problem stringlengths 10 5.15k | answer dict |
|---|---|
Express the given value of $22$ nanometers in scientific notation. | {
"answer": "2.2\\times 10^{-8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $2\sin^2 \frac{A+B}{2}-\cos 2C=1$, and the radius of the circumcircle $R=2$.
$(1)$ Find $C$;
$(2)$ Find the maximum value of $S_{\triangle ABC}$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In base \( R_1 \), the fractional expansion of \( F_1 \) is \( 0.373737 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.737373 \cdots \). In base \( R_2 \), the fractional expansion of \( F_1 \) is \( 0.252525 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.525252 \cdots \). What is the sum of \( R_1 \) and \( R_2 \) (both expressed in decimal)? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$? | {
"answer": "\\frac{7}{33}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of a regular triangular prism is $8$, the base edge length that minimizes the surface area of the prism is __________. | {
"answer": "2\\sqrt[3]{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular table measures $12'$ in length and $9'$ in width and is currently placed against one side of a rectangular room. The owners desire to move the table to lay diagonally in the room. Determine the minimum length of the shorter side of the room, denoted as $S$, in feet, for the table to fit without tilting or taking it apart. | {
"answer": "15'",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular prism has dimensions 10 inches by 3 inches by 30 inches. If a cube has the same volume as this prism, what is the surface area of the cube, in square inches? | {
"answer": "6 \\times 900^{2/3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\((a+b-c)(a+b+c)=3ab\\) and \\(c=4\\), the maximum area of \\(\Delta ABC\\) is \_\_\_\_\_\_\_. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$ , $y$ , $z$ are nonnegative real numbers such that $x+y+z=1$ . | {
"answer": "\\frac{7}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $XOY$, there is a line $l:\begin{cases} & x=t \\ & y=-\sqrt{3}t \\ \end{cases}(t$ is a parameter$)$, and a curve ${C_{1:}}\begin{cases} & x=\cos \theta \\ & y=1+\sin \theta \\ \end{cases}(\theta$ is a parameter$)$. Establish a polar coordinate system with the origin $O$ of this rectangular coordinate system as the pole and the non-negative half-axis of the $X$-axis as the polar axis. The equation of the curve ${C_{2}}$ is $\rho=4\sin (\theta -\frac{\pi }{6})$.
1. Find the polar coordinate equation of the curve ${C_{1}}$ and the rectangular coordinate equation of the curve ${C_{2}}$.
2. Suppose the line $l$ intersects the curve ${C_{1}}$ at points $O$ and $A$, and intersects the curve ${C_{2}}$ at points $O$ and $B$. Find the length of $|AB|$. | {
"answer": "4- \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence of even counting numbers starting from $0$, find the sum of the first $1500$ terms. Then, given the sequence of odd counting numbers, find the sum of the first $1500$ terms, and calculate their difference. | {
"answer": "1500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M = 123456789101112\dots4950$ be the $95$-digit number formed by writing integers from $1$ to $50$ in order, one after the other. What is the remainder when $M$ is divided by $45$? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 8.$ Find the maximum value of
\[\sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}.\] | {
"answer": "3\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? | {
"answer": "115",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = |c| = 1$ and $\arg(c) = \arg(a) + \arg(b)$. Suppose that
\[ a z^2 + b z + c = 0. \]
Find the largest possible value of $|z|$. | {
"answer": "\\frac{1 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of triangle $MNP$ given below:
[asy]
unitsize(1inch);
pair M,N,P;
M = (0,0);
N= (sqrt(3),0);
P = (0,1);
draw (M--N--P--M, linewidth(0.9));
draw(rightanglemark(N,M,P,3));
label("$M$",M,S);
label("$N$",N,S);
label("$P$",P,N);
label("$15$",(N+P)/2,NE);
label("$60^\circ$",(0,0.75),E);
[/asy] | {
"answer": "28.125\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n = \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. For how many positive integers $b_1 \le 1500$ is it true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$? | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a circle with radius $4$, and there are numerous line segments of length $6$ that are tangent to the circle at their midpoints. Compute the area of the region consisting of all such line segments.
A) $8\pi$
B) $7\pi$
C) $9\pi$
D) $10\pi$ | {
"answer": "9\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be a right triangle at $A$ with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 25$, $BC = 34$, and $TX^2 + TY^2 + XY^2 = 1975$. Find $XY^2$. | {
"answer": "987.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a sunny day, 3000 people, including children, boarded a cruise ship. Two-fifths of the people were women, and a third were men. If 25% of the women and 15% of the men were wearing sunglasses, and there were also 180 children on board with 10% wearing sunglasses, how many people in total were wearing sunglasses? | {
"answer": "530",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular quadrilateral pyramid $S-ABCD$, with a base side length of $2$ and a volume of $\frac{{4\sqrt{3}}}{3}$, the length of the lateral edge of this quadrilateral pyramid is ______. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(\frac{a}{b} = 5\), \(\frac{b}{c} = \frac{1}{4}\), and \(\frac{c^2}{d} = 16\), then what is \(\frac{d}{a}\)? | {
"answer": "\\frac{1}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Read the following material: Expressing a fraction as the sum of two fractions is called expressing the fraction as "partial fractions."<br/>Example: Express the fraction $\frac{{1-3x}}{{{x^2}-1}}$ as partial fractions. Solution: Let $\frac{{1-3x}}{{{x^2}-1}}=\frac{M}{{x+1}}+\frac{N}{{x-1}}$, cross multiply on the right side of the equation, we get $\frac{{M(x-1)+N(x+1)}}{{(x+1)(x-1)}}=\frac{{(M+N)x+(N-M)}}{{{x^2}-1}}$. According to the question, we have $\left\{\begin{array}{l}M+N=3\\ N-M=1\end{array}\right.$, solving this system gives $\left\{\begin{array}{l}M=-2\\ N=-1\end{array}\right.$, so $\frac{{1-3x}}{{{x^2}-1}}=\frac{{-2}}{{x+1}}+\frac{{-1}}{{x-1}}$. Please use the method learned above to solve the following problems:<br/>$(1)$ Express the fraction $\frac{{2n+1}}{{{n^2}+n}}$ as partial fractions;<br/>$(2)$ Following the pattern in (1), find the value of $\frac{3}{{1×2}}-\frac{5}{{2×3}}+\frac{7}{{3×4}}-\frac{9}{{4×5}}+⋯+\frac{{39}}{{19×20}}-\frac{{41}}{{20×21}}$. | {
"answer": "\\frac{20}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dima calculated the factorials of all natural numbers from 80 to 99, found the numbers that are reciprocals to them, and printed the resulting decimal fractions on 20 infinite ribbons (for example, the last ribbon had printed the number $\frac{1}{99!}=0, \underbrace{00 \ldots 00}_{155 \text { zeros! }} 10715$.. ). Sasha wants to cut out a piece from one ribbon that contains $N$ consecutive digits without a comma. What is the maximum value of $N$ for which Dima will not be able to determine from this piece which ribbon Sasha spoiled? | {
"answer": "155",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the power function $y=f(x)$ passes through the point $\left( -2,-\frac{1}{8} \right)$, find the value(s) of $x$ that satisfy $f(x)=27$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. The area of the triangle is given by $S= \frac{a^{2}+b^{2}-c^{2}}{4}$ and $\sin A= \frac{3}{5}$.
1. Find $\sin B$.
2. If side $c=5$, find the area of $\triangle ABC$, denoted as $S$. | {
"answer": "\\frac{21}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back.
*Proposed by William Yue* | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square $EFGH$ is inscribed in the region bounded by the parabola $y = x^2 - 6x + 5$ and the $x$-axis. Find the area of square $EFGH$.
\[
\text{[asy]}
unitsize(0.8 cm);
real parab (real x) {
return(x^2 - 6*x + 5);
}
pair E, F, G, H;
real x = -1 + sqrt(3);
E = (3 - x,0);
F = (3 + x,0);
G = (3 + x,-2*x);
H = (3 - x,-2*x);
draw(graph(parab,0,6));
draw(E--H--G--F);
draw((0,0)--(6,0));
label("$E$", E, N);
label("$F$", F, N);
label("$G$", G, SE);
label("$H$", H, SW);
\text{[/asy]}
\] | {
"answer": "24 - 8\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\frac{S_8}{S_4}$. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $b$ and $c$ be real numbers. If the polynomial $x^3 + bx^2 + cx + d$ has exactly one real root and $d = c + b + 1$, find the value of the product of all possible values of $c$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | {
"answer": "\\frac{335}{2011}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y = \lg(-x^2 + x + 2)$ with domain $A$, find the range $B$ for the exponential function $y = a^x$ $(a>0$ and $a \neq 1)$ where $x \in A$.
1. If $a=2$, determine $A \cup B$;
2. If $A \cap B = (\frac{1}{2}, 2)$, find the value of $a$. | {
"answer": "a = 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 2) & \quad \text{ otherwise}. \end{aligned} \right.\] Compute $g(50).$ | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the scores of a math exam follow a normal distribution N(102, 4²), the percentage of scores 114 and above is _______ (Note: P(μ-σ<X≤μ+σ)=0.6826, P(μ-2σ<X≤μ+2σ)=0.9544, P(μ-3σ<X≤μ+3σ)=0.9974). | {
"answer": "0.13\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two strips of width 2 overlap at an angle of 60 degrees inside a rectangle of dimensions 4 units by 3 units. Find the area of the overlap, considering that the angle is measured from the horizontal line of the rectangle.
A) $\frac{2\sqrt{3}}{3}$
B) $\frac{8\sqrt{3}}{9}$
C) $\frac{4\sqrt{3}}{3}$
D) $3\sqrt{3}$
E) $\frac{12}{\sqrt{3}}$ | {
"answer": "\\frac{4\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence? | {
"answer": "236",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f^{-1}(g(x))=x^4-4$ and $g$ has an inverse, find $g^{-1}(f(15))$. | {
"answer": "\\sqrt[4]{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tetrahedron has a triangular base with sides all equal to 2, and each of its three lateral faces are squares. A smaller tetrahedron is placed within the larger one so that its base is parallel to the base of the larger tetrahedron and its vertices touch the midpoints of the lateral faces of the larger tetrahedron. Calculate the volume of this smaller tetrahedron. | {
"answer": "\\frac{\\sqrt{2}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The solid \( T \) consists of all points \((x,y,z)\) such that \( |x| + |y| \le 2 \), \( |x| + |z| \le 1 \), and \( |z| + |y| \le 1 \). Find the volume of \( T \). | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\tan \beta= \frac{4}{3}$, $\sin (\alpha+\beta)= \frac{5}{13}$, and both $\alpha$ and $\beta$ are within $(0, \pi)$, find the value of $\sin \alpha$. | {
"answer": "\\frac{63}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $y = x^2 + 2x - 1$ attains its minimum value on the interval $[0, 3]$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many 5-digit numbers beginning with $2$ are there that have exactly three identical digits which are not $2$? | {
"answer": "324",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle \(ABCD\) has length 9 and width 5. Diagonal \(AC\) is divided into 5 equal parts at \(W, X, Y\), and \(Z\). Determine the area of the shaded region. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, $\left(a+c\right)\sin A=\sin A+\sin C$, $c^{2}+c=b^{2}-1$. Find:<br/>
$(1)$ $B$;<br/>
$(2)$ Given $D$ is the midpoint of $AC$, $BD=\frac{\sqrt{3}}{2}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the quadratic equation $x^2 + ax + b = 0$ with roots $r_1$ and $r_2$, find an equation where the roots are three times those of $x^2 + cx + a = 0$ and provide the value of $b/c$. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, angles $\alpha$ and $\beta$ both start from $Ox$, and their terminal sides are symmetric about the $y$-axis. If the terminal side of angle $\alpha$ passes through the point $(3,4)$, then $\tan (\alpha-\beta)=$ ______. | {
"answer": "- \\dfrac {24}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the graph of the linear function $y=kx+b$ ($k \neq 0$) intersects the positive half-axes of the $x$-axis and $y$-axis at points $A$ and $B$, respectively, and the area of $\triangle OAB$ is equal to $|OA|+|OB|+3$.
(1) Express $k$ in terms of $b$;
(2) Find the minimum value of the area of $\triangle OAB$. | {
"answer": "7+2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Ming forgot the last two digits of his WeChat login password. He only remembers that the last digit is one of the letters \\(A\\), \\(a\\), \\(B\\), or \\(b\\), and the other digit is one of the numbers \\(4\\), \\(5\\), or \\(6\\). The probability that Xiao Ming can successfully log in with one attempt is \_\_\_\_\_\_. | {
"answer": "\\dfrac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that points A and B are on the x-axis, and the two circles with centers at A and B intersect at points M $(3a-b, 5)$ and N $(9, 2a+3b)$, find the value of $a^{b}$. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A strip of size $1 \times 10$ is divided into unit squares. The numbers $1, 2, \ldots, 10$ are written in these squares. First, the number 1 is written in one of the squares, then the number 2 is written in one of the neighboring squares, then the number 3 is written in one of the squares neighboring those already occupied, and so on (the choice of the first square and the choice of neighbor at each step are arbitrary). In how many ways can this be done? | {
"answer": "512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $50x$ is divisible by 100 and $kx$ is not divisible by 100 for all $k=1,2,\cdots, 49$ Find number of solutions for $x$ when $x$ takes values $1,2,\cdots 100$ .
[list=1]
[*] 20
[*] 25
[*] 15
[*] 50
[/list] | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the region defined by the equation $x^2 + y^2 - 10 = 4y - 10x + 4$? | {
"answer": "43\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If two lines \( l \) and \( m \) have equations \( y = -2x + 8 \), and \( y = -3x + 9 \), what is the probability that a point randomly selected in the 1st quadrant and below \( l \) will fall between \( l \) and \( m \)? Express your answer as a decimal to the nearest hundredth. | {
"answer": "0.16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers \( p, q, r \) form an equilateral triangle with a side length of 24 in the complex plane. If \( |p + q + r| = 48 \), find \( |pq + pr + qr| \). | {
"answer": "768",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of line $l$ is $y=x+4$, and the parametric equation of circle $C$ is $\begin{cases} x=2\cos \theta \\ y=2+2\sin \theta \end{cases}$ (where $\theta$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis. Establish a polar coordinate system.
- (I) Find the polar coordinates of the intersection points of line $l$ and circle $C$.
- (II) If $P$ is a moving point on circle $C$, find the maximum value of the distance $d$ from $P$ to line $l$. | {
"answer": "\\sqrt{2}+2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the sides \( AB \) and \( AD \) of the square \( ABCD \), points \( E \) and \( F \) are marked such that \( BE : EA = AF : FD = 2022 : 2023 \). The segments \( EC \) and \( FC \) intersect the diagonal of the square \( BD \) at points \( G \) and \( H \) respectively. Find \( \frac{GH}{BD} \). | {
"answer": "\\frac{12271519}{36814556}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bob buys four burgers and three sodas for $\$5.00$, and Carol buys three burgers and four sodas for $\$5.40$. How many cents does a soda cost? | {
"answer": "94",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list? | {
"answer": "147",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles with centers $A$ and $B$ intersect at points $X$ and $Y$ . The minor arc $\angle{XY}=120$ degrees with respect to circle $A$ , and $\angle{XY}=60$ degrees with respect to circle $B$ . If $XY=2$ , find the area shared by the two circles. | {
"answer": "\\frac{10\\pi - 12\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a diagram, the grid is composed of 1x1 squares. What is the area of the shaded region if the overall width of the grid is 15 units and its height is 5 units? Some parts are shaded in the following manner: A horizontal stretch from the left edge (6 units wide) that expands 3 units upward from the bottom, and another stretch that begins 6 units from the left and lasts for 9 units horizontally, extending from the 3 units height to the top of the grid. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1\leq x < y \leq 26$ there is exactly one passenger that goes from the $x$ -th stop to the $y$ -th one. If every passenger wants to take a seat during his journey, find the minimum number of seats that must be available on the train.
*Proposed by **FedeX333X*** | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A local community group sells 180 event tickets for a total of $2652. Some tickets are sold at full price, while others are sold at a discounted rate of half price. Determine the total revenue generated from the full-price tickets.
A) $960
B) $984
C) $1008
D) $1032 | {
"answer": "984",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\tan A= \frac {3}{4}$ and $\tan (A-B)=- \frac {1}{3}$, find the value of $\tan C$. | {
"answer": "\\frac {79}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $O$ is the coordinate origin, the complex numbers $z_1$ and $z_2$ correspond to the vectors $\overrightarrow{OZ_1}$ and $\overrightarrow{OZ_2}$, respectively. $\bar{z_1}$ is the complex conjugate of $z_1$. The vectors are represented as $\overrightarrow{OZ_1} = (10 - a^2, \frac{1}{a + 5})$ and $\overrightarrow{OZ_2} = (2a - 5, 2 - a)$, where $a \in \mathbb{R}$, and $(z_2 - z_1)$ is a purely imaginary number.
(1) Determine the quadrant in which the point corresponding to the complex number $\bar{z_1}$ lies in the complex plane.
(2) Calculate $|z_1 \cdot z_2|$. | {
"answer": "\\frac{\\sqrt{130}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice and Bob each arrive at a gathering at a random time between 12:00 noon and 1:00 PM. If Alice arrives after Bob, what is the probability that Bob arrived before 12:45 PM? | {
"answer": "0.5625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If point P is one of the intersections of the hyperbola with foci A(-√10,0), B(√10,0) and a real axis length of 2√2, and the circle x^2 + y^2 = 10, calculate the value of |PA| + |PB|. | {
"answer": "6\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a square arrangement of tiles comprising 12 black and 23 white square tiles. A border consisting of an alternating pattern of black and white tiles is added around the square. The border follows the sequence: black, white, black, white, and so on. What is the ratio of black tiles to white tiles in the newly extended pattern?
A) $\frac{25}{37}$
B) $\frac{26}{36}$
C) $\frac{26}{37}$
D) $\frac{27}{37}$ | {
"answer": "\\frac{26}{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line passing through the point P(3/2, 1/2) intersects the ellipse x^2/6 + y^2/2 = 1 at points A and B, satisfying PA + PB = 0. If M is any point on the line AB and O is the origin, find the minimum value of |OM|. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ratio of the legs of a right triangle is $3: 4$, determine the ratio of the corresponding segments of the hypotenuse created by dropping a perpendicular from the opposite vertex of the right angle onto the hypotenuse. | {
"answer": "\\frac{16}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the diameter of the circle inscribed in triangle $DEF$ if $DE = 13,$ $DF = 8,$ and $EF = 15$? Express your answer in simplest radical form. | {
"answer": "\\frac{10\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shape was cut out from a regular hexagon as shown in the picture. The marked points on both the perimeter and inside the hexagon divide the respective line segments into quarters.
What is the ratio of the areas of the original hexagon to the cut-out shape? | {
"answer": "4:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , and $CA=15$ . Points $P$ , $Q$ , and $R$ are chosen on segments $BC$ , $CA$ , and $AB$ , respectively, such that triangles $AQR$ , $BPR$ , $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $PQR$ . What is the perimeter of $PQR$ ?
*2021 CCA Math Bonanza Individual Round #2* | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle of radius 3, find the area of the region consisting of all line segments of length 6 that are tangent to the circle at their midpoints.
A) $3\pi$
B) $6\pi$
C) $9\pi$
D) $12\pi$
E) $15\pi$ | {
"answer": "9\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\sin ^{2}A+\sin A\sin C+\sin ^{2}C+\cos ^{2}B=1$.
$(1)$ Find the measure of angle $B$;
$(2)$ If $a=5$, $b=7$, find $\sin C$. | {
"answer": "\\frac{3\\sqrt{3}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a certain crosswalk, the pedestrian signal alternates between red and green lights, with the red light lasting for $40s$. If a pedestrian arrives at the crosswalk and encounters a red light, the probability that they need to wait at least $15s$ for the green light to appear is ______. | {
"answer": "\\dfrac{5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
China was the first country in the world to use negative numbers. Li Heng, in the book "Fa Jing" written during the Warring States period, already used negative numbers. If the year 500 BC is written as $-500$ years, then the year 2024 AD should be written as ______ years. | {
"answer": "+2024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\), \(b\), and \(c\) be real numbers such that \(9a^2 + 4b^2 + 25c^2 = 4\). Find the maximum value of
\[6a + 3b + 10c.\] | {
"answer": "\\sqrt{41}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least possible sum of two bases, $c$ and $d$, such that the numeral $29$ in base $c$ represents the same number as $92$ in base $d$, where $c$ and $d$ are positive integers. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sqrt{3}\sin x\cos x-{\cos }^2x$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ If $f(x)=-1$, find the value of $\cos \left(\dfrac{2\pi }{3}-2x\right)$. | {
"answer": "-\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The South China tiger is a first-class protected animal in our country. To save the species from the brink of extinction, the country has established a South China tiger breeding base. Due to scientific artificial cultivation, the relationship between the number of South China tigers $y$ (individuals) and the breeding time $x$ (years) can be approximately described by $y=a\log_{2}(x+1)$. If there were 20 tigers in the first year of breeding (2012), then by 2015, it is predicted that there will be approximately how many tigers? | {
"answer": "46",
"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. Given vectors $\overrightarrow{m}=(\sin B+\sin C,\sin A+\sin B)$, $\overrightarrow{n}=(\sin B-\sin C,\sin A)$, and $\overrightarrow{m}\perp \overrightarrow{n}$.
(1) Find the measure of angle $C$;
(2) If $\triangle ABC$ is an isosceles triangle and its circumcircle is a unit circle, find the perimeter $L$ of $\triangle ABC$. | {
"answer": "2+\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 19 candy boxes arranged in a row, with the middle box containing $a$ candies. Moving to the right, each box contains $m$ more candies than the previous one; moving to the left, each box contains $n$ more candies than the previous one ($a$, $m$, and $n$ are all positive integers). If the total number of candies is 2010, then the sum of all possible values of $a$ is. | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Equilateral triangle $ABC$ has a side length of $\sqrt{144}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{12}$. Additionally, $BD_1$ and $BD_2$ are placed such that $\angle ABD_1 = 30^\circ$ and $\angle ABD_2 = 150^\circ$. Determine the sum $\sum_{k=1}^4 (CE_k)^2$. | {
"answer": "576",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate \(7 \cdot 9\frac{2}{5}\). | {
"answer": "65\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance on the map is 3.6 cm, and the actual distance is 1.2 mm. What is the scale of this map? | {
"answer": "30:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(2\sin x,-\cos x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x,2\cos x)$, and function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$.
(I) Find the smallest positive period of function $f(x)$, and find the range of $f(x)$ when $x\in\left[\dfrac{\pi}{12},\dfrac{2\pi}{3}\right]$;
(II) Translate the graph of function $f(x)$ to the left by $\dfrac{\pi}{3}$ unit to obtain the graph of function $g(x)$. In triangle $ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$, respectively. If $g\left(\dfrac{A}{2}\right)=1$, $a=2$, and $b+c=4$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001$. Let $y_{k}=\frac{1}{k} \sum_{i=1}^{k} x_{i}$ for $k=1,2, \cdots, 2001$. Find the maximum possible value of $\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|$. | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that 60% of all students in Ms. Hanson's class answered "Yes" to the question "Do you love science" at the beginning of the school year, 40% answered "No", 80% answered "Yes" and 20% answered "No" at the end of the school year, calculate the difference between the maximum and the minimum possible values of y%, the percentage of students that gave a different answer at the beginning and end of the school year. | {
"answer": "40\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$ , find $100a+b$ .
*Proposed by Michael Tang* | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be an interior point of triangle $ABC$ . Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$ taking into consideration all possible choices of triangle $ABC$ and of point $P$ .
by Elton Bojaxhiu, Albania | {
"answer": "\\frac{2}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tetrahedron $ABCD$, with $AD$ perpendicular to plane $BCD$, $BC$ perpendicular to $CD$, $AD=2$, $BD=4$, calculate the surface area of the circumscribed sphere of tetrahedron $ABCD$. | {
"answer": "20\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$? | {
"answer": "\\frac{35}{132}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a 5-digit number $\overline{x a x a x}$ is divisible by 15, calculate the sum of all such numbers. | {
"answer": "220200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a point $(-4,a)$ lies on the terminal side of an angle of $600^{\circ}$, determine the value of $a$. | {
"answer": "-4 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangular pyramid $P-ABC$, $PA \perp$ the base $ABC$, $AB=1$, $AC=2$, $\angle BAC=60^{\circ}$, the volume is $\frac{\sqrt{3}}{3}$, then the volume of the circumscribed sphere of the triangular pyramid is $\_\_\_\_\_\_\_\_\_\_.$ | {
"answer": "\\frac{8 \\sqrt{2}}{3} \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sqrt{3}\cos x\sin x - \frac{1}{2}\cos 2x$.
(1) Find the smallest positive period of $f(x)$.
(2) Find the maximum and minimum values of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ and the corresponding values of $x$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of cos $$\frac {π}{11}$$cos $$\frac {2π}{11}$$cos $$\frac {3π}{11}$$cos $$\frac {4π}{11}$$cos $$\frac {5π}{11}$$\=\_\_\_\_\_\_． | {
"answer": "\\frac {1}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Within a triangular piece of paper, there are 100 points, along with the 3 vertices of the triangle, making it a total of 103 points, and no three of these points are collinear. If these points are used as vertices to create triangles, and the paper is cut into small triangles, then the number of such small triangles is ____. | {
"answer": "201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If point A $(3,1)$ lies on the line $mx+ny+1=0$, where $mn>0$, then the maximum value of $\frac {3}{m}+ \frac {1}{n}$ is \_\_\_\_\_. | {
"answer": "-16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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