problem stringlengths 10 5.15k | answer dict |
|---|---|
Let $\triangle ABC$ be an acute isosceles triangle with circumcircle $\omega$. The tangents to $\omega$ at vertices $B$ and $C$ intersect at point $T$. Let $Z$ be the projection of $T$ onto $BC$. Assume $BT = CT = 20$, $BC = 24$, and $TZ^2 + 2BZ \cdot CZ = 478$. Find $BZ \cdot CZ$. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a city, from 7:00 to 8:00, is a peak traffic period, during which all vehicles travel at half their normal speed. Every morning at 6:50, two people, A and B, start from points A and B respectively and travel towards each other. They meet at a point 24 kilometers from point A. If person A departs 20 minutes later, they meet exactly at the midpoint of the route between A and B. If person B departs 20 minutes earlier, they meet at a point 20 kilometers from point A. What is the distance between points A and B in kilometers? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express 2.175 billion yuan in scientific notation. | {
"answer": "2.175 \\times 10^9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? | {
"answer": "\\frac{6}{2 + 3\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with side length $2$ is sliced by a plane that passes through a vertex $A$, the midpoint $M$ of an adjacent edge, and the midpoint $P$ of the face diagonal of the top face, not containing vertex $A$. Find the area of the triangle $AMP$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle with center $O$, the measure of $\angle TIQ$ is $45^\circ$ and the radius $OT$ is 12 cm. Find the number of centimeters in the length of arc $TQ$. Express your answer in terms of $\pi$. | {
"answer": "6\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three forces in space, $\overrightarrow {F_{1}}$, $\overrightarrow {F_{2}}$, and $\overrightarrow {F_{3}}$, each with a magnitude of 2, and the angle between any two of them is 60°, the magnitude of their resultant force $\overrightarrow {F}$ is ______. | {
"answer": "2 \\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 630 = 0$ has integral solutions? | {
"answer": "160",
"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": "192",
"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. It is known that $\frac {2a+b}{c}= \frac {\cos (A+C)}{\cos C}$.
(I) Find the magnitude of angle $C$,
(II) If $c=2$, find the maximum area of $\triangle ABC$. | {
"answer": "\\frac { \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the variables $a$ and $b$ satisfy the equation $b=-\frac{1}{2}a^2 + 3\ln{a} (a > 0)$, and that point $Q(m, n)$ lies on the line $y = 2x + \frac{1}{2}$, find the minimum value of $(a - m)^2 + (b - n)^2$. | {
"answer": "\\frac{9}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$ , where $n$ is an integer that is not a multiple of $3$ . What is the largest integer that is a divisor of every number in $S$ ? | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A particle is placed on the curve $y = x^3 - 3x^2 - x + 3$ at a point $P$ whose $y$-coordinate is $5$. It is allowed to roll along the curve until it reaches the nearest point $Q$ whose $y$-coordinate is $-2$. Compute the horizontal distance traveled by the particle.
A) $|\sqrt{6} - \sqrt{3}|$
B) $\sqrt{3}$
C) $\sqrt{6}$
D) $|1 - \sqrt{3}|$ | {
"answer": "|\\sqrt{6} - \\sqrt{3}|",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equations $\left(625\right)^{0.24}$ and $\left(625\right)^{0.06}$, find the value of their product. | {
"answer": "5^{6/5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We place planes through each edge and the midpoint of the edge opposite to it in a tetrahedron. Into how many parts do these planes divide the tetrahedron, and what are the volumes of these parts? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, with $g(x) \neq 0$, $f(x)g'(x) > f'(x)g(x)$, and $f(x) = a^{x}g(x)$ ($a > 0$ and $a \neq 1$), $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$. For the finite sequence $\frac{f(n)}{g(n)} = (n = 1, 2, \ldots, 0)$, find the probability that the sum of the first $k$ terms is greater than $\frac{15}{16}$ for any positive integer $k$ ($1 \leq k \leq 10$). | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer $m$ such that the following is true?
*Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$* \[\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm e. ~1415\] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\frac{x^8 + 16x^4 + 64 + 4x^2}{x^4 + 8}$ when $x = 3$. | {
"answer": "89 + \\frac{36}{89}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two identical cups, A and B. Cup A is half-filled with pure water, and cup B is fully filled with a 50% alcohol solution. First, half of the alcohol solution from cup B is poured into cup A and mixed thoroughly. Then, half of the alcohol solution in cup A is poured back into cup B. How much of the solution in cup B is alcohol at this point? | {
"answer": "3/7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$, $q$, $r$, $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s$, and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q$. Additionally, $p + q + r + s = 201$. Find the value of $pq + rs$. | {
"answer": "-\\frac{28743}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, if $a^{2} + b^{2} - 6c^{2}$, then the value of $(\cot A + \cot B) \tan C$ is equal to: | {
"answer": "$\\frac{2}{5}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the extension of side $AD$ of rhombus $ABCD$, point $K$ is taken beyond point $D$. The lines $AC$ and $BK$ intersect at point $Q$. It is known that $AK=14$ and that points $A$, $B$, and $Q$ lie on a circle with a radius of 6, the center of which belongs to segment $AA$. Find $BK$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Below is the graph of \( y = a \sin(bx + c) \) for some constants \( a > 0 \), \( b > 0 \), and \( c \). The graph reaches its maximum value at \( 3 \) and completes one full cycle by \( 2\pi \). There is a phase shift where the maximum first occurs at \( \pi/6 \). Find the values of \( a \), \( b \), and \( c \). | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mrs. Crabapple now teaches two different classes of British Literature. Her first class has 12 students and meets three times a week, while her second class has 9 students and meets twice a week. How many different sequences of crabapple recipients are possible in a week for both classes combined? | {
"answer": "139,968",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terms of the geometric sequence $\{a_n\}$ are positive, and the common ratio is $q$, if $q^2 = 4$, then $$\frac {a_{3}+a_{4}}{a_{4}+a_{5}}$$ equals \_\_\_\_\_\_. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $c$ be a real number randomly selected from the interval $[-20,20]$. Then, $p$ and $q$ are two relatively prime positive integers such that $\frac{p}{q}$ is the probability that the equation $x^4 + 36c^2 = (9c^2 - 15c)x^2$ has at least two distinct real solutions. Find the value of $p + q$. | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \\(x,y\\) satisfy the constraint conditions \\(\begin{cases} & x+y-2\geqslant 0 \\\\ & x-y+1\geqslant 0 \\\\ & x\leqslant 3 \end{cases}\\). If the minimum value of \\(z=mx+y\\) is \\(-3\\), then the value of \\(m\\) is . | {
"answer": "- \\dfrac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
- $D(1) = 0$ ;
- $D(p)=1$ for all primes $p$ ;
- $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$ .
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$ . | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.
The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.
a) Find the probability that under this condition, event \( B \) did not occur at all.
b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \). | {
"answer": "\\frac{16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos \alpha = \dfrac{1}{7}$, $\cos (\alpha - \beta) = \dfrac{13}{14}$, and $0 < \beta < \alpha < \dfrac{\pi}{2}$.
1. Find the value of $\tan 2\alpha$.
2. Find the value of $\cos \beta$. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, $c$, $a+b-c$, $a+c-b$, $b+c-a$, $a+b+c$ be seven distinct prime numbers, and among $a$, $b$, $c$, the sum of two numbers is 800. Let $d$ be the difference between the largest and the smallest of these seven prime numbers. Find the maximum possible value of $d$. | {
"answer": "1594",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the symbols $\diamond$ and $\circ$ represents an operation in the set $\{+,-,\times,\div\}$, and $\frac{15 \diamond 3}{8 \circ 2} = 3$. What is the value of $\frac{9 \diamond 4}{14 \circ 7}$? Express your answer as a common fraction. | {
"answer": "\\frac{13}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a set of integers $\{1, 2, 3, \ldots, 12\}$, eight distinct integers are chosen at random. What is the probability that, among those selected, the third smallest number is $4$? | {
"answer": "\\frac{56}{165}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs share a common element. Each sum $a_i+b_i$ must be distinct and less than or equal to $3009$. Determine the maximum possible value of $k$. | {
"answer": "1504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tim continues the prank into the next week after a successful first week. This time, he starts on Monday with two people willing to do the prank, on Tuesday there are three options, on Wednesday everyone from Monday and Tuesday refuses but there are six new people, on Thursday four of Wednesday's people can't participate but two additional new ones can, and on Friday two people from Monday are again willing to help along with one new person. How many different combinations of people could Tim involve in this prank across the week? | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function $g$ is defined by $g(z) = (3 - 2i) z^2 + \beta z + \delta$ for all complex numbers $z$, where $\beta$ and $\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\beta| + |\delta|$? | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If for any real numbers $u,v$, the inequality ${{(u+5-2v)}^{2}}+{{(u-{{v}^{2}})}^{2}}\geqslant {{t}^{2}}(t > 0)$ always holds, then the maximum value of $t$ is | {
"answer": "2 \\sqrt{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. It is given that \\(a+b=5\\), \\(c=\sqrt{7}\\), and \\(4{{\left( \sin \frac{A+B}{2} \right)}^{2}}-\cos 2C=\frac{7}{2}\\).
\\((1)\\) Find the magnitude of angle \\(C\\);
\\((2)\\) Find the area of \\(\triangle ABC\\). | {
"answer": "\\frac {3 \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Use \((a, b)\) to represent the greatest common divisor of \(a\) and \(b\). Let \(n\) be an integer greater than 2021, and \((63, n+120) = 21\) and \((n+63, 120) = 60\). What is the sum of the digits of the smallest \(n\) that satisfies the above conditions? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$ and $y$ that satisfy the equation $x^{2}+y^{2}-4x+6y+12=0$, find the minimum value of $|2x-y-2|$. | {
"answer": "5-\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < β < \dfrac{π}{2} < α < π$, and $\cos (α- \dfrac{β}{2} )= \dfrac{5}{13} $, $\sin ( \dfrac{α}{2}-β)= \dfrac{3}{5} $. Find the values of:
$(1) \tan (α- \dfrac{β}{2} )$
$(2) \cos ( \dfrac{α+β}{2} )$ | {
"answer": "\\dfrac{56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an equilateral triangle $PQR$ with a side length of 8 units, a process similar to the previous one is applied, but here each time, the triangle is divided into three smaller equilateral triangles by joining the midpoints of its sides, and the middle triangle is shaded each time. If this procedure is repeated 100 times, what is the total area of the shaded triangles?
A) $6\sqrt{3}$
B) $8\sqrt{3}$
C) $10\sqrt{3}$
D) $12\sqrt{3}$
E) $14\sqrt{3}$ | {
"answer": "8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$, $y$ satisfying $x > y > 0$, and $x + y \leqslant 2$, the minimum value of $\dfrac{2}{x+3y}+\dfrac{1}{x-y}$ is | {
"answer": "\\dfrac {3+2 \\sqrt {2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is inscribed in a circle. The number of inches in the perimeter of the square equals the number of square inches in the area of the circumscribed circle. What is the radius, in inches, of the circle? Express your answer in terms of pi. | {
"answer": "\\frac{4\\sqrt{2}}{\\pi}",
"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 that $c = a \cos B + 2b \sin^2 \frac{A}{2}$.
(1) Find angle $A$.
(2) If $b=4$ and the length of median drawn to side $AC$ is $\sqrt{7}$, find $a$. | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the distribution list of the random variable $X$, $P(X=\frac{k}{5})=ak$, where $k=1,2,3,4,5$.
1. Find the value of the constant $a$.
2. Find $P(X\geqslant\frac{3}{5})$.
3. Find $P(\frac{1}{10}<X<\frac{7}{10})$. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Out of the digits 0 through 9, three digits are randomly chosen to form a three-digit number without repeating any digits. What is the probability that this number is not divisible by 3? | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the four numbers $0,1,2,3$, we want to select $3$ digits to form a three-digit number with no repeating digits. What is the probability that this three-digit number is divisible by $3$? | {
"answer": "\\dfrac{5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any real number $a$, let $\left[a\right]$ denote the largest integer not exceeding $a$. For example, $\left[4\right]=4$, $[\sqrt{3}]=1$. Now, for the number $72$, the following operations are performed: $72\stackrel{1st}{→}[\sqrt{72}]=8\stackrel{2nd}{→}[\sqrt{8}]=2\stackrel{3rd}{→}[\sqrt{2}]=1$. In this way, the number $72$ becomes $1$ after $3$ operations. Similarly, among all positive integers that become $2$ after $3$ operations, the largest one is ____. | {
"answer": "6560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a triangle with integral sides is isosceles and has a perimeter of 12, find the area of the triangle. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line \( x = \frac{\pi}{4} \) intercepts the curve \( C: (x - \arcsin a)(x - \arccos a) + (y - \arcsin a)(y + \arccos a) = 0 \) at a chord of length \( d \), find the minimum value of \( d \) as \( a \) varies. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $∀x∈(0,+\infty)$, $ln2x-\frac{ae^{x}}{2}≤lna$, then find the minimum value of $a$. | {
"answer": "\\frac{2}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 20 rooms, with some lights on and some lights off. The people in these rooms want to have their lights in the same state as the majority of the other rooms. Starting with the first room, if the majority of the remaining 19 rooms have their lights on, the person will turn their light on; otherwise, they will turn their light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone has had their turn, how many rooms will have their lights off? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a real number $x$ , let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$ .
(1) Find the minimum value of $f(x)$ .
(2) Evaluate $\int_0^1 f(x)\ dx$ .
*2011 Tokyo Institute of Technology entrance exam, Problem 2* | {
"answer": "\\frac{1}{4} + \\frac{1}{2} \\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $b$ for which $x^2 + bx + 1760$ factors into a product of two polynomials, each having integer coefficients. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the coefficients of the first three terms of the expansion of $(x+ \frac {1}{2})^{n}$ form an arithmetic sequence. Let $(x+ \frac {1}{2})^{n} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{n}x^{n}$. Find:
(1) The value of $n$;
(2) The value of $a_{5}$;
(3) The value of $a_{0} - a_{1} + a_{2} - a_{3} + \ldots + (-1)^{n}a_{n}$. | {
"answer": "\\frac {1}{256}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The term containing \(x^7\) in the expansion of \((1 + 2x - x^2)^4\) arises when \(x\) is raised to the power of 3 in three factors and \(-x^2\) is raised to the power of 1 in one factor. | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle ABC with sides 9, 12, and 15, a small circle with center Q and radius 2 rolls around the inside of the triangle, 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"
} |
Simplify $\frac{1}{1+\sqrt{3}} \cdot \frac{1}{1-\sqrt{5}}$. | {
"answer": "\\frac{1}{1 - \\sqrt{5} + \\sqrt{3} - \\sqrt{15}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ice cream shop offers 8 different flavors of ice cream. What is the greatest number of sundaes that can be made if each sundae can consist of 1, 2, or 3 scoops, with each scoop possibly being a different type of ice cream and no two sundaes having the same combination of flavors? | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system on the plane, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of the curve C₁ is ρ²-6ρcosθ+5=0, and the parametric equation of the curve C₂ is $$\begin{cases} x=tcos \frac {π}{6} \\ y=tsin \frac {π}{6}\end{cases}$$ (t is the parameter).
(1) Find the rectangular coordinate equation of the curve C₁ and explain what type of curve it is.
(2) If the curves C₁ and C₂ intersect at points A and B, find the value of |AB|. | {
"answer": "\\sqrt {7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides opposite to the internal angles $A$, $B$, and $C$ of $\triangle ABC$ are $a$, $b$, and $c$ respectively. It is given that $b\sin C + c\sin B = 4a\sin B\sin C$ and $b^2 + c^2 - a^2 = 8$. The area of $\triangle ABC$ is __________. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $EFGH$, we have $E=(1,1)$, $F=(101,21)$, and $H=(3,y)$ for some integer $y$. What is the area of rectangle $EFGH$?
A) 520
B) 1040
C) 2080
D) 2600 | {
"answer": "1040",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x) = \sqrt{3}\sin\omega x - 2\sin^2\left(\frac{\omega x}{2}\right)$ ($\omega > 0$) has a minimum positive period of $3\pi$,
(I) Find the maximum and minimum values of the function $f(x)$ on the interval $[-\pi, \frac{3\pi}{4}]$;
(II) In $\triangle ABC$, where $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $a < b < c$, with $\sqrt{3}a = 2c\sin A$, find the measure of angle $C$;
(III) Under the conditions of (II), if $f\left(\frac{3}{2}A + \frac{\pi}{2}\right) = \frac{11}{13}$, find the value of $\cos B$. | {
"answer": "\\frac{12 + 5\\sqrt{3}}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
No math tournament exam is complete without a self referencing question. What is the product of
the smallest prime factor of the number of words in this problem times the largest prime factor of the
number of words in this problem | {
"answer": "1681",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M \) be a subset of the set \(\{1, 2, 3, \cdots, 15\}\), and suppose that the product of any three different elements in \( M \) is not a perfect square. Let \( |M| \) denote the number of elements in the set \( M \). Find the maximum value of \( |M| \). | {
"answer": "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 $\sin x = \sin (n^2 x)$ on the interval $[0, 2\pi]$. What is $\sum_{n=2}^{100} G(n)$? | {
"answer": "676797",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin(2x+\varphi)$ where $(0 < \varphi < \pi)$ satisfies $f(x) \leq |f(\frac{\pi}{6})|$, and $f(x_{1}) = f(x_{2}) = -\frac{3}{5}$, calculate the value of $\sin(x_{2}-x_{1})$. | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos x\cdot\sin \left(x+ \frac {\pi}{3}\right)- \sqrt {3}\cos ^{2}x+ \frac { \sqrt {3}}{4}$, $x\in\mathbb{R}$.
(I) Find the smallest positive period of $f(x)$.
(II) Find the maximum and minimum values of $f(x)$ on the closed interval $\left[- \frac {\pi}{4}, \frac {\pi}{4}\right]$. | {
"answer": "- \\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$ , $x_2y_1-x_1y_2=5$ , and $x_1y_1+5x_2y_2=\sqrt{105}$ . Find the value of $y_1^2+5y_2^2$ | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assume every 7-digit whole number is a possible telephone number except those that begin with a digit less than 3. What fraction of telephone numbers begin with $9$ and have $3$ as their middle digit (i.e., fourth digit)?
A) $\frac{1}{60}$
B) $\frac{1}{70}$
C) $\frac{1}{80}$
D) $\frac{1}{90}$
E) $\frac{1}{100}$ | {
"answer": "\\frac{1}{70}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=2+3\cos\alpha,}\\{y=3\sin\alpha}\end{array}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative $x$-axis as the polar axis to establish a polar coordinate system, the polar coordinate equation of the line $l$ is $2\rho \cos \theta -\rho \sin \theta -1=0$.
$(1)$ Find the Cartesian equation of curve $C$ and the rectangular coordinate equation of line $l$;
$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and point $P(0,-1)$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$. | {
"answer": "\\frac{3\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, and $b \sin A = \sqrt{3} a \cos B$.
1. Find the measure of angle $B$.
2. If $b = 3$ and $\sin C = 2 \sin A$, find the values of $a$ and $c$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), a line with an angle of $60^\circ$ passes through one of the foci and intersects the y-axis and the right branch of the hyperbola. Find the eccentricity of the hyperbola if the point where the line intersects the y-axis bisects the line segment between one of the foci and the point of intersection with the right branch of the hyperbola. | {
"answer": "2 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\alpha$ and $\beta$ are acute angles, and $\sin \alpha = \frac{\sqrt{5}}{5}$, $\cos \beta = \frac{3\sqrt{10}}{10}$, then $\sin (\alpha + \beta) =$____, $\alpha + \beta =$____. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfying $2a_{n+1}+a_{n}=3$ for $n\geqslant 1$, $a_{1}=10$, and $b_{n}=a_{n}-1$. Find the smallest integer $n$ that satisfies the inequality $|{{S_n}-6}|＜\frac{1}{{170}}$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $$\begin{cases} x=3\cos \theta \\ y=2\sin \theta \end{cases} (\theta \text{ is the parameter}),$$ and the parametric equation of the line $l$ is $$\begin{cases} x=t-1 \\ y=2t-a-1 \end{cases} (t \text{ is the parameter}).$$
(Ⅰ) If $a=1$, find the length of the line segment cut off by line $l$ from curve $C$.
(Ⅱ) If $a=11$, find a point $M$ on curve $C$ such that the distance from $M$ to line $l$ is minimal, and calculate the minimum distance. | {
"answer": "2\\sqrt{5}-2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle PQR$ is isosceles with $PQ = PR = 39$ and $\triangle SQR$ is equilateral with side length 30. The area of $\triangle PQS$ is closest to: | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=1+t\cos\alpha}\\{y=t\sin\alpha}\end{array}}\right.$ ($t$ is the parameter, $0\leqslant \alpha\ \ \lt \pi$). Taking the origin $O$ as the pole and the non-negative $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is ${\rho^2}=\frac{{12}}{{3+{{\sin}^2}\theta}}$. <br/>$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian equation of $C_{2}$; <br/>$(2)$ Given $F(1,0)$, the intersection points $A$ and $B$ of curve $C_{1}$ and $C_{2}$ satisfy $|BF|=2|AF|$ (point $A$ is in the first quadrant), find the value of $\cos \alpha$. | {
"answer": "\\frac{2}{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": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos^2x-\sin^2x+\frac{1}{2}, x \in (0,\pi)$.
$(1)$ Find the interval of monotonic increase for $f(x)$;
$(2)$ Suppose $\triangle ABC$ is an acute triangle, with the side opposite to angle $A$ being $a=\sqrt{19}$, and the side opposite to angle $B$ being $b=5$. If $f(A)=0$, find the area of $\triangle ABC$. | {
"answer": "\\frac{15\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x$, $y \in \mathbb{R}^{+}$ and $2x+3y=1$, find the minimum value of $\frac{1}{x}+ \frac{1}{y}$. | {
"answer": "5+2 \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin (2x+ \frac {\pi}{4})$, let $f_1(x)$ denote the function after translating and transforming $f(x)$ to the right by $φ$ units and compressing every point's abscissa to half its original length, then determine the minimum value of $φ$ for which $f_1(x)$ is symmetric about the line $x= \frac {\pi}{4}$. | {
"answer": "\\frac{3\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dice is repeatedly rolled, and the upward-facing number is recorded for each roll. The rolling stops once three different numbers are recorded. If the sequence stops exactly after five rolls, calculate the total number of distinct recording sequences for these five numbers. | {
"answer": "840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If for any ${x}_{1},{x}_{2}∈[1,\frac{π}{2}]$, $x_{1} \lt x_{2}$, $\frac{{x}_{2}sin{x}_{1}-{x}_{1}sin{x}_{2}}{{x}_{1}-{x}_{2}}＞a$ always holds, then the maximum value of the real number $a$ is ______. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains seven cards, each with a different integer from 1 to 7 written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh "I know the sum of the numbers on your cards is even." What is the sum of the numbers on Avani's cards?
A 6
B 9
C 10
D 11
E 12 | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1500$ are not factorial tails? | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A woman buys a property for $15,000, aiming for a $6\%$ return on her investment annually. She sets aside $15\%$ of the rent each month for maintenance, and pays $360 annually in taxes. What must be the monthly rent to meet her financial goals?
A) $110.00$
B) $123.53$
C) $130.45$
D) $142.86$
E) $150.00$ | {
"answer": "123.53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $\sqrt[3]{1+27} + \sqrt[3]{1+\sqrt[3]{27}}$. | {
"answer": "\\sqrt[3]{28} + \\sqrt[3]{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many values of \( k \) is \( 18^{18} \) the least common multiple of the positive integers \( 9^9 \), \( 12^{12} \), and \( k \)? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cowboy is 5 miles north of a stream which flows due west. He is also 10 miles east and 6 miles south of his cabin. He wishes to water his horse at the stream and then return home. Determine the shortest distance he can travel to accomplish this.
A) $5 + \sqrt{256}$ miles
B) $5 + \sqrt{356}$ miles
C) $11 + \sqrt{356}$ miles
D) $5 + \sqrt{116}$ miles | {
"answer": "5 + \\sqrt{356}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers between 100 and 999 (inclusive) have digits that form an arithmetic progression when read from left to right?
A sequence of three numbers \( a, b, c \) is said to form an arithmetic progression if \( a + c = 2b \).
A correct numerical answer without justification will earn 4 points. For full points, a detailed reasoning is expected. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$. | {
"answer": "\\frac{1}{72}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $A=\{x\in \mathbb{R} | ax^2-3x+2=0, a\in \mathbb{R}\}$.
1. If $A$ is an empty set, find the range of values for $a$.
2. If $A$ contains only one element, find the value of $a$ and write down this element. | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is nine times smaller than the other. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the geometric sequence $\{a_n\}$, $a_5a_7=2$, $a_2+a_{10}=3$, determine the value of $\frac{a_{12}}{a_4}$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $c$ denote the number of positive integers $n \leq 1000$ with $S(n)$ odd, and let $d$ denote the number of positive integers $n \leq 1000$ with $S(n)$ even. Find $|c-d|.$ | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus. | {
"answer": "\\frac{90\\sqrt{261}}{261}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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