problem stringlengths 10 5.15k | answer dict |
|---|---|
At Central Middle School, the $150$ students who participate in the Math Club plan to gather for a game night. Due to an overlapping school event, attendance is expected to drop by $40\%$. Walter and Gretel are preparing cookies using a new recipe that makes $18$ cookies per batch. The other details remain the same. How many full recipes should they prepare? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation $x^2 + 14x = 72$. The positive solution has the form $\sqrt{c} - d$ for positive natural numbers $c$ and $d$. What is $c + d$? | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a truncated cone with a volume of 91 from the original cone. Find the volume of the original cone. | {
"answer": "94.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. Additionally, the letter $C$ must appear at least once in the word. How many six-letter good words are there? | {
"answer": "94",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola $C: y^2 = 2px (p > 0)$, and a circle $M: (x-2)^2 + y^2 = 4$, the distance from the center $M$ of the circle to the directrix of the parabola is $3$. Point $P(x_0, y_0)(x_0 \geqslant 5)$ is a point on the parabola in the first quadrant. Through point $P$, two tangent lines to circle $M$ are drawn, intersecting the x-axis at points $A$ and $B$.
$(1)$ Find the equation of the parabola $C$;
$(2)$ Find the minimum value of the area of $\triangle PAB$. | {
"answer": "\\frac{25}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g$ be a function taking the positive integers to the positive integers, such that:
(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$),
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $g(m) = n^2$ or $g(n) = m^2.$
Find the sum of all possible values of $g(18).$ | {
"answer": "104976",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
4. Find the biggest positive integer $n$ , lesser thar $2012$ , that has the following property:
If $p$ is a prime divisor of $n$ , then $p^2 - 1$ is a divisor of $n$ . | {
"answer": "1944",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 15 rectangular sheets of paper. In each move, one of the sheets is chosen and cut with a straight line, not passing through its vertices, into two sheets. After 60 moves, it turned out that all the sheets are triangles or hexagons. How many hexagons are there? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu, 45]=180$? | {
"answer": "292",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\alpha\) and \(\beta\) be angles such that
\[
\frac{\cos^2 \alpha}{\cos \beta} + \frac{\sin^2 \alpha}{\sin \beta} = 2,
\]
Find the sum of all possible values of
\[
\frac{\sin^2 \beta}{\sin \alpha} + \frac{\cos^2 \beta}{\cos \alpha}.
\] | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two rectangles, one measuring \(8 \times 10\) and the other \(12 \times 9\), are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction. | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Form a five-digit number with no repeated digits using the numbers 0, 1, 2, 3, and 4, where exactly one even number is sandwiched between two odd numbers. The total number of such five-digit numbers is | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the modular inverse of \( 31 \), modulo \( 45 \).
Express your answer as an integer from \( 0 \) to \( 44 \), inclusive. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a\cdot (a-b)=3$, find the angle between $a$ and $b$. | {
"answer": "\\frac{\\pi }{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The expression $10y^2 - 51y + 21$ can be written as $(Cy - D)(Ey - F)$, where $C, D, E, F$ are integers. What is $CE + C$? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest common divisor of $8!$ and $(6!)^2.$ | {
"answer": "2880",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and a point $$P(1, \frac {3}{2})$$ on the ellipse such that the line connecting $P$ and the right focus of the ellipse is perpendicular to the x-axis.
(1) Find the equation of ellipse C;
(2) Find the minimum value of the slope of line MN, where line l, tangent to the parabola $y^2=4x$ in the first quadrant, intersects ellipse C at points A and B, intersects the x-axis at point M, and the perpendicular bisector of segment AB intersects the y-axis at point N. | {
"answer": "- \\frac { \\sqrt {3}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$ , find the sum of digits of $E$ | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
9 kg of toffees cost less than 10 rubles, and 10 kg of the same toffees cost more than 11 rubles. How much does 1 kg of these toffees cost? | {
"answer": "1.11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c= \sqrt {7}$, $C= \frac {\pi}{3}$.
(1) If $2\sin A=3\sin B$, find $a$ and $b$;
(2) If $\cos B= \frac {3 \sqrt {10}}{10}$, find the value of $\sin 2A$. | {
"answer": "\\frac {3-4 \\sqrt {3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the point corresponding to the complex number (2-i)z is in the second quadrant of the complex plane, calculate the value of z. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the binary operations $\clubsuit$ and $\spadesuit$ defined as $\clubsuit$ : $a^{\log_5(b)}$ and $\spadesuit$ : $a^{\frac{1}{\log_5(b)}}$ for real numbers $a$ and $b$ where these expressions are defined, a sequence $(b_n)$ is defined recursively as $b_4 = 4 \spadesuit 2$ and $b_n = (n \spadesuit (n-2)) \clubsuit b_{n-1}$ for all integers $n \geq 5$. Find $\log_5(b_{2023})$ to the nearest integer. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The residents of an accommodation need to pay the rent for the accommodation. If each of them contributes $10 \mathrm{Ft}$, the amount collected falls $88 \mathrm{Ft}$ short of the rent. However, if each of them contributes $10.80 \mathrm{Ft}$, then the total amount collected exceeds the rent by $2.5 \%$. How much should each resident contribute to collect exactly the required rent? | {
"answer": "10.54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $k$ be the product of every third positive integer from $2$ to $2006$ , that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$ . Find the number of zeros there are at the right end of the decimal representation for $k$ . | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Wang left home at 8:30 to visit a housing exhibition and returned home at 12:00, while his alarm clock at home was showing 11:46 when he got back. Calculate the time in minutes until the alarm clock will point to 12:00 exactly. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the product of the sums of the squares and the cubes of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 1 = 0,\] given that all roots are real and nonnegative. | {
"answer": "13754",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g$ be a function taking the positive integers to the positive integers, such that:
(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$,
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$,
(iv) $g(2) = 3$.
Find the sum of all possible values of $g(18)$. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of functions of the form \( f(x) = ax^3 + bx^2 + cx + d \) such that
\[ f(x)f(-x) = f(x^3). \] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCDEFGH$ is a cube. Find $\sin \angle BAE$, where $E$ is the top vertex directly above $A$. | {
"answer": "\\frac{1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$? | {
"answer": "\\frac{1}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tom adds up all the even integers from 2 to 600, inclusive. Lara adds up all the integers from 1 to 200, inclusive. What is Tom's sum divided by Lara's sum? | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ be such that for every positive integer $n$ , followings are satisfied.
i. $f(n+1) > f(n)$ ii. $f(f(n)) = 2n+2$ Find the value of $f(2013)$ .
(Here, $\mathbb{N}$ is the set of all positive integers.) | {
"answer": "4026",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integer divisors of $1800^{1800}$ are divisible by exactly 180 positive integers? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the coordinates of a direction vector of line $l$ are $({-1,\sqrt{3}})$, the inclination angle of line $l$ is ____. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$ with right angle at $Z$, $\angle XYZ < 45^\circ$ and $XY = 6$. A point $Q$ on $\overline{XY}$ is chosen such that $\angle YQZ = 3\angle XQZ$ and $QZ = 2$. Determine the ratio $\frac{XQ}{YQ}$ in simplest form. | {
"answer": "\\frac{7 + 3\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
First, factorize 42 and 30 into prime factors, then answer the following questions:
(1) 42= , 30= .
(2) The common prime factors of 42 and 30 are .
(3) The unique prime factors of 42 and 30 are .
(4) The greatest common divisor (GCD) of 42 and 30 is .
(5) The least common multiple (LCM) of 42 and 30 is .
(6) From the answers above, you can conclude that . | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left(x\right)=\cos x+\left(x+1\right)\sin x+1$ on the interval $\left[0,2\pi \right]$, find the minimum and maximum values of $f(x)$. | {
"answer": "\\frac{\\pi}{2}+2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the distinct prime factors of $7^7 - 7^4$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair coin is flipped 8 times. What is the probability that at least 6 consecutive flips come up heads? | {
"answer": "\\frac{3}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The New Year's gala has a total of 8 programs, 3 of which are non-singing programs. When arranging the program list, it is stipulated that the non-singing programs are not adjacent, and the first and last programs are singing programs. How many different ways are there to arrange the program list? | {
"answer": "720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $ABCD$ , $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$ . Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$ .
*Proposed by Lewis Chen* | {
"answer": "260",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle whose side lengths are all positive integers, and only one side length is 5, which is not the shortest side, the number of such triangles is . | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the original largest square? | {
"answer": "\\frac{3\\pi}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 boys and 3 girls standing in a row. (You must write down the formula before calculating the result to score points)
(Ⅰ) If the 3 girls must stand together, how many different arrangements are there?
(Ⅱ) If no two girls are next to each other, how many different arrangements are there?
(Ⅲ) If there are exactly three people between person A and person B, how many different arrangements are there? | {
"answer": "720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? | {
"answer": "$-\\frac{2\\sqrt{3}}{3}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\] | {
"answer": "7\\sqrt{2}.",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \( p_1, p_2, p_3, q_1, q_2, q_3, r_1, r_2, r_3 \) are equal to the numbers \( 1, 2, 3, \dots, 9 \) in some order. Find the smallest possible value of
\[
P = p_1 p_2 p_3 + q_1 q_2 q_3 + r_1 r_2 r_3.
\] | {
"answer": "214",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $M(m,n)$ is any point on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.
(I) Find the maximum and minimum values of $|MQ|$;
(II) Find the maximum and minimum values of $\frac{n-3}{m+2}$. | {
"answer": "2-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks into the warehouse and performs exactly one of the
following moves:
(a) He adds one piece of rubbish to each non-empty pile.
(b) He creates a new pile with one piece of rubbish.
What is the first morning when Angel can guarantee to have cleared all the rubbish from the
warehouse? | {
"answer": "199",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than $1000$ are either a perfect cube or a perfect square? | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is cut into three equal wedges. The circumference of the sphere is $18\pi$ inches. What is the volume of the intersection between one wedge and the top half of the sphere? Express your answer in terms of $\pi$. | {
"answer": "162\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A large square region is paved with $n^2$ square black tiles, where each tile measures $t$ inches on each side. Surrounding each tile is a white border that is $w$ inches wide. When $n=30$, it's given that the black tiles cover $81\%$ of the area of the large square region. Find the ratio $\frac{w}{t}$ in this scenario.
A) $\frac{1}{8}$
B) $\frac{1}{9}$
C) $\frac{2}{9}$
D) $\frac{1}{10}$
E) $\frac{1}{11}$ | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a rectangular grid where grid lines are spaced $1$ unit apart, the acronym XYZ is depicted below. The X is formed by two diagonal lines crossing, the Y is represented with a 'V' shape starting from a bottom point going up to join two endpoints with horizontal lines, the Z is drawn with a top horizontal line, a diagonal from top right to bottom left and a bottom horizontal line. Calculate the sum of lengths of the line segments that form the acronym XYZ.
A) $6 + 3\sqrt{2}$
B) $4 + 5\sqrt{2}$
C) $3 + 6\sqrt{2}$
D) $5 + 4\sqrt{2}$ | {
"answer": "4 + 5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $JKLM$ and $NOPM$ are squares each of area 25. If $Q$ is the midpoint of both $KL$ and $NO$, find the total area of polygon $JMQPON$.
[asy]
unitsize(3 cm);
pair J, K, L, M, N, O, P, Q;
O = (0,0);
P = (1,0);
M = (1,1);
N = (0,1);
Q = (N + O)/2;
J = reflect(M,Q)*(P);
K = reflect(M,Q)*(O);
L = reflect(M,Q)*(N);
draw(J--K--L--M--cycle);
draw(M--N--O--P--cycle);
label("$J$", J, N);
label("$K$", K, W);
label("$L$", L, S);
label("$M$", M, NE);
label("$N$", N, NW);
label("$O$", O, SW);
label("$P$", P, SE);
label("$Q$", Q, SW);
[/asy] | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The storage capacity of two reservoirs, A and B, changes over time. The relationship between the storage capacity of reservoir A (in hundred tons) and time $t$ (in hours) is: $f(t) = 2 + \sin t$, where $t \in [0, 12]$. The relationship between the storage capacity of reservoir B (in hundred tons) and time $t$ (in hours) is: $g(t) = 5 - |t - 6|$, where $t \in [0, 12]$. The question is: When do the combined storage capacities of reservoirs A and B reach their maximum value? And what is this maximum value?
(Reference data: $\sin 6 \approx -0.279$). | {
"answer": "6.721",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \( KLMN \), diagonal \( KM \) is equal to 1 and is also its height. From points \( K \) and \( M \), perpendiculars \( KP \) and \( MQ \) are drawn to sides \( MN \) and \( KL \), respectively. Find \( LM \) if \( KN = MQ \) and \( LM = MP \). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the drawing, 5 lines intersect at a single point. One of the resulting angles is $34^\circ$. What is the sum of the four angles shaded in gray, in degrees? | {
"answer": "146",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ be real numbers, where $y > x > 0$, such that
\[
\frac{x}{y} + \frac{y}{x} = 4.
\]
Find the value of
\[
\frac{x + y}{x - y}.
\] | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers. | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum:
\[
\sum_{n=1}^\infty \frac{n^3 + n^2 + n - 1}{(n+3)!}
\] | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many irreducible fractions with numerator 2015 exist that are less than \( \frac{1}{2015} \) and greater than \( \frac{1}{2016} \)? | {
"answer": "1440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student research group at a school found that the attention index of students during class changes with the listening time. At the beginning of the lecture, students' interest surges; then, their interest remains in a relatively ideal state for a while, after which students' attention begins to disperse. Let $f(x)$ represent the student attention index, which changes with time $x$ (minutes) (the larger $f(x)$, the more concentrated the students' attention). The group discovered the following rule for $f(x)$ as time $x$ changes:
$$f(x)= \begin{cases} 100a^{ \frac {x}{10}}-60, & (0\leqslant x\leqslant 10) \\ 340, & (10 < x\leqslant 20) \\ 640-15x, & (20 < x\leqslant 40)\end{cases}$$
where $a > 0, a\neq 1$.
If the attention index at the 5th minute after class starts is 140, answer the following questions:
(Ⅰ) Find the value of $a$;
(Ⅱ) Compare the concentration of attention at the 5th minute after class starts and 5 minutes before class ends, and explain the reason.
(Ⅲ) During a class, how long can the student's attention index remain at least 140? | {
"answer": "\\dfrac {85}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle in which $AB=AC$ . Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$ . | {
"answer": "3/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four boxes with ball capacity 3, 5, 7, and 8 are given. Find the number of ways to distribute 19 identical balls into these boxes. | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f_1(x)=x^2-1$ , and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$ . How many distinct real roots does the polynomial $f_{2004}$ have? | {
"answer": "2005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
22. Let the function $f : \mathbb{Z} \to \mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$ , f satisfies $$ f(x) + f(y) = f(x + 1) + f(y - 1) $$ If $f(2016) = 6102$ and $f(6102) = 2016$ , what is $f(1)$ ?
23. Let $d$ be a randomly chosen divisor of $2016$ . Find the expected value of $$ \frac{d^2}{d^2 + 2016} $$ 24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$ ? | {
"answer": "8117",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the positive solution to
\[\sqrt{x + 2 + \sqrt{x + 2 + \dotsb}} = \sqrt{x \sqrt{x \dotsm}}.\] | {
"answer": "1 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $x$ is an integer that satisfies the following congruences:
\[
4 + x \equiv 3^2 \pmod{2^3}, \\
6 + x \equiv 2^3 \pmod{3^3}, \\
8 + x \equiv 7^2 \pmod{5^3}.
\]
What is the remainder when $x$ is divided by $30$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left and right foci \(F_{1}\) and \(F_{2}\) respectively, a line \(l\) passes through the right focus and intersects the ellipse at points \(P\) and \(Q\). Find the maximum area of the inscribed circle of \(\triangle F_{1}PQ\). | {
"answer": "\\frac{9\\pi}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The longest seminar session and the closing event lasted a total of $4$ hours and $45$ minutes plus $135$ minutes, plus $500$ seconds. Convert this duration to minutes and determine the total number of minutes. | {
"answer": "428",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with incenter $I$, centroid $G$, and $|AC|>|AB|$. If $IG\parallel BC$, $|BC|=2$, and $\text{Area}(ABC)=3\sqrt{5}/8$, calculate $|AB|$. | {
"answer": "\\frac{9}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any 2016 complex numbers \( z_1, z_2, \ldots, z_{2016} \), it holds that
\[
\sum_{k=1}^{2016} |z_k|^2 \geq \lambda \min_{1 \leq k \leq 2016} \{ |z_{k+1} - z_k|^2 \},
\]
where \( z_{2017} = z_1 \). Find the maximum value of \( \lambda \). | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Estimate the population of the island of Thalassa in the year 2050, knowing that its population doubles every 20 years and increases by an additional 500 people every decade thereafter, given that the population in the year 2000 was 250. | {
"answer": "1500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$A$ and $B$ travel around an elliptical track at uniform speeds in opposite directions, starting from the vertices of the major axis. They start simultaneously and meet first after $B$ has traveled $150$ yards. They meet a second time $90$ yards before $A$ completes one lap. Find the total distance around the track in yards.
A) 600
B) 720
C) 840
D) 960
E) 1080 | {
"answer": "720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a\_n\}$, where $a\_n= \sqrt {5n-1}$, $n\in\mathbb{N}^*$, arrange the integer terms of the sequence $\{a\_n\}$ in their original order to form a new sequence $\{b\_n\}$. Find the value of $b_{2015}$. | {
"answer": "5037",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$ and $y$ satisfy the equation $x^2+y^2-4x+1=0$.
(1) Find the maximum and minimum value of $\frac {y}{x}$.
(2) Find the maximum and minimum value of $y-x$.
(3) Find the maximum and minimum value of $x^2+y^2$. | {
"answer": "7-4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider that Henry's little brother now has 10 identical stickers and 5 identical sheets of paper. How many ways can he distribute all the stickers on the sheets of paper, if only the number of stickers on each sheet matters and no sheet can remain empty? | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For what smallest natural $k$ is the number \( 2016 \cdot 20162016 \cdot 201620162016 \cdot \ldots \cdot 20162016\ldots2016 \) (with $k$ factors) divisible by \(3^{67}\)? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ and $y$ satisfy the equation $(x-1)^{2}+(y+2)^{2}=4$, find the maximum and minimum values of $S=3x-y$. | {
"answer": "5 - 2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $U$ be a square with side length 1. Two points are randomly chosen on the sides of $U$. The probability that the distance between these two points is at least $\frac{1}{2}$ is $\frac{a - b \pi}{c}\left(a, b, c \in \mathbf{Z}_{+}, (a, b, c)=1\right)$. Find the value of $a + b + c$. | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a finite sequence $S=(2, 2x, 2x^2,\ldots ,2x^{200})$ of $n=201$ real numbers, let $A(S)$ be the sequence $\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{200}+a_{201}}{2}\right)$ of $n-1=200$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le 150$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(2, 2x, 2x^2,\ldots ,2x^{200})$. If $A^{150}(S)=(2 \cdot 2^{-75})$, then what is $x$?
A) $1 - \frac{\sqrt{2}}{2}$
B) $2^{3/8} - 1$
C) $\sqrt{2} - 2$
D) $2^{1/5} - 1$ | {
"answer": "2^{3/8} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An organization has a structure where there is one president, two vice-presidents (VP1 and VP2), and each vice-president supervises two managers. If the organization currently has 12 members, in how many different ways can the leadership (president, vice-presidents, and managers) be chosen? | {
"answer": "554400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads? | {
"answer": "\\frac{3}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\). | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The store has 89 gold coins with numbers ranging from 1 to 89, each priced at 30 yuan. Among them, only one is a "lucky coin." Feifei can ask an honest clerk if the number of the lucky coin is within a chosen subset of numbers. If the answer is "Yes," she needs to pay a consultation fee of 20 yuan. If the answer is "No," she needs to pay a consultation fee of 10 yuan. She can also choose not to ask any questions and directly buy some coins. What is the minimum amount of money (in yuan) Feifei needs to pay to guarantee she gets the lucky coin? | {
"answer": "130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $ABCD$ has sides of length 4. Set $T$ is the set of all line segments that have length 4 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $T$ enclose a region whose area to the nearest hundredth is $m$. Find $100m$. | {
"answer": "343",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.
$(1)$ Find the equation of ellipse $C$ and its eccentricity;
$(2)$ A line $y=kx (k \in \mathbb{R}, k \neq 0)$ intersects ellipse $C$ at points $A$ and $B$, point $D$ is a moving point on ellipse $C$, and $|AD| = |BD|$. Does the area of $\triangle ABD$ have a minimum value? If it exists, find the equation of line $AB$; if not, explain why. | {
"answer": "\\dfrac{8}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a square $R_1$ with area 81, an equilateral triangle $T_1$ is inscribed such that each vertex of $T_1$ touches one side of $R_1$. Each midpoint of $T_1’s$ sides is connected to form a smaller triangle $T_2$. The process is repeated with $T_2$ to form $T_3$. Find the area of triangle $T_3$. | {
"answer": "\\frac{81\\sqrt{3}}{256}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle ABC$ is right-angled. Side $AB$ is extended in each direction to points $D$ and $G$ such that $DA = AB = BG$. Similarly, $BC$ is extended to points $F$ and $K$ so that $FB = BC = CK$, and $AC$ is extended to points $E$ and $H$ so that $EA = AC = CH$. Find the ratio of the area of the hexagon $DEFGHK$ to the area of $\triangle ABC$. | {
"answer": "13:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Aquatic plants require a specific type of nutrient solution. Given that each time $a (1 \leqslant a \leqslant 4$ and $a \in R)$ units of the nutrient solution are released, its concentration $y (\text{g}/\text{L})$ changes over time $x (\text{days})$ according to the function $y = af(x)$, where $f(x)=\begin{cases} \frac{4+x}{4-x} & 0\leqslant x\leqslant 2 \\ 5-x & 2\prec x\leqslant 5 \end{cases}$. If the nutrient solution is released multiple times, the concentration at a given moment is the sum of the concentrations released at the corresponding times. According to experience, the nutrient solution is effective only when its concentration is not less than $4(\text{g}/\text{L})$.
(1) If $4$ units of the nutrient solution are released only once, how many days can it be effective?
(2) If $2$ units of the nutrient solution are released first, and then $b$ units are released after $3$ days. In order to keep the nutrient solution continuously effective in the next $2$ days, find the minimum value of $b$. | {
"answer": "24-16\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line segment is divided into four parts by three randomly selected points. What is the probability that these four parts can form the four sides of a quadrilateral? | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two rectangles, one $8 \times 10$ and the other $12 \times 9$, are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction. | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with an edge length of 6 is cut into smaller cubes with integer edge lengths. If the total surface area of these smaller cubes is \(\frac{10}{3}\) times the surface area of the original larger cube before cutting, how many of these smaller cubes have an edge length of 1? | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( T = 1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + \cdots + 2023 + 2024 - 2025 - 2026 \). What is the residue of \( T \), modulo 2027? | {
"answer": "2026",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $2310$, then what is the sum of the numbers written on the faces? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a^2 = \frac{9}{25}$ and $b^2 = \frac{(3+\sqrt{7})^2}{14}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a-b)^2$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, there are several triangles formed by connecting points in a shape. If each triangle has the same probability of being selected, what is the probability that a selected triangle includes a vertex marked with a dot? Express your answer as a common fraction.
[asy]
draw((0,0)--(2,0)--(1,2)--(0,0)--cycle,linewidth(1));
draw((0,0)--(1,1)--(1,2)--(0,0)--cycle,linewidth(1));
dot((1,2));
label("A",(0,0),SW);
label("B",(2,0),SE);
label("C",(1,2),N);
label("D",(1,1),NE);
label("E",(1,0),S);
[/asy] | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many pairs of positive integers \((x, y)\) are there such that \(x < y\) and \(\frac{x^{2}+y^{2}}{x+y}\) is a divisor of 2835? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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