problem stringlengths 10 5.15k | answer dict |
|---|---|
Given $$(5x- \frac {1}{ \sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion. | {
"answer": "375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt {3}\sin ^{2}x+\sin x\cos x- \frac { \sqrt {3}}{2}$ $(x\in\mathbb{R})$.
$(1)$ If $x\in(0, \frac {\pi}{2})$, find the maximum value of $f(x)$;
$(2)$ In $\triangle ABC$, if $A < B$ and $f(A)=f(B)= \frac {1}{2}$, find the value of $\frac {BC}{AB}$. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, three identical circles touch each other, and each circle has a circumference of 24. Calculate the perimeter of the shaded region within the triangle formed by the centers of the circles. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 10$, $BC = 5$, and the height of the pyramid, $PA$, while not perpendicular to the plane of $ABCD$, ends at the center of rectangle $ABCD$ and is twice the length of $BC$. What is the volume of $PABCD$? | {
"answer": "\\frac{500}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ is an angle in the third quadrant and $\cos 2α=-\frac{3}{5}$, find $\tan (\frac{π}{4}+2α)$. | {
"answer": "-\\frac{1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a dodecahedron, which is made up of 12 pentagonal faces. An ant starts at one of the top vertices and walks to one of the three adjacent vertices (vertex A). From vertex A, the ant walks again to one of its adjacent vertices (vertex B). What is the probability that vertex B is one of the bottom vertices? There are three bottom vertices in total in a dodecahedron. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 2,$ and $x \ge -\frac{1}{2},$ $y \ge -2,$ and $z \ge -3.$ Find the maximum value of:
\[
\sqrt{4x + 2} + \sqrt{4y + 8} + \sqrt{4z + 12}.
\] | {
"answer": "3\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin\alpha + \cos\alpha = \frac{1}{5}$, and $0 \leq \alpha < \pi$, find the value of $\tan\alpha$. | {
"answer": "- \\frac {4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus ride. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a point on the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{20} = 1$, and let $F_{1}$ and $F_{2}$ be the left and right foci, respectively. If $|PF_{1}| = 9$, then find $|PF_{2}|$. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$ .
\[ \begin{tabular}{|c|c|c|}\hline A & 1 & B \hline 5 & C & 13 \hline D & E & 3 \hline\end{tabular} \] | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The mean of the set of numbers $\{106, 102, 95, 103, 100, y, x\}$ is 104. What is the median of this set of seven numbers? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at $O$, $BO=4$, $OD=5$, $AO=9$, $OC=2$, and $AB=7$, find the length of $AD$. | {
"answer": "\\sqrt{166}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Read the material first, then answer the question.
$(1)$ Xiao Zhang encountered a problem when simplifying a quadratic radical: simplify $\sqrt{5-2\sqrt{6}}$.
After thinking about it, Xiao Zhang's process of solving this problem is as follows:
$\sqrt{5-2\sqrt{6}}=\sqrt{2-2\sqrt{2\times3}+3}$①
$=\sqrt{{(\sqrt{2})}^2}-2\sqrt{2}\times\sqrt{3}+{(\sqrt{3})}^2$②
$=\sqrt{{(\sqrt{2}-\sqrt{3})}^2}$③
$=\sqrt{2}-\sqrt{3}$④
In the above simplification process, an error occurred in step ____, and the correct result of the simplification is ____;
$(2)$ Please simplify $\sqrt{8+4\sqrt{3}}$ based on the inspiration you obtained from the above material. | {
"answer": "\\sqrt{6}+\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the following equations using appropriate methods:
(1) $x^2=49$;
(2) $(2x+3)^2=4(2x+3)$;
(3) $2x^2+4x-3=0$ (using the formula method);
(4) $(x+8)(x+1)=-12$. | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\frac{1}{2}x^{2}-a\ln x+b$ where $a\in R$.
(I) If the equation of the tangent line to the curve $y=f(x)$ at $x=1$ is $3x-y-3=0$, find the values of the real numbers $a$ and $b$.
(II) If $x=1$ is the extreme point of the function $f(x)$, find the value of the real number $a$.
(III) If $-2\leqslant a < 0$, for any $x_{1}$, $x_{2}\in(0,2]$, the inequality $|f(x_{1})-f(x_{2})|\leqslant m| \frac{1}{x_{1}}- \frac{1}{x_{2}}|$ always holds. Find the minimum value of $m$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\tan (\theta-\pi)=2$, then $\sin ^{2}\theta+\sin \theta\cos \theta-2\cos ^{2}\theta=$ \_\_\_\_\_\_ . | {
"answer": "\\frac {4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Anton colors a cell in a \(4 \times 50\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola $C$ that shares the same foci with the ellipse $\frac{x^{2}}{27}+ \frac{y^{2}}{36}=1$ and passes through the point $(\sqrt{15},4)$.
(I) Find the equation of the hyperbola $C$.
(II) If $F\_1$ and $F\_2$ are the two foci of the hyperbola $C$, and point $P$ is on the hyperbola $C$ such that $\angle F\_1 P F\_2 = 120^{\circ}$, find the area of $\triangle F\_1 P F\_2$. | {
"answer": "\\frac{5\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, BR = RC, CS = 3SA, and (AT)/(TB) = p/q. If the area of △RST is twice the area of △TBR, determine the value of p/q. | {
"answer": "\\frac{7}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of \[\cot(\cot^{-1}5 + \cot^{-1}11 + \cot^{-1}17 + \cot^{-1}23).\] | {
"answer": "\\frac{97}{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable X follows a normal distribution N(2, σ²) and P(X≤4)=0.88, find P(0<X<4). | {
"answer": "0.76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a four-digit positive integer $\overline{abcd}$, if $a+c=b+d=11$, then this number is called a "Shangmei number". Let $f(\overline{abcd})=\frac{{b-d}}{{a-c}}$ and $G(\overline{abcd})=\overline{ab}-\overline{cd}$. For example, for the four-digit positive integer $3586$, since $3+8=11$ and $5+6=11$, $3586$ is a "Shangmei number". Also, $f(3586)=\frac{{5-6}}{{3-8}}=\frac{1}{5}$ and $G(M)=35-86=-51$. If a "Shangmei number" $M$ has its thousands digit less than its hundreds digit, and $G(M)$ is a multiple of $7$, then the minimum value of $f(M)$ is ______. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) If 7 students stand in a row, and students A and B must stand next to each other, how many different arrangements are there?
(2) If 7 students stand in a row, and students A, B, and C must not stand next to each other, how many different arrangements are there?
(3) If 7 students stand in a row, with student A not standing at the head and student B not standing at the tail, how many different arrangements are there? | {
"answer": "3720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jason wishes to purchase some comic books. He has $15 and each comic book costs $1.20, tax included. Additionally, there is a discount of $0.10 on each comic book if he buys more than 10 comic books. What is the maximum number of comic books he can buy? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To solve the problem, we need to find the value of $\log_{4}{\frac{1}{8}}$.
A) $-\frac{1}{2}$
B) $-\frac{3}{2}$
C) $\frac{1}{2}$
D) $\frac{3}{2}$ | {
"answer": "-\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a}$=（sinx，cosx）, $\overrightarrow {b}$=（1，$\sqrt {3}$）.
(1) If $\overrightarrow {a}$$∥ \overrightarrow {b}$, find the value of tanx;
(2) Let f(x) = $\overrightarrow {a}$$$\cdot \overrightarrow {b}$, stretch the horizontal coordinates of each point on the graph of f(x) to twice their original length (vertical coordinates remain unchanged), then shift all points to the left by φ units (0 < φ < π), obtaining the graph of function g(x). If the graph of g(x) is symmetric about the y-axis, find the value of φ. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=ax + a^{-x}$ ($a>0$ and $a\neq1$), and $f(1)=3$, find the value of $f(0)+f(1)+f(2)$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the scores for innovation capability, innovation value, and innovation impact are $8$ points, $9$ points, and $7$ points, respectively, and the total score is calculated based on the ratio of $5:3:2$ for the three scores, calculate the total score of the company. | {
"answer": "8.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $x$ such that $\log_x 49 = \log_2 32$. | {
"answer": "7^{2/5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C$: $mx^{2}+ny^{2}=1(mn < 0)$, one of its asymptotes is tangent to the circle $x^{2}+y^{2}-6x-2y+9=0$. Determine the eccentricity of $C$. | {
"answer": "\\dfrac {5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer that is both an integer power of 7 and is not a palindrome. | {
"answer": "2401",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. If $a_5=5S_5=15$, find the sum of the first $100$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$. | {
"answer": "\\frac{100}{101}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the shortest distance from a point on the ellipse $\frac{y^2}{16} + \frac{x^2}{9} = 1$ to the line $y = x + m$ is $\sqrt{2}$, find the minimum value of $m$. | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABC$ is triangle. $l_1$ - line passes through $A$ and parallel to $BC$ , $l_2$ - line passes through $C$ and parallel to $AB$ . Bisector of $\angle B$ intersect $l_1$ and $l_2$ at $X,Y$ . $XY=AC$ . What value can take $\angle A- \angle C$ ? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$ ? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We flip a fair coin 12 times. What is the probability that we get exactly 9 heads and all heads occur consecutively? | {
"answer": "\\dfrac{1}{1024}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Chelsea is ahead by 60 points halfway through a 120-shot archery contest, with each shot scoring 10, 8, 5, 3, or 0 points and Chelsea scoring at least 5 points on every shot, determine the smallest number of bullseyes (10 points) Chelsea needs to shoot in her next n attempts to ensure victory, assuming her opponent can score a maximum of 10 points on each remaining shot. | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cagney can frost a cupcake every 15 seconds, while Lacey can frost every 40 seconds. They take a 10-second break after every 10 cupcakes. Calculate the number of cupcakes that they can frost together in 10 minutes. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is given by
$$\begin{cases}
x= \dfrac { \sqrt {2}}{2}t \\
y= \dfrac { \sqrt {2}}{2}t+4 \sqrt {2}
\end{cases}
(t \text{ is the parameter}),$$
establishing a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of circle $C$ is $ρ=2\cos \left(θ+ \dfrac {π}{4}\right)$.
(Ⅰ) Find the Cartesian coordinates of the center $C$ of the circle;
(Ⅱ) From any point on line $l$, draw a tangent to the circle $C$, and find the minimum length of the tangent line. | {
"answer": "2 \\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Harry has 4 sisters and 6 brothers, and his sister Harriet has S sisters and B brothers, calculate the product of S and B. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a periodic sequence $\left\{x_{n}\right\}$ that satisfies $x_{n}=\left|x_{n-1}-x_{n-2}\right|(n \geqslant 3)$, if $x_{1}=1$ and $x_{2}=a \geqslant 0$, calculate the sum of the first 2002 terms when the period of the sequence is minimized. | {
"answer": "1335",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin x\cos x+2 \sqrt {3}\cos ^{2}x- \sqrt {3}$, where $x\in\mathbb{R}$.
(Ⅰ) Find the smallest positive period and the intervals of monotonic decrease for the function $y=f(-3x)+1$;
(Ⅱ) Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If the acute angle $A$ satisfies $f\left( \frac {A}{2}- \frac {\pi}{6}\right)= \sqrt {3}$, and $a=7$, $\sin B+\sin C= \frac {13 \sqrt {3}}{14}$, find the area of $\triangle ABC$. | {
"answer": "10 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a bustling street, a middle-aged man is shouting "giving away money" while holding a small black cloth bag in his hand. Inside the bag, there are 3 yellow and 3 white ping-pong balls (which are identical in volume and texture). Next to him, there's a small blackboard stating:
Method of drawing balls: Randomly draw 3 balls from the bag. If the 3 balls drawn are of the same color, the stall owner will give the drawer $10; if the 3 balls drawn are not of the same color, the drawer will pay the stall owner $2.
(1) What is the probability of drawing 3 yellow balls?
(2) What is the probability of drawing 2 yellow balls and 1 white ball?
(3) Assuming there are 80 draws per day, estimate how much money the stall owner can make in a month (30 days) from a probabilistic perspective? | {
"answer": "1920",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f\left(x\right)=\frac{1}{2}\left(m-2\right){x}^{2}+\left(n-8\right)x+1\left(m\geqslant 0,n\geqslant 0\right)$ is monotonically decreasing in the interval $\left[\frac{1}{2},2\right]$, find the maximum value of $mn$. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be colonized, assuming each planet is unique? | {
"answer": "136",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x = 151$ and $x^3y - 3x^2y + 3xy = 3423000$, what is the value of $y$? | {
"answer": "\\frac{3423000}{3375001}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the cyclist encounters red lights at each of 4 intersections with probability $\frac{1}{3}$ and the events of encountering red lights are independent, calculate the probability that the cyclist does not encounter red lights at the first two intersections and encounters the first red light at the third intersection. | {
"answer": "\\frac{4}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a_n\}$ where $a_n > 0$, $a_1=1$, $a_{n+2}= \frac {1}{a_n+1}$, and $a_{100}=a_{96}$, find the value of $a_{2014}+a_3$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $0 < \beta < \alpha < \frac{\pi}{2}$, point $P(1,4 \sqrt{3})$ is a point on the terminal side of angle $\alpha$, and $\sin \alpha \sin \left(\frac{\pi}{2}-\beta \right)+\cos \alpha \cos \left(\frac{\pi}{2}+\beta \right)= \frac{3 \sqrt{3}}{14}$, calculate the value of angle $\beta$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Read the text below and answer the questions. Everyone knows that $\sqrt{2}$ is an irrational number, and irrational numbers are infinite non-repeating decimals. Therefore, we cannot write out all the decimal parts of $\sqrt{2}$, but since $1 \lt \sqrt{2} \lt 2$, the integer part of $\sqrt{2}$ is $1$. Subtracting the integer part $1$ from $\sqrt{2}$ gives the decimal part as $(\sqrt{2}-1)$. Answer the following questions:
$(1)$ The integer part of $\sqrt{10}$ is ______, and the decimal part is ______;
$(2)$ If the decimal part of $\sqrt{6}$ is $a$, and the integer part of $\sqrt{13}$ is $b$, find the value of $a+b-\sqrt{6}$;
$(3)$ Given $12+\sqrt{3}=x+y$, where $x$ is an integer and $0 \lt y \lt 1$, find the opposite of $x-y$. | {
"answer": "\\sqrt{3} - 14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The domain of the function \( f(x) \) is \( D \). If for any \( x_{1}, x_{2} \in D \), when \( x_{1} < x_{2} \), it holds that \( f(x_{1}) \leq f(x_{2}) \), then \( f(x) \) is called a non-decreasing function on \( D \). Suppose that the function \( f(x) \) is non-decreasing on \( [0,1] \) and satisfies the following three conditions:
1. \( f(0)=0 \);
2. \( f\left(\frac{x}{3}\right)=\frac{1}{2}f(x) \);
3. \( f(1-x)=1-f(x) \).
What is \( f\left(\frac{5}{12}\right) + f\left(\frac{1}{8}\right) \)? | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average of seven numbers in a list is 62. The average of the first four numbers is 55. What is the average of the last three numbers? | {
"answer": "71.\\overline{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circle I is externally tangent to Circle II and passes through the center of Circle II. Given that the area of Circle I is increased to 16 square inches, determine the area of Circle II, in square inches. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $60\%$ of students like dancing and the rest dislike it, $80\%$ of those who like dancing say they like it and the rest say they dislike it, also $90\%$ of those who dislike dancing say they dislike it and the rest say they like it. Calculate the fraction of students who say they dislike dancing but actually like it. | {
"answer": "25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the first eight prime numbers that have a units digit of 3. | {
"answer": "394",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence ${a_{n}}$ where ${a_{1}}=1$, ${a_{2}}=2$, and ${a_{n+2}}-{a_{n}}=2-2{(-1)^{n}}$, $n\in {N^{*}}$, find the value of ${S_{2017}}$. | {
"answer": "2017\\times1010-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dormitory of a certain high school senior class has 8 people. In a health check, the weights of 7 people were measured to be 60, 55, 60, 55, 65, 50, 50 (in kilograms), respectively. One person was not measured due to some reasons, and it is known that the weight of this student is between 50 and 60 kilograms. The probability that the median weight of the dormitory members in this health check is 55 is __. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person's commute times (in minutes) for 5 trips were 12, 8, 10, 11, and 9, respectively. The standard deviation of this set of data is ______. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of a triangle with angles $\frac{1}{7} \pi$ , $\frac{2}{7} \pi$ , and $\frac{4}{7} \pi $ , and radius of its circumscribed circle $R=1$ . | {
"answer": "\\frac{\\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's consider two positive real numbers $a$ and $b$, where an operation $a \, \blacktriangle \, b$ is defined such that $(ab) \, \blacktriangle \, b = a(b \, \blacktriangle \, b)$ and $(a \, \blacktriangle \, 1) \, \blacktriangle \, a = a \, \blacktriangle \, 1$ for all $a,b>0$. Additionally, it is given that $1 \, \blacktriangle \, 1 = 2$. Find the value of $23 \, \blacktriangle \, 45$. | {
"answer": "2070",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit positive integers $y$ satisfy $5678y + 123 \equiv 890 \pmod{29}$? | {
"answer": "310",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $f(x)$ and $g(x)$ are functions satisfying $f(g(x)) = x^2$ and $g(f(x)) = x^4$ for all $x \ge 1.$ If $g(81) = 81,$ compute $[g(9)]^4.$ | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tangent line is drawn to the moving circle $C: x^2 + y^2 - 2ay + a^2 - 2 = 0$ passing through the fixed point $P(2, -1)$. If the point of tangency is $T$, then the minimum length of the line segment $PT$ is \_\_\_\_\_\_. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many times does the digit 9 appear in the list of all integers from 1 to 1000? | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center $D$ and radius four feet is tangent at $E$ to a circle with center $F$, as shown. If point $F$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),8),gray,linewidth(2));
filldraw(circle(4dir(-30),4),white,linewidth(2));
dot((0,0));
dot(4dir(-30));
dot(8dir(-30));
label("$F$",(0,0),NW);
label("$D$",4dir(-30),NE);
label("$E$",8dir(-30),SE);
[/asy] | {
"answer": "48\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive real number $x$ is such that \[
\sqrt[3]{1-x^4} + \sqrt[3]{1+x^4} = 1.
\]Find $x^8.$ | {
"answer": "\\frac{28}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A math test consists of 12 multiple-choice questions, each worth 5 points. It is known that a student is confident in correctly answering 6 of these questions. For another three questions, the student can eliminate one incorrect option. For two questions, the student can eliminate two incorrect options. For the last question, due to a lack of understanding, the student has to guess randomly. Estimate the score of this student in this test. | {
"answer": "41.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determine the average of the first and last numbers. | {
"answer": "5.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the complex plane, the distance between the points corresponding to the complex numbers $-3+i$ and $1-i$ is $\boxed{\text{answer}}$. | {
"answer": "\\sqrt{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a cube $ABCDEFGH$, the coordinates of vertices are set in a conventional cube alignment with $A(0, 0, 0)$, $B(2, 0, 0)$, $C(2, 0, 2)$, $D(0, 0, 2)$, $E(0, 2, 0)$, $F(2, 2, 0)$, $G(2, 2, 2)$, and $H(0, 2, 2)$. Let $M$ and $N$ be the midpoints of the segments $\overline{EB}$ and $\overline{HD}$, respectively. Determine the ratio $S^2$ where $S$ is the ratio of the area of triangle $MNC$ to the total surface area of the cube.
A) $\frac{1}{144}$
B) $\frac{17}{2304}$
C) $\frac{1}{48}$
D) $\frac{1}{96}$ | {
"answer": "\\frac{17}{2304}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence {a<sub>n</sub>} is an arithmetic sequence, if $\frac {a_{11}}{a_{10}}$ < -1, and its first n terms sum S<sub>n</sub> has a maximum value, determine the maximum value of n that makes S<sub>n</sub> > 0. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A volunteer organizes a spring sports event and wants to form a vibrant and well-trained volunteer team. They plan to randomly select 3 people from 4 male volunteers and 3 female volunteers to serve as the team leader. The probability of having at least one female volunteer as the team leader is ____; given the condition that "at least one male volunteer is selected from the 3 people drawn," the probability of "all 3 people drawn are male volunteers" is ____. | {
"answer": "\\frac{2}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, $(4,-2)$, and $(3,4)$. | {
"answer": "\\frac{11}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $$\sin\alpha= \frac {4}{7} \sqrt {3}$$ and $$\cos(\alpha+\beta)=- \frac {11}{14}$$, and $\alpha$, $\beta$ are acute angles, then $\beta= \_\_\_\_\_\_$. | {
"answer": "\\frac {\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiaoming has several 1-yuan, 2-yuan, and 5-yuan banknotes. He wants to use no more than 10 banknotes to buy a kite priced at 18 yuan, and he must use at least two different denominations. How many different payment methods are possible? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with its center at point $M$ on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) is tangent to the $x$-axis exactly at one of the foci $F$ of the hyperbola, and intersects the $y$-axis at points $P$ and $Q$. If $\triangle MPQ$ is an equilateral triangle, calculate the eccentricity of the hyperbola. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hairdresser moved from Vienna to Debrecen to continue his trade. Over the course of 3 years, he became impoverished despite having some money originally. In the first year, he had to spend half of his money. In the second year, he spent a third of what he initially took with him. In the third year, he spent 200 forints, leaving him with only 50 forints for returning. How many forints did he have when he moved, and how much did he spend each year? | {
"answer": "1500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Filling the gas tank of a small car cost, in updated values, $\mathrm{R} \$ 29.90$ in 1972 and $\mathrm{R} \$ 149.70$ in 1992. Which of the following values best approximates the percentage increase in the price of gasoline during this 20-year period?
(a) $20 \%$
(b) $125 \%$
(c) $300 \%$
(d) $400 \%$
(e) $500 \%$ | {
"answer": "400\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\(a, b, c > 0\\), the minimum value of \\(\frac{a^{2} + b^{2} + c^{2}}{ab + 2bc}\\) is \_\_\_\_\_\_. | {
"answer": "\\frac{2 \\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Read the following material before solving the problem: In mathematics, there are numbers with square roots that contain another square root, which can be simplified by using the complete square formula and the properties of quadratic surds. For example, $\sqrt{3+2\sqrt{2}}=\sqrt{3+2×1×\sqrt{2}}=\sqrt{{1^2}+2×1×\sqrt{2}+{{({\sqrt{2}})}^2}}=\sqrt{{{({1+\sqrt{2}})}^2}}=|1+\sqrt{2}|=1+\sqrt{2}$.
Solve the following problems:
$(1) \sqrt{7+4\sqrt{3}}$;
$(2) \sqrt{9-4\sqrt{5}}$. | {
"answer": "\\sqrt{5}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to both the radius lines $OA$ and $OB$, and the arc $AB$. Determine the radius of the inscribed circle in centimeters. Express your answer in simplest radical form. | {
"answer": "6\\sqrt{2} - 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three players are playing table tennis. The player who loses a game gives up their spot to the player who did not participate in that game. In the end, it turns out that the first player played 10 games, and the second player played 21 games. How many games did the third player play? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Read the following text and answer the questions:<br/>$\because \sqrt{1}＜\sqrt{2}＜\sqrt{4}$, which means $1＜\sqrt{2}＜2$,<br/>$\therefore$ The integer part of $\sqrt{2}$ is $1$, and the decimal part is $\sqrt{2}-1$.<br/>Please answer:<br/>$(1)$ The integer part of $\sqrt{33}$ is ______, and the decimal part is ______;<br/>$(2)$ If the decimal part of $\sqrt{143}$ is $a$, and the integer part of $\sqrt{43}$ is $b$, find the value of $a+|2b-\sqrt{143}|$;<br/>$(3)$ Given: $10+\sqrt{5}=2x+y$, where $x$ is an integer, and $0 \lt y \lt 1$, find the opposite of $x-y$. | {
"answer": "\\sqrt{5} - 8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(3-\frac{2}{\cos x}\right)^{\operatorname{cosec}^{2} x}
$$ | {
"answer": "e^{-1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\(x \geqslant 0\\), \\(y \geqslant 0\\), \\(x\\), \\(y \in \mathbb{R}\\), and \\(x+y=2\\), find the minimum value of \\( \dfrac {(x+1)^{2}+3}{x+2}+ \dfrac {y^{2}}{y+1}\\). | {
"answer": "\\dfrac {14}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.
$(1)$ If $2a\sin B = \sqrt{3}b$, find the measure of angle $A$.
$(2)$ If the altitude on side $BC$ is equal to $\frac{a}{2}$, find the maximum value of $\frac{c}{b} + \frac{b}{c}$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$ , and $C$ . What is the magnitude of $\angle B AC$ in degrees? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\mathbf{v}\) be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of \(\|\mathbf{v}\|\). | {
"answer": "10 - 2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ meets side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, then what is $RS$? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$ . What is the minimum possible value of $a + b$ ? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define two binary operations on real numbers where $a \otimes b = \frac{a+b}{a-b}$ and $b \oplus a = \frac{b-a}{b+a}$. Compute the value of $(8\otimes 6) \oplus 2$.
A) $\frac{5}{9}$
B) $\frac{7}{9}$
C) $\frac{12}{9}$
D) $\frac{1}{9}$
E) $\frac{14}{9}$ | {
"answer": "\\frac{5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\alpha= \frac{1}{2}+\cos\alpha$, and $\alpha\in(0, \frac{\pi}{2})$, then $\sin2\alpha= \_\_\_\_\_\_$, $\cos2\alpha= \_\_\_\_\_\_$. | {
"answer": "-\\frac{\\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area of the triangle bounded by the axes and the curve $y = (x-5)^2 (x+3)$. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the areas of two squares is $\frac{300}{147}$. Find the simplified form of the ratio of their side lengths, expressed as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Additionally, if the perimeter of the larger square is 60 units, determine the side length of the smaller square. | {
"answer": "10.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, given the curve $C: \rho = 2\cos \theta$, the line $l: \left\{ \begin{array}{l} x = \sqrt{3}t \\ y = -1 + t \end{array} \right.$ (where $t$ is a parameter), and the line $l$ intersects the curve $C$ at points $A$ and $B$.
$(1)$ Find the rectangular coordinate equation of curve $C$ and the general equation of line $l$.
$(2)$ Given the polar coordinates of point $P$ as $({1, \frac{3\pi}{2}})$, find the value of $\left(|PA|+1\right)\left(|PB|+1\right)$. | {
"answer": "3 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\cos\alpha =\frac{\sqrt{5}}{5}$ and $\sin\beta =\frac{3\sqrt{10}}{10}$, with $0 < \alpha$, $\beta < \frac{\pi}{2}$, determine the value of $\alpha +\beta$. | {
"answer": "\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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