{2} + 2\sin(C-B)$, find the maximum value of $y$ and determine the shape of $\triangle ABC$ when $y$ reaches its maximum value. | {
"answer": "2-2 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $u$ and $v$ are complex numbers such that $|u+v|=2$ and $|u^2+v^2|=8,$ find the smallest possible value of $|u^3+v^3|$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^{2}=2px$ with its directrix equation $x=-2$, let point $P$ be a point on the parabola. Find the minimum distance from point $P$ to the line $y=x+3$. | {
"answer": "\\frac { \\sqrt{2} }{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mrs. Riley revised her data after realizing that there was an additional score bracket and a special bonus score for one of the brackets. Recalculate the average percent score for the $100$ students given the updated table:
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&5\\\hline
95&12\\\hline
90&20\\\hline
80&30\\\hline
70&20\\\hline
60&8\\\hline
50&4\\\hline
40&1\\\hline
\end{tabular}
Furthermore, all students scoring 95% receive a 5% bonus, which effectively makes their score 100%. | {
"answer": "80.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the argument of the sum:
\[ e^{2\pi i/40} + e^{6\pi i/40} + e^{10\pi i/40} + e^{14\pi i/40} + e^{18\pi i/40} + e^{22\pi i/40} + e^{26\pi i/40} + e^{30\pi i/40} + e^{34\pi i/40} + e^{38\pi i/40} \]
and express it in the form \( r e^{i \theta} \), where \( 0 \le \theta < 2\pi \). | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers \(a\), \(b\), \(c\) form an equilateral triangle with side length 24 in the complex plane. If \(|a + b + c| = 48\), find \(|ab + ac + bc|\). | {
"answer": "768",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=e^{-x}+ \frac {nx}{mx+n}$.
$(1)$ If $m=0$, $n=1$, find the minimum value of the function $f(x)$.
$(2)$ If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac {m}{n}$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and it is known that $2\cos C(a\cos C+c\cos A)+b=0$.
$(1)$ Find the magnitude of angle $C$;
$(2)$ If $b=2$ and $c=2\sqrt{3}$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student used the "five-point method" to draw the graph of the function $f(x)=A\sin(\omega x+\varphi)$ ($\omega\ \gt 0$, $|\varphi|＜\frac{π}{2}$) within one period. The student listed and filled in some of the data in the table below:
| $\omega x+\varphi$ | $0$ | $\frac{π}{2}$ | $\pi$ | $\frac{{3π}}{2}$ | $2\pi$ |
|-------------------|-----|---------------|-------|------------------|-------|
| $x$ | | | $\frac{{3π}}{8}$ | $\frac{{5π}}{8}$ | |
| $A\sin(\omega x+\varphi)$ | $0$ | $2$ | | $-2$ | $0$ |
$(1)$ Please complete the data in the table and write the analytical expression of the function $f(x)$ on the answer sheet.
$(2)$ Move the graph of $f(x)$ to the left by $\theta$ units to obtain the graph of $g(x)$. If the graph of $g(x)$ is symmetric about the line $x=\frac{π}{3}$, find the minimum value of $\theta$. | {
"answer": "\\frac{7\\pi}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality $$ \left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) $$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur?
*(Proposed by Walther Janous)* | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x_1, x_2, \ldots, x_n$ be integers, satisfying:
(1) $-1 \leq x_i \leq 2$, for $i=1, 2, \ldots, n$;
(2) $x_1 + x_2 + \ldots + x_n = 19$;
(3) $x_1^2 + x_2^2 + \ldots + x_n^2 = 99$.
Find the maximum and minimum values of $x_1^3 + x_2^3 + \ldots + x_n^3$. | {
"answer": "133",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A basketball player made the following number of successful free throws in 10 successive games: 8, 17, 15, 22, 14, 12, 24, 10, 20, and 16. He attempted 10, 20, 18, 25, 16, 15, 27, 12, 22, and 19 free throws in those respective games. Calculate both the median number of successful free throws and the player's best free-throw shooting percentage game. | {
"answer": "90.91\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the enclosed shape formed by the line $y=x-2$ and the curve $y^2=x$ can be calculated. | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the imaginary unit $i$, let $z=1+i+i^{2}+i^{3}+\ldots+i^{9}$, then $|z|=$______. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius 3 are centered at $(0,3)$ and at $(3,0)$. What is the area of the intersection of the interiors of these two circles? Express your answer in terms of $\pi$ in its simplest form. | {
"answer": "\\frac{9\\pi - 18}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of 20 consecutive integers is a triangular number. What is the smallest such sum? | {
"answer": "190",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Place the sequence $\{2n+1\}$ in parentheses sequentially, with the first parenthesis containing one number, the second two numbers, the third three numbers, the fourth four numbers, the fifth one number again, and then continuing in this cycle. Determine the sum of the numbers in the 104th parenthesis. | {
"answer": "2104",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers $x$ and $y$ satisfy
\begin{align*}
x^2 + y^2 &= 2023
(x-2)(y-2) &= 3.
\end{align*}
Find the largest possible value of $|x-y|$ .
*Proposed by Howard Halim* | {
"answer": "13\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A spherical balloon collapses into a wet horizontal surface and settles into a shape of a hemisphere while keeping the same volume. The minor radius of the original balloon, when viewed as an ellipsoid due to unequal pressure distribution, was $4\sqrt[3]{3}$ cm. Find the major radius of the original balloon, assuming the major and minor axes were proportional and the proportionality constant is 2 before it became a hemisphere. | {
"answer": "8\\sqrt[3]{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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