problem stringlengths 10 5.15k | answer dict |
|---|---|
In tetrahedron $S\-(ABC)$, $SA$ is perpendicular to plane $ABC$, $\angle BAC=120^{\circ}$, $SA=AC=2$, $AB=1$, find the surface area of the circumscribed sphere of the tetrahedron. | {
"answer": "\\frac{40\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cone is placed on a table, pointing upwards. The vertical cross-section triangle, perpendicular to the base, has a vertex angle of 90 degrees. The diameter of the cone's base is 16 inches. A sphere is placed inside the cone so that it touches the sides of the cone and rests on the table. Find the volume of the sphere, expressed in terms of \(\pi\). | {
"answer": "\\frac{256}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A team consisting of Petya, Vasya, and a single-seat scooter is participating in a race. The distance is divided into 42 equal-length segments, with a checkpoint at the beginning of each segment. Petya completes a segment in 9 minutes, Vasya in 11 minutes, and either of them can cover a segment on the scooter in 3 minutes. They all start simultaneously, and the finishing time is determined by whoever finishes last. The boys agree that one will cover the first part of the distance on the scooter and the rest by running, while the other does the opposite (the scooter can be left at any checkpoint). How many segments should Petya cover on the scooter for the team to show the best possible time?
| {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < \alpha < \frac{\pi}{2}$, $-\frac{\pi}{2} < \beta < 0$, $\cos\left(\frac{\pi}{4}+\alpha\right) = \frac{1}{3}$, $\cos\left(\frac{\pi}{4}-\frac{\beta}{2}\right) = \frac{\sqrt{3}}{3}$,
find
$(1)$ the value of $\cos \alpha$;
$(2)$ the value of $\cos\left(\alpha+\frac{\beta}{2}\right)$. | {
"answer": "\\frac{5\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a machine that when a positive integer $N$ is entered, the machine's processing rule is:
- If $N$ is odd, output $4N + 2$.
- If $N$ is even, output $N / 2$.
Using the above rule, if starting with an input of $N = 9$, after following the machine's process for six times the output is $22$. Calculate the sum of all possible integers $N$ such that when $N$ undergoes this 6-step process using the rules above, the final output is $10$.
A) 320
B) 416
C) 540
D) 640
E) 900 | {
"answer": "640",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circular park 20 feet in diameter with a straight walking path 5 feet wide passing through its center, calculate the area of the park covered by grass after the path is laid. | {
"answer": "100\\pi - 100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vanya had a certain amount of cookies; he ate some, and then Tanya came to visit him and they divided the remaining cookies equally. It turned out that Vanya ate five times more cookies than Tanya did. What fraction of all the cookies did Vanya eat by the time Tanya arrived? | {
"answer": "\\frac{5}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $$x^{5}=a_{0}+a_{1}(2-x)+a_{2}(2-x)^{2}+…+a_{5}(2-x)^{5}$$, find the value of $$\frac {a_{0}+a_{2}+a_{4}}{a_{1}+a_{3}}$$. | {
"answer": "- \\frac {61}{60}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two circles that intersect at two points $(2,3)$ and $(m,2)$, and both circle centers lie on the line $x+y+n=0$. Find the value of $m+n$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The vertices of a tetrahedron in the spatial rectangular coordinate system O-xyz are located at the coordinates (1,0,1), (1,1,0), (0,1,0), and (1,1,1). Find the volume of the circumscribed sphere of the tetrahedron. | {
"answer": "\\frac{\\sqrt{3}}{2} \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Recently, many cities in China have been intensifying efforts to develop the "night economy" to meet the diverse consumption needs of different groups and to boost employment, drive entrepreneurship, and enhance regional economic development vitality. A handicraft seller at a night market found through a survey of daily sales over the past month (30 days) that the selling price $P(x)$ per item (unit: yuan per item) approximately follows the function with respect to the $x$-th day $(1\leqslant x\leqslant 30, x\in \mathbb{N})$ as $P(x)=5+\frac{k}{x}$ (where $k$ is a positive constant). The daily sales volume $Q(x)$ per day (unit: items) for some days is shown in the table below:
| $x$ | $10$ | $15$ | $20$ | $25$ | $30$ |
|-------|------|------|------|------|------|
| $Q(x)$| $90$ | $95$ | $100$| $95$ | $90$ |
It is known that the daily sales revenue on the 10th day is $459$ yuan.
$(1)$ Find the value of $k$;
$(2)$ Given the following four function models:
① $Q(x)=ax+b$;
② $Q(x)=a|x-m|+b$;
③ $Q(x)=a^{x}+b$;
④ $Q(x)=b\cdot \log ax$.
Based on the data in the table, choose the most suitable function model to describe the relationship between the daily sales volume $Q(x)$ and the day $x$, and find the analytical expression of that function;
$(3)$ Let the daily sales revenue of the handicraft item be a function $y=f(x)$ (unit: yuan). Find the minimum value of this function. | {
"answer": "441",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, there is an ellipse $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity $e=\frac{\sqrt{2}}{2}$. Also, point $P(2,1)$ is on the ellipse $(C)$.
1. Find the equation of the ellipse $(C)$.
2. If points $A$ and $B$ are both on the ellipse $(C)$, and the midpoint $M$ of $AB$ is on the line segment $OP$ (not including the endpoints), find the maximum value of the area of triangle $AOB$. | {
"answer": "\\frac{3 \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sophie has collected one of each of the first 25 U.S. state quarters based on the order in which the states joined the union. The following graph indicates the number of states that joined the union in each decade. Determine the fraction of Sophie's 25 coins that represent states that joined the union during the decade 1790 through 1799. Express your answer as a common fraction.
[asy]size(200);
label("1790",(6,0),S);
label("1810",(12,-12),S);
label("1830",(18,0),S);
label("1850",(24,-12),S);
label("1870",(30,0),S);
label("1890",(36,-12),S);
label("1910",(42,0),S);
label("1960",(48,-12),S);
label("to",(6,-4),S);
label("to",(12,-16),S);
label("to",(18,-4),S);
label("to",(24,-16),S);
label("to",(30,-4),S);
label("to",(36,-16),S);
label("to",(42,-4),S);
label("to",(48,-16),S);
label("1799",(6,-8),S);
label("1809",(12,-20),S);
label("1829",(18,-8),S);
label("1849",(24,-20),S);
label("1869",(30,-8),S);
label("1889",(36,-20),S);
label("1909",(42,-8),S);
label("1959",(48,-20),S);
draw((0,0)--(50,0));
draw((0,2)--(50,2));
draw((0,4)--(50,4));
draw((0,6)--(50,6));
draw((0,8)--(50,8));
draw((0,10)--(50,10));
draw((0,12)--(50,12));
draw((0,14)--(50,14));
draw((0,16)--(50,16));
draw((0,18)--(50,18));
fill((4,0)--(8,0)--(8,10)--(4,10)--cycle,gray(0.8));
fill((10,0)--(14,0)--(14,7)--(10,7)--cycle,gray(0.8));
fill((16,0)--(20,0)--(20,8)--(16,8)--cycle,gray(0.8));
fill((22,0)--(26,0)--(26,5)--(22,5)--cycle,gray(0.8));
fill((28,0)--(32,0)--(32,6)--(28,6)--cycle,gray(0.8));
fill((34,0)--(38,0)--(38,4)--(34,4)--cycle,gray(0.8));
fill((40,0)--(44,0)--(44,3)--(40,3)--cycle,gray(0.8));
[/asy] | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ and $β$ are both acute angles, $cosα= \frac {3}{5}$, and $cos(α+β)=- \frac {5}{13}$, find the value of $sinβ$. | {
"answer": "\\frac {56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A company allocates 5 employees to 3 different departments, with each department being allocated at least one employee. Among them, employees A and B must be allocated to the same department. Calculate the number of different allocation methods. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $m=(\sin x,-1)$ and $n=\left( \sqrt{3}\cos x,-\frac{1}{2}\right)$, and the function $f(x)=(m+n)\cdot m$.
1. Find the interval where the function $f(x)$ is monotonically decreasing.
2. Given $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, with $A$ being an acute angle, $a=2\sqrt{3}$, $c=4$. If $f(A)$ is the maximum value of $f(x)$ in the interval $\left[0,\frac{\pi}{2}\right]$, find $A$, $b$, and the area $S$ of $\triangle ABC$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin2α + \sinα = 0, α ∈ (\frac{π}{2}, π)$, find the value of $\tan(α + \frac{π}{4})$. | {
"answer": "\\sqrt{3} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^{2}=4x$, a line $l$ passing through its focus $F$ intersects the parabola at points $A$ and $B$ (with point $A$ in the first quadrant), such that $\overrightarrow{AF}=3\overrightarrow{FB}$. A line passing through the midpoint of $AB$ and perpendicular to $l$ intersects the $x$-axis at point $G$. Calculate the area of $\triangle ABG$. | {
"answer": "\\frac{32\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number of tourists want to take a cruise, and it is required that the number of people on each cruise ship is the same. If each cruise ship carries 12 people, there will be 1 person left who cannot board. If one cruise ship leaves empty, then all tourists can be evenly distributed among the remaining ships. It is known that each cruise ship can accommodate up to 15 people. Please calculate how many tourists there are in total. | {
"answer": "169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})^{2}=\_\_\_\_\_\_$. | {
"answer": "\\sqrt{3}-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Liz had no money initially, and her friends gave her one-sixth, one-fifth, and one-fourth of their respective amounts, find the fractional part of the group's total money that Liz has. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find how many integer values of \( x \) are there such that \( \lceil{\sqrt{x}}\rceil=18 \)? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Wang Hong's father deposited 20,000 yuan in the bank for a fixed term of three years, with an annual interest rate of 3.33%. How much money, including the principal and interest, can Wang Hong's father withdraw at the end of the term? | {
"answer": "21998",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest positive integer $m$ such that an $m \times m$ square can be exactly divided into 7 rectangles with pairwise disjoint interiors, and the lengths of the 14 sides of these 7 rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14$. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $8\cos ^{2}25^{\circ}-\tan 40^{\circ}-4$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the volume in cubic inches of a right, rectangular prism where the areas of the side, front, and bottom faces are 20 square inches, 12 square inches, and 8 square inches, respectively. | {
"answer": "8\\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A particle begins at a point P on the parabola y = x^2 - 2x - 8 where the y-coordinate is 8. It rolls along the parabola to the nearest point Q where the y-coordinate is -8. Calculate the horizontal distance traveled by the particle, defined as the absolute difference between the x-coordinates of P and Q. | {
"answer": "\\sqrt{17} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a bus station, there are three buses departing to a school between 6:30 AM and 7:30 AM each day. The ticket prices for the buses are the same, but the comfort levels vary. Xiao Jie, a student, observes before boarding. When the first bus arrives, he does not get on but carefully observes its comfort level. If the comfort level of the second bus is better than the first, he will board the second bus; otherwise, he will take the third bus. Given that the comfort levels of the buses can be classified as high, medium, and low, the probability of Xiao Jie boarding a bus with a high comfort level is ______. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 15x^2 + 50x - 60$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 15s^2 + 50s - 60} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
A) 133
B) 134
C) 135
D) 136
E) 137 | {
"answer": "135",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle PQR,$ $PQ=PR=30$ and $QR=28.$ Points $M, N,$ and $O$ are located on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{MN}$ and $\overline{NO}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PMNO$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct non-consecutive positive integers $x$ and $y$ are factors of 48. If $x\cdot y$ is not a factor of 48, what is the smallest possible value of $x\cdot y$? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the integrals:
1) \(\int_{0}^{\frac{\pi}{2}} \sin ^{3} x \, dx\);
2) \(\int_{0}^{\ln 2} \sqrt{e^{x}-1} \, dx\);
3) \(\int_{-a}^{a} x^{2} \sqrt{a^{2}-x^{2}} \, dx\);
4) \(\int_{1}^{2} \frac{\sqrt{x^{2}-1}}{x} \, dx\). | {
"answer": "\\sqrt{3} - \\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point P is a point on the graph of the function f(x)=e^(2x), find the minimum distance from point P to the line l:y=2x. | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylindrical can has a circumference of 24 inches and a height of 7 inches. A spiral strip is painted on the can such that it winds around the can precisely once, reaching from the bottom to the top. However, instead of reaching directly above where it started, it ends 3 inches horizontally to the right. What is the length of the spiral strip? | {
"answer": "\\sqrt{778}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$ , such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A space probe travels $5,555,555,555,555$ kilometers in its journey towards a distant star. After reaching its destination, it has traveled $3,333,333,333,333$ kilometers from its last refueling point. How many kilometers did the probe travel before the last refueling point?
A) $2,111,111,111,111$
B) $2,222,222,222,222$
C) $2,333,333,333,333$
D) $2,444,444,444,444$
E) $2,555,555,555,555$ | {
"answer": "2,222,222,222,222",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the curve $C_{1}$ is defined by $\begin{cases} x=-2+\cos \alpha \\ y=-1+\sin \alpha \end{cases}$ (where $\alpha$ is a parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the curve $C_{2}$ is defined by $\rho\cos \theta-3=0$. Point $P$ is a moving point on the curve $C_{1}$.
$(1)$ Find the maximum distance from point $P$ to the curve $C_{2}$;
$(2)$ If the curve $C_{3}$: $\theta= \frac {\pi}{4}$ intersects curve $C_{1}$ at points $A$ and $B$, find the area of $\triangle ABC_{1}$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\theta \in \left[\dfrac{\pi}{4}, \dfrac{\pi}{2}\right]$ and $\sin 2\theta = \dfrac{3\sqrt{7}}{8}$, find the value of $\sin \theta$. | {
"answer": "\\dfrac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 300$. The points corresponding to $a,$ $b,$ and $c$ on the complex plane are the vertices of a right triangle. Find the square of the length of the hypotenuse, $h^2$, given that the triangle's centroid is at the origin. | {
"answer": "450",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \[P(x) = (3x^4 - 39x^3 + ax^2 + bx + c)(4x^4 - 96x^3 + dx^2 + ex + f),\] where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\{1, 2, 2, 3, 3, 4, 6\}.$ Find $P(7).$ | {
"answer": "86400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right-angled triangle, one of whose acute angles is $\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\alpha$ this ratio will be the smallest. | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The five books "Poetry," "Documents," "Rites," "Changes," and "Spring and Autumn" all have different numbers of pages. The differences in the number of pages between the books are as follows:
1. "Poetry" and "Documents" differ by 24 pages.
2. "Documents" and "Rites" differ by 17 pages.
3. "Rites" and "Changes" differ by 27 pages.
4. "Changes" and "Spring and Autumn" differ by 19 pages.
5. "Spring and Autumn" and "Poetry" differ by 15 pages.
Determine the difference in the number of pages between the book with the most pages and the book with the least pages. | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When arranging the performance order of 6 singers, requiring that both singers B and C are either before or after singer A, calculate the total number of different arrangements. | {
"answer": "480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The robotics club has 30 members: 12 boys and 18 girls. A 6-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | {
"answer": "\\frac{574,287}{593,775}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b=2\left(a\cos B-c\right)$. Find:<br/>
$(1)$ The value of angle $A$;<br/>
$(2)$ If $a\cos C=\sqrt{3}$ and $b=1$, find the value of $c$. | {
"answer": "2\\sqrt{3} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular octagon, find the ratio of the length of the shortest diagonal to the longest diagonal. Express your answer as a common fraction in simplest radical form. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that square $ABCE$ has side lengths $AF = 3FE$ and $CD = 3DE$, calculate the ratio of the area of $\triangle AFD$ to the area of square $ABCE$. | {
"answer": "\\frac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 4 qualified and 2 defective products, determine the probability of finding the last defective product exactly on the fourth inspection when selectins products one at a time and not returning them after each selection. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Choose one of the following conditions from (1) $a\sin \left(B+C\right)+c\sin C-b\sin B=2a\sin C\sin B$, (2) $\frac{cosB}{cosC}+\frac{b}{c-\sqrt{2}a}=0$, (3) $2a^{2}=(a^{2}+b^{2}-c^{2})(1+\tan C)$, and fill in the blank in the question below, and answer the corresponding questions.
Given $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, and satisfying ____.
$(1)$ Find the measure of angle $B$;
$(2)$ If the area of $\triangle ABC$ is $3$ and $a=2\sqrt{2}$, find the value of $\sin A\sin C$. | {
"answer": "\\frac{3\\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify \[\frac{1}{\dfrac{3}{\sqrt{5}+2} + \dfrac{4}{\sqrt{7}-2}}.\] | {
"answer": "\\frac{9\\sqrt{5} + 4\\sqrt{7} + 10}{(9\\sqrt{5} + 4\\sqrt{7})^2 - 100}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic function $f(x) = ax^2 + bx + c$ (where $a$, $b$, and $c$ are constants). If the solution set of the inequality $f(x) \geq 2ax + b$ is $\mathbb{R}$ (the set of all real numbers), then the maximum value of $\frac{b^2}{a^2 + c^2}$ is __________. | {
"answer": "2\\sqrt{2} - 2",
"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, with $c=2$ and $b=\sqrt{2}a$. Find the maximum value of the area of $\triangle ABC$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. What is the number of square units in the area of the fourth square?
Note that the diagram is not provided, but imagine it similarly to the reference where:
- $AB$ and $CD$ have squares on them (inside the triangles $ABC$ and $ACD$ respectively),
- $AC$ is a common hypotenuse for triangles $ABC$ and $ACD$,
- A square is drawn on $AD$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center \( Q \) and radius 2 rolls around the inside of a right triangle \( DEF \) with side lengths 9, 12, and 15, always remaining tangent to at least one side of the triangle. When \( Q \) first returns to its original position, through what distance has \( Q \) traveled? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the power function $f(x) = kx^a$ whose graph passes through the point $\left( \frac{1}{3}, 81 \right)$, find the value of $k + a$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $z=1+i$, then $|{iz+3\overline{z}}|=\_\_\_\_\_\_$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the sequence ${a_{n}}$, $a_{1}=1$, $a_{n+2}+(-1)^{n}a_{n}=1$. Let $s_{n}$ be the sum of the first $n$ terms of the sequence ${a_{n}}$. Find $s_{100}$ = \_\_\_\_\_\_. | {
"answer": "1300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by $\begin{cases}x=1+\frac{\sqrt{2}}{2}t\\y=\frac{\sqrt{2}}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the $x$-axis as the polar axis, the polar equation of curve $C$ is $\rho=4\sin\theta$.
(1) Find the Cartesian equation of line $l$ and the polar equation of curve $C$.
(2) Let $M$ be a moving point on curve $C$, and $P$ be the midpoint of $OM$. Find the minimum distance from point $P$ to line $l$. | {
"answer": "\\sqrt{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\operatorname{log}_{8}(p) = \operatorname{log}_{12}(q) = \operatorname{log}_{18}(p-q)$, calculate the value of $\frac{q}{p}$. | {
"answer": "\\frac{\\sqrt{5} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$ , and $AB$ , respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the line segments $ PA, PB$ . and $PC$ is $x^o$ , determine the value of $x$ . | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that all edges of a tetrahedron have a length of $\sqrt{2}$, and the four vertices are on the same sphere, calculate the surface area of this sphere. | {
"answer": "3 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos^4x-2\sin x\cos x-\sin^4x.$
$(1)$ Find the smallest positive period of $f(x)$.
$(2)$ When $x\in\left[0, \frac{\pi}{2}\right]$, find the minimum value of $f(x)$ and the set of $x$ values for which this minimum is achieved. | {
"answer": "\\frac{3\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function be $$f(x)=1-2\sin^{2}x-\cos(2x+ \frac {\pi}{3})$$
(1) Find the smallest positive period of the function $f(x)$.
(2) For triangle ABC, the sides $a$, $b$, $c$ are opposite to the angles $A$, $B$, $C$, respectively. Given $b=5$, and $$f\left( \frac {B}{2}\right)=1$$, find the maximum area of triangle ABC. | {
"answer": "\\frac {25 \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $α∈(\frac{\pi}{2},π)$, $\sin α =\frac{\sqrt{5}}{5}$.
(Ⅰ) Find the value of $\tan\left( \frac{\pi}{4}+2α \right)$;
(Ⅱ) Find the value of $\cos\left( \frac{5\pi}{6}-2α \right)$. | {
"answer": "-\\frac{3\\sqrt{3}+4}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n+1)x$ on the interval $[0, 2\pi]$. Calculate $\sum_{n=2}^{100} G(n)$. | {
"answer": "10296",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle \(WXYZ\) is divided into four smaller rectangles. The perimeters of three of these smaller rectangles are 11, 16, and 19. The perimeter of the fourth rectangle lies between 11 and 19. What is the length of the perimeter of \(WXYZ\)?
Options:
A) 28
B) 30
C) 32
D) 38
E) 40 | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the palindromes between 1000 and 2000 contain at least one 3 or 5, except in the first digit? | {
"answer": "36\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a_n\}$ satisfies $a_{n+1}=(2|\sin \frac{n\pi}{2}|-1)a_{n}+n$, then the sum of the first $100$ terms of the sequence $\{a_n\}$ is __________. | {
"answer": "2550",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $1,000,000,000,000 - 888,777,888,777$. | {
"answer": "111,222,111,223",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a \in \mathbb{R}$, if the real part and the imaginary part of the complex number $\frac{a + i}{1 + i}$ (where $i$ is the imaginary unit) are equal, then $\_\_\_\_\_\_$, $| \overline{z}| = \_\_\_\_\_\_$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many whole numbers between $200$ and $500$ contain the digit $3$? | {
"answer": "138",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$ , $ b$ , $ c$ , we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\] | {
"answer": "\\frac{3\\sqrt{2} - 4}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum positive period of $y=\tan(4x+ \frac{\pi}{3})$ is $\pi$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the interior angles $A$, $B$, and $C$, respectively, and $a\cos C+\left(2b+c\right)\cos A=0$.
$(1)$ Find the value of angle $A$.
$(2)$ If $D$ is the midpoint of segment $BC$ and $AD=\frac{7}{2}$, $AC=3$, find the area of triangle $ABC$. | {
"answer": "6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular hexagon $ABCDEF$ has an area of $n$. Let $m$ be the area of triangle $ACE$. What is $\tfrac{m}{n}?$
A) $\frac{1}{2}$
B) $\frac{2}{3}$
C) $\frac{3}{4}$
D) $\frac{1}{3}$
E) $\frac{3}{2}$ | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a batch of parts, 50 are drawn, and then 40 out of these 50 are inspected. It is found that there are 38 qualified products. Calculate the pass rate of this batch of products as a percentage. | {
"answer": "95\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each integer \( n \geq 2 \), let \( A(n) \) be the area of the region in the coordinate plane defined by the inequalities \( 1 \leq x \leq n \) and \( 0 \leq y \leq x \left\lfloor \log_2{x} \right\rfloor \), where \( \left\lfloor \log_2{x} \right\rfloor \) is the greatest integer not exceeding \( \log_2{x} \). Find the number of values of \( n \) with \( 2 \leq n \leq 100 \) for which \( A(n) \) is an integer. | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)?
(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).) | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the largest constant $K\geq 0$ such that $$ \frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 $$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$ .
*Proposed by Orif Ibrogimov (Czech Technical University of Prague).* | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $x$ and $y$ are positive real numbers such that $x^2 - 2xy + 3y^2 = 9$. Find the maximum possible value of $x^2 + 2xy + 3y^2$. | {
"answer": "18 + 9\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ is an acute angle and satisfies $\cos(\alpha+\frac{\pi}{4})=\frac{\sqrt{3}}{3}$.
$(1)$ Find the value of $\sin(\alpha+\frac{7\pi}{12})$.
$(2)$ Find the value of $\cos(2\alpha+\frac{\pi}{6})$. | {
"answer": "\\frac{2\\sqrt{6}-1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $|\omega^2 + 7\omega + 40|$ if $\omega = 4 + 3i$. | {
"answer": "15\\sqrt{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that z and w are complex numbers with a modulus of 1, and 1 ≤ |z + w| ≤ √2, find the minimum value of |z - w|. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let be the set $ \mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right.\quad\exists x_1,x_2\in [0,1]\quad x_1\neq x_2\wedge \left( f\left(
x_1 \right) = f\left( x_2 \right) =0\vee f\left(
x_1 \right) = f'\left( x_1 \right) = 0\right) \wedge f''<1 \right\} , $ and $ f^*\in\mathcal{C} $ such that $ \int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx . $ Find $ \int_0^1\left| f^*(x) \right| dx $ and describe $ f^*. $ | {
"answer": "1/12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the non-empty subsets of the set \( A = \{1, 2, \cdots, 10\} \), how many subsets have the sum of their elements being a multiple of 10? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Aaron takes a square sheet of paper, with one corner labeled $A$ . Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$ . Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$ .
*Proposed by Aaron Lin* | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A lattice point is a point whose coordinates are integers. How many lattice points are on the boundary or inside the region bounded by \( y = |x| \) and \( y = -x^2 + 8 \)? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of curve C is $\rho - 2\cos\theta - 6\sin\theta + \frac{1}{\rho} = 0$, and the parametric equation of line l is $\begin{cases} x=3+ \frac{1}{2}t \\ y=3+ \frac{\sqrt{3}}{2}t \end{cases}$ (t is the parameter).
(1) Find the standard equation of curve C;
(2) If line l intersects curve C at points A and B, and the coordinates of point P are (3, 3), find the value of $|PA|+|PB|$. | {
"answer": "2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that real numbers $a$ and $b$ satisfy $\frac{1}{a} + \frac{2}{b} = \sqrt{ab}$, calculate the minimum value of $ab$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A horizontal line with equation $y=s$ intersects line segment $\overline{AB}$ at $P$ and line segment $\overline{AC}$ at $Q$, forming $\triangle APQ$ with area 18. Compute $s$. | {
"answer": "10 - 2\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases} 2x-10, & x\leqslant 7 \\ \frac {1}{f(x-2)}, & x > 7 \end{cases}$, and the sequence ${a_{n}}={f(n)}$ where $n\in\mathbb{N}^{*}$, find the sum of the first 50 terms of the sequence ${a_{n}}$. | {
"answer": "\\frac {225}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively. Given that a = 3, cosC = $- \frac{1}{15}$, and 5sin(B + C) = 3sin(A + C).
(1) Find the length of side c.
(2) Find the value of sin(B - $\frac{\pi}{3}$). | {
"answer": "\\frac{2\\sqrt{14} - 5\\sqrt{3}}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
 | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A frustum of a cone has a lower base radius of 8 inches, an upper base radius of 4 inches, and a height of 5 inches. Calculate its lateral surface area and total surface area. | {
"answer": "(80 + 12\\sqrt{41})\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $b\sin C + c\sin B = 4a\sin B\sin C$ and $b^2 + c^2 - a^2 = 8$. Find the area of $\triangle ABC$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral triangle ABC has a side length of 4. A right isosceles triangle DBE, where $DB=EB=1$ and angle $D\hat{B}E = 90^\circ$, is cut from triangle ABC. Calculate the perimeter of the remaining quadrilateral. | {
"answer": "10 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, given $A(1,4)$, $B(4,1)$, $C(0,-4)$, find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA}$. | {
"answer": "- \\dfrac {62}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Around a circular table, there are 18 girls seated, 11 dressed in blue and 7 dressed in red. Each girl is asked if the girl to her right is dressed in blue, and each one responds with either yes or no. It is known that a girl tells the truth only when both of her neighbors, the one to her right and the one to her left, are dressed in the same color. How many girls will respond yes? If there is more than one possibility, state all. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If each of the four numbers $3, 4, 6,$ and $7$ replaces a $\square$, what is the largest possible sum of the fractions shown? | {
"answer": "$\\frac{23}{6}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the angle $\theta$ for the sum
\[e^{3\pi i/60} + e^{11\pi i/60} + e^{19\pi i/60} + e^{27\pi i/60} + e^{35\pi i/60} + e^{43\pi i/60} + e^{51\pi i/60} + e^{59\pi i/60}\]
when expressed in the form of $r e^{i\theta}$, where $0 \leq \theta < 2\pi$. | {
"answer": "\\dfrac{31\\pi}{60}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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