problem stringlengths 10 5.15k | answer dict |
|---|---|
Let $n$ be a positive integer with $k\ge22$ divisors $1=d_{1}< d_{2}< \cdots < d_{k}=n$ , all different. Determine all $n$ such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\] | {
"answer": "2^3 * 3 * 5 * 17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On eight cards, the numbers $1, 1, 2, 2, 3, 3, 4, 4$ are written. Is it possible to arrange these cards in a row such that there is exactly one card between the ones, two cards between the twos, three cards between the threes, and four cards between the fours? | {
"answer": "41312432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < a \leqslant \frac{5}{4}$, find the range of real number $b$ such that all real numbers $x$ satisfying the inequality $|x - a| < b$ also satisfy the inequality $|x - a^2| < \frac{1}{2}$. | {
"answer": "\\frac{3}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Henry constructs a model involving four spherical clay pieces of radii 1 inch, 4 inches, 6 inches, and 3 inches. Calculate the square of the total volume of the clay used, subtracting 452 from the squared result, and express your answer in terms of $\pi$. | {
"answer": "168195.11\\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube is suspended in space with its top and bottom faces horizontal. The cube has one top face, one bottom face, and four side faces. Determine the number of ways to move from the top face to the bottom face, visiting each face at most once, without moving directly from the top face to the bottom face, and not moving from side faces back to the top face. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the shortest chord is cut by the line $y = kx + 1$ on the circle $C: x^2 + y^2 - 2x - 3 = 0$, then $k = \boxed{\_\_\_\_\_\_\_\_}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a>0$ and $b>0,$ a new operation $\Delta$ is defined as follows: $$a \Delta b = \frac{a^2 + b^2}{1 + ab}.$$ Calculate $(2 \Delta 3) \Delta 4$. | {
"answer": "\\frac{6661}{2891}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function f(x) = sin(2x) is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of the function g(x). Find the analytical expression for g(x). Also, find the minimum value of $|x_1 - x_2|$ for $x_1$ and $x_2$ that satisfy $|f(x_1) - g(x_2)| = 2$. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follows:<br/>
| | First-class | Second-class | Total |
|----------|-------------|--------------|-------|
| Machine A | 150 | 50 | 200 |
| Machine B | 120 | 80 | 200 |
| Total | 270 | 130 | 400 |
$(1)$ What are the frequencies of first-class products produced by Machine A and Machine B, respectively?<br/>
$(2)$ Can we be $99\%$ confident that there is a difference in the quality of the products produced by Machine A and Machine B?<br/>
Given: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.<br/>
| $P(K^{2}\geqslant k)$ | 0.050 | 0.010 | 0.001 |
|-----------------------|-------|-------|-------|
| $k$ | 3.841 | 6.635 | 10.828| | {
"answer": "99\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sufficiently large positive integer \( n \), which can be divided by all the integers from 1 to 250 except for two consecutive integers \( k \) and \( k+1 \), find \( k \). | {
"answer": "127",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(a) In a tennis tournament with 64 players, how many matches are played?
(b) In a tournament with 2011 players, how many matches are played? | {
"answer": "113",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(15) Given the following propositions:
(1) "If $x > 2$, then $x > 0$" - the negation of the proposition
(2) "For all $a \in (0, +\infty)$, the function $y = a^x$ is strictly increasing on its domain" - the negation
(3) "$π$ is a period of the function $y = \sin x$" or "$2π$ is a period of the function $y = \sin 2x$"
(4) "$x^2 + y^2 = 0$" is a necessary condition for "$xy = 0$"
The sequence number(s) of the true proposition(s) is/are _______. | {
"answer": "(2)(3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number \( x \) in a base \( r \) system (\( r \leq 36 \)) is represented as \( \overline{ppqq} \), where \( 2q = 5p \). It turns out that the base-\( r \) representation of \( x^2 \) is a seven-digit palindrome with a middle digit of zero. (A palindrome is a number that reads the same from left to right and from right to left). Find the sum of the digits of the number \( x^2 \) in base \( r \). | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\ln x$, $g(x)= \frac {1}{3}x^{3}+ \frac {1}{2}x^{2}+mx+n$, and line $l$ is tangent to both the graphs of $f(x)$ and $g(x)$ at point $(1,0)$.
1. Find the equation of line $l$ and the expression for $g(x)$.
2. If $h(x)=f(x)-g′(x)$ (where $g′(x)$ is the derivative of $g(x)$), find the range of the function $h(x)$. | {
"answer": "\\frac {1}{4}- \\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 15 married couples among 30 people. Calculate the total number of handshakes that occurred among these people. | {
"answer": "301",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a rectangular sheet of paper. Points $E$ and $F$ are located on edges $AB$ and $CD$, respectively, such that $BE < CF$. The rectangle is folded over line $EF$ so that point $C$ maps to $C'$ on side $AD$ and point $B$ maps to $B'$ on side $AD$ such that $\angle{AB'C'} \cong \angle{B'EA}$ and $\angle{B'C'A} = 90^\circ$. If $AB' = 3$ and $BE = 12$, compute the area of rectangle $ABCD$ in the form $a + b\sqrt{c}$, where $a$, $b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Compute $a + b + c$. | {
"answer": "57",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair coin is tossed 4 times. What is the probability of at least two consecutive heads? | {
"answer": "\\frac{5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer that satisfies the congruence $5x \equiv 17 \pmod{31}$? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=|2x-1|$.
(1) Solve the inequality $f(x) < 2$;
(2) If the minimum value of the function $g(x)=f(x)+f(x-1)$ is $a$, and $m+n=a$ $(m > 0,n > 0)$, find the minimum value of $\frac{m^{2}+2}{m}+\frac{n^{2}+1}{n}$. | {
"answer": "2\\sqrt{2}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which of the following numbers is not an integer? | {
"answer": "$\\frac{2014}{4}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$ . | {
"answer": "843",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( 6x + t = 4x - 9 \), what is the value of \( x + 4 \)? | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum:
\[\sum_{N = 1}^{2048} \lfloor \log_3 N \rfloor.\] | {
"answer": "12049",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ with a common ratio of $2$ and the sum of the first $n$ terms denoted by $S_n$. If $a_2= \frac{1}{2}$, find the expression for $a_n$ and the value of $S_5$. | {
"answer": "\\frac{31}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \) with \( AB = 8 \) and \( AC = 10 \), the incenter \( I \) is reflected across side \( AB \) to point \( X \) and across side \( AC \) to point \( Y \). Given that segment \( XY \) bisects \( AI \), compute \( BC^2 \). (The incenter \( I \) is the center of the inscribed circle of triangle \( ABC \).) | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is inscribed in quadrilateral $EFGH$, tangent to $\overline{EF}$ at $R$ and to $\overline{GH}$ at $S$. Given that $ER=24$, $RF=31$, $GS=40$, and $SH=29$, find the square of the radius of the circle. | {
"answer": "945",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A polyhedron has 12 faces and is such that:
(i) all faces are isosceles triangles,
(ii) all edges have length either \( x \) or \( y \),
(iii) at each vertex either 3 or 6 edges meet, and
(iv) all dihedral angles are equal.
Find the ratio \( x / y \). | {
"answer": "3/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of a square \(ABCD\), and the point of tangency cuts off a segment of length \(6 - 2\sqrt{5}\) cm from vertex \(A\). Two tangents are drawn to this circle from point \(C\). Find the side length of the square, given that the angle between the tangents is \(36^{\circ}\), and it is known that \(\sin 18^{\circ} = \frac{\sqrt{5} - 1}{4}\). | {
"answer": "(\\sqrt{5} - 1)(2\\sqrt{2} - \\sqrt{5} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Person A and Person B start simultaneously from points A and B respectively, walking towards each other. Person A starts from point A, and their speed is 4 times that of Person B. The distance between points A and B is \( S \) kilometers, where \( S \) is a positive integer with 8 factors. The first time they meet at point C, the distance \( AC \) is an integer. The second time they meet at point D, the distance \( AD \) is still an integer. After the second meeting, Person B feels too slow, so they borrow a motorbike from a nearby village near point D. By the time Person B returns to point D with the motorbike, Person A has reached point E, with the distance \( AE \) being an integer. Finally, Person B chases Person A with the motorbike, which travels at 14 times the speed of Person A. Both arrive at point A simultaneously. What is the distance between points A and B?
\[ \text{The distance between points A and B is } \qquad \text{kilometers.} \] | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, a rectangle has a perimeter of $40$, and a triangle has a height of $40$. If the rectangle and the triangle have the same area, what is the value of $x?$ Assume the length of the rectangle is twice its width. [asy]
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);
draw((4,0)--(7,0)--(7,5)--cycle);
draw((6.8,0)--(6.8,.2)--(7,.2));
label("$x$",(5.5,0),S);
label("40",(7,2.5),E);
[/asy] | {
"answer": "4.4445",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube with an unknown volume, two of its dimensions are increased by $1$ and the third is decreased by $2$, and the volume of the resulting rectangular solid is $27$ less than that of the cube. Determine the volume of the original cube. | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(AB\) is a diameter of a circle with center \(O\). \(C\) and \(D\) are points on the circle. \(OD\) intersects \(AC\) at \(P\), \(OC\) intersects \(BD\) at \(Q\), and \(AC\) intersects \(BD\) at \(R\). If \(\angle BOQ = 60^{\circ}\) and \(\angle APO = 100^{\circ}\), calculate the measure of \(\angle BQO\). | {
"answer": "95",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each number in the list $1,2,3,\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . What is $m+n$ ?
*2021 CCA Math Bonanza Team Round #9* | {
"answer": "455",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of positive integers $a$, $b$, and $c$ equals 3960. What is the minimum possible value of the sum $a + b + c$? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Integers $x$ and $y$ with $x > y > 0$ satisfy $x + y + xy = 119$. What is $x$? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = (\sin x + \cos x)^2 + \cos 2x - 1$.
(1) Find the smallest positive period of the function $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \(a, b, c\) and positive number \(\lambda\) make the function \(f(x) = x^3 + ax^2 + bx + c\) 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 \(\frac{2a^3 + 27c + 9ab}{\lambda^3}\). | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, $QR = 24$. An incircle of the triangle trisects the median $PS$ from $P$ to side $QR$. Given that the area of the triangle is $k \sqrt{p}$, where $k$ and $p$ are integers and $p$ is square-free, find $k+p$. | {
"answer": "106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression \[ \frac{a+3}{a+1} \cdot \frac{b-2}{b-3} \cdot \frac{c + 9}{c+7} , \] given that $c = b-11$, $b = a+3$, $a = 5$, and none of the denominators are zero. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a triangle with vertices at points \( (0, 0), (30, 0), \) and \( (18, 26) \). The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the original triangle along the sides of its midpoint triangle. What is the volume of this pyramid? | {
"answer": "3380",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the curves $C_{1}: \begin{cases} x=2+\cos t \\ y=\sin t-1 \end{cases}$ (with $t$ as the parameter), and $C_{2}: \begin{cases} x=4\cos \alpha \\ y=\sin \alpha \end{cases}$ (with $\alpha$ as the parameter), in the polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, there is a line $C_{3}: \theta= \frac {\pi}{4} (\rho \in \mathbb{R})$.
1. Find the standard equations of curves $C_{1}$ and $C_{2}$, and explain what curves they respectively represent;
2. If the point $P$ on $C_{2}$ corresponds to the parameter $\alpha= \frac {\pi}{2}$, and $Q$ is a point on $C_{1}$, find the minimum distance $d$ from the midpoint $M$ of $PQ$ to the line $C_{3}$. | {
"answer": "\\frac { \\sqrt {2}-1}{ \\sqrt {2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be the set of all positive integer divisors of $144,000$. Calculate the number of numbers that are the product of two distinct elements of $T$. | {
"answer": "451",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest integer that must divide the product of any 5 consecutive integers? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer that ends in 3 and is divisible by 11? | {
"answer": "113",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence of integers $\{a_i\}_{i = 0}^{\infty}$ satisfies $a_0 = 3$ , $a_1 = 4$ , and
\[a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil\]
for $n \ge 0$ . Evaluate the sum
\[\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).\] | {
"answer": "0.518",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number \(n\) is a three-digit positive integer and is the product of the three factors \(x\), \(y\), and \(5x+2y\), where \(x\) and \(y\) are integers less than 10 and \((5x+2y)\) is a composite number. What is the largest possible value of \(n\) given these conditions? | {
"answer": "336",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many even divisors does \(8!\) have, and how many of those are also multiples of both 2 and 3? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a quadrilateral inscribed in a unit circle with center \(O\). Suppose that \(\angle AOB = \angle COD = 135^\circ\), and \(BC = 1\). Let \(B'\) and \(C'\) be the reflections of \(A\) across \(BO\) and \(CO\) respectively. Let \(H_1\) and \(H_2\) be the orthocenters of \(AB'C'\) and \(BCD\), respectively. If \(M\) is the midpoint of \(OH_1\), and \(O'\) is the reflection of \(O\) about the midpoint of \(MH_2\), compute \(OO'\). | {
"answer": "\\frac{1}{4}(8-\\sqrt{6}-3\\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a, b,\) and \(c\) (not necessarily integers) satisfy the conditions
\[
a + b + c = 0 \quad \text{and} \quad \frac{a}{b} + \frac{b}{c} + \frac{c}{a} = 100
\]
What is the value of \(\frac{b}{a} + \frac{c}{b} + \frac{a}{c}\)? | {
"answer": "-101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On graph paper, a stepwise right triangle was drawn with legs equal to 6 cells each. Then, all grid lines inside the triangle were outlined. What is the maximum number of rectangles that can be found in this drawing? | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight distinct pieces of candy are to be distributed among three bags: red, blue, and white, with each bag receiving at least one piece of candy. Determine the total number of arrangements possible. | {
"answer": "846720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges start at the vertices of a square, and this configuration stores 20 Joules of energy. How much more energy, in Joules, would be stored if one of these charges was moved to the center of the square? | {
"answer": "5(3\\sqrt{2} - 3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $\alpha$ passes through point $P$((-$ sqrt{3}$, $m$)), and $\sin \alpha$ = $\frac{\sqrt{3}}{4}$ $m$($m$($m$ne 0)), determine in which quadrant angle $\alpha$ lies, and find the value of $\tan \alpha$. | {
"answer": "\\frac{\\sqrt{7}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given integers $a$ and $b$ whose sum is $20$, determine the number of distinct integers that can be written as the sum of two, not necessarily different, fractions $\frac{a}{b}$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parallelogram \(A B C D\) with \(\angle B = 111^\circ\) and \(B C = B D\). On the segment \(B C\), there is a point \(H\) such that \(\angle B H D = 90^\circ\). Point \(M\) is the midpoint of side \(A B\). Find the angle \(A M H\). Provide the answer in degrees. | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a_{1}, a_{2}, \cdots, a_{2014} \) be a permutation of the positive integers \( 1, 2, \cdots, 2014 \). Define
\[ S_{k} = a_{1} + a_{2} + \cdots + a_{k} \quad (k=1, 2, \cdots, 2014). \]
What is the maximum number of odd numbers among \( S_{1}, S_{2}, \cdots, S_{2014} \)? | {
"answer": "1511",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \(P, Q, R,\) and \(S\) lie in the plane of the square \(EFGH\) such that \(EPF\), \(FQG\), \(GRH\), and \(HSE\) are equilateral triangles. If \(EFGH\) has an area of 25, find the area of quadrilateral \(PQRS\). Express your answer in simplest radical form. | {
"answer": "100 + 50\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $a, b, c$ , and $d$ are real numbers simultaneously satisfying $a + b - c - d = 3$ $ab - 3bc + cd - 3da = 4$ $3ab - bc + 3cd - da = 5$ Find $11(a - c)^2 + 17(b -d)^2$ . | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integers \(a\) and \(b\) are such that the numbers \(15a + 165\) and \(16a - 155\) are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | {
"answer": "481",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $1-2+3-4+ \dots -100+101$. | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=x+\sqrt{6}$, the circle $(O)$: $x^2+y^2=5$, and the ellipse $(E)$: $\frac{y^2}{a^2}+\frac{x^2}{b^2}=1$ $(b > 0)$ with an eccentricity of $e=\frac{\sqrt{3}}{3}$. The length of the chord intercepted by line $(l)$ on circle $(O)$ is equal to the length of the major axis of the ellipse. Find the product of the slopes of the two tangent lines to ellipse $(E)$ passing through any point $P$ on circle $(O)$, if the tangent lines exist. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest number, all of whose digits are 1 or 2, and whose digits add up to $10$? | {
"answer": "111111112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2014 points marked on a circle. A grasshopper sits on one of these points and makes jumps either 57 divisions or 10 divisions clockwise. It is known that he visited all the marked points, making the minimum number of jumps. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\mathbf{v}$ such that
\[\begin{pmatrix} 3 & 4 \\ 6 & 3 \end{pmatrix} \mathbf{v} = k \mathbf{v}.\] | {
"answer": "3 - 2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If three different numbers are selected from 2, 3, 4, 5, 6 to be $a$, $b$, $c$ such that $N = abc + ab + bc + a - b - c$ reaches its maximum value, then this maximum value is. | {
"answer": "167",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given circles $P$, $Q$, and $R$ each have radius 2, circle $P$ and $Q$ are tangent to each other, and circle $R$ is tangent to the midpoint of $\overline{PQ}$. Calculate the area inside circle $R$ but outside circle $P$ and circle $Q$. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let triangle $PQR$ be a right triangle with $\angle PRQ = 90^\circ$. A circle is tangent to the sides $PQ$ and $PR$ at points $S$ and $T$ respectively. The points on the circle diametrically opposite $S$ and $T$ both lie on side $QR$. Given that $PQ = 12$, find the area of the portion of the circle that lies outside triangle $PQR$. | {
"answer": "4\\pi - 8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest number which when successively divided by \( 45,454,4545 \) and 45454 leaves remainders of \( 4, 45,454 \) and 4545 respectively. | {
"answer": "35641667749",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $N$ such that any "hydra" with 100 necks, where each neck connects two heads, can be defeated by cutting at most $N$ strikes. Here, one strike can sever all the necks connected to a particular head $A$, and immediately after, $A$ grows new necks to connect with all previously unconnected heads (each head connects to one neck). The hydra is considered defeated when it is divided into two disconnected parts. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two positive integers \(m\) and \(n\) are chosen such that \(m\) is the smallest positive integer with only two positive divisors, and \(n\) is the largest integer less than 200 that has exactly four positive divisors. What is \(m+n\)? | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$, $y$, $z$ satisfying $\begin{cases} xy+2z=1 \\ x^{2}+y^{2}+z^{2}=5 \end{cases}$, the minimum value of $xyz$ is \_\_\_\_\_\_. | {
"answer": "9 \\sqrt {11}-32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadrilateral $MARE$ inscribed in a unit circle $\omega,$ $AM$ is a diameter of $\omega,$ and $E$ lies on the angle bisector of $\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$ | {
"answer": "\\frac{8\\sqrt{2}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, $PQ = 4$, $PR = 8$, and $\cos \angle P = \frac{1}{10}$. Find the length of angle bisector $\overline{PS}$. | {
"answer": "4.057",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Approximate the reading indicated by the arrow in the diagram of a measuring device. | {
"answer": "42.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The measure of angle $ACB$ is 30 degrees. If ray $CA$ is rotated 510 degrees about point $C$ clockwise, what will be the positive measure of the acute angle $ACB$, in degrees? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that P and Q are two points on the unit circle centered at the origin O, and they are located in the first and fourth quadrants, respectively. The x-coordinate of point P is $\frac{4}{5}$, and the x-coordinate of point Q is $\frac{5}{13}$. Then, $\cos \angle POQ = \_\_\_\_\_\_$. | {
"answer": "\\frac{56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two students were asked to add two positive integers. Alice subtracted the two numbers by mistake and obtained 3. Bob mistakenly multiplied the same two integers and got 63. What was the correct sum of the two integers? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle touches the extensions of two sides \(AB\) and \(AD\) of square \(ABCD\), and the point of tangency cuts off a segment of length \(2 + \sqrt{5 - \sqrt{5}}\) cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square, given that the angle between the tangents is \(72^\circ\), and it is known that \(\sin 36^\circ = \frac{\sqrt{5 - \sqrt{5}}}{2\sqrt{2}}\). | {
"answer": "\\frac{\\sqrt{\\sqrt{5} - 1} \\cdot \\sqrt[4]{125}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a department store, they received 10 suitcases and 10 keys separately in an envelope. Each key opens only one suitcase, and every suitcase can be matched with a corresponding key.
A worker in the department store, who received the suitcases, sighed:
- So much hassle with matching keys! I know how stubborn inanimate objects can be!! You start matching the key to the first suitcase, and it always turns out that only the tenth key fits. You'll try the keys ten times because of one suitcase, and because of ten - a whole hundred times!
Let’s summarize the essence briefly. A salesperson said that the number of attempts is no more than \(10+9+8+\ldots+2+1=55\), and another employee proposed to reduce the number of attempts since if the key does not fit 9 suitcases, it will definitely fit the tenth one. Thus, the number of attempts is no more than \(9+8+\ldots+1=45\). Moreover, they stated that this will only occur in the most unfortunate scenario - when each time the key matches the last suitcase. It should be expected that in reality the number of attempts will be roughly
\[\frac{1}{2} \times \text{the maximum possible number of attempts} = 22.5.\]
Igor Fedorovich Akulich from Minsk wondered why the expected number of attempts is half the number 45. After all, the last attempt is not needed only if the key does not fit any suitcase except the last one, but in all other cases, the last successful attempt also takes place. Akulich assumed that the statement about 22.5 attempts is unfounded, and in reality, it is a bit different.
**Problem:** Find the expected value of the number of attempts (all attempts to open the suitcases are counted - unsuccessful and successful, in the case where there is no clarity). | {
"answer": "29.62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value $2^{10} - 1$ is divisible by several prime numbers. What is the sum of these prime numbers? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a_{1}, a_{2}, \cdots, a_{10}\) be any 10 distinct positive integers such that \(a_{1} + a_{2} + \cdots + a_{10} = 1995\). Find the minimum value of \(a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{10} a_{1}\). | {
"answer": "6044",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Formulas for shortened multiplication (other).
Common fractions | {
"answer": "198719871987",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g: \mathbb{R} \to \mathbb{R}$ be a function such that
\[g((x + y)^2) = g(x)^2 - 2xg(y) + 2y^2\]
for all real numbers $x$ and $y.$ Find the number of possible values of $g(1)$ and the sum of all possible values of $g(1)$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\frac{\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region enclosed by the trajectory of the moving point $P$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.
Two neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ competes with each of the 20 fighters from village $B$. The stronger fighter wins. We say that village $A$ is stronger than village $B$ if a fighter from village $A$ wins at least $k$ of the 400 fights.
It turns out that each village is stronger than its neighboring village in a clockwise direction. Determine the maximum value of $k$ so that this can be the case. | {
"answer": "290",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arithmetic sequence $\{a_n\}$, $a_3+a_6+a_9=54$. Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Then, determine the value of $S_{11}$. | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\triangle ABC$ is an equilateral triangle with side length $s$, determine the value of $s$ when $AP = 2$, $BP = 2\sqrt{3}$, and $CP = 4$. | {
"answer": "\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by at least $3$ from all multiples of $p$. Considering the prime numbers $5$, $7$, and $11$, find the number of positive integers less than or equal to $15,000$ which are simultaneously $5$-safe, $7$-safe, and $11$-safe. | {
"answer": "975",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $16^n$ inclusive, considering a hexadecimal system in which digits range from 1 to 15. Find the integer $n$ for which $T_n$ becomes an integer. | {
"answer": "15015",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set of digits {1, 2, 3, 4, 5}, find the number of three-digit numbers that can be formed with the digits 2 and 3, where 2 is positioned before 3. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that \( m, n, \) and \( k \) are distinct natural numbers greater than 1, the number \( \log_{m} n \) is rational, and additionally,
$$
k^{\sqrt{\log_{m} n}} = m^{\sqrt{\log_{n} k}}
$$
Find the minimum possible value of the sum \( k + 5m + n \). | {
"answer": "278",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\ln x+\frac{1}{2}x^2-ax$, where $a\in \mathbb{R}$, it has two extreme points at $x=x_1$ and $x=x_2$, with $x_1 < x_2$.
(I) When $a=3$, find the extreme values of the function $f(x)$.
(II) If $x_2\geqslant ex_1$ ($e$ is the base of the natural logarithm), find the maximum value of $f(x_2)-f(x_1)$. | {
"answer": "1-\\frac{e}{2}+\\frac{1}{2e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the top vertex of the uppermost cube, determine the length of the portion of $\overline{XY}$ contained in the cube with edge length $4$. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For $\{1, 2, 3, \ldots, 10\}$ and each of its non-empty subsets, a unique alternating sum is defined similarly as before. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such alternating sums for $n=10$. | {
"answer": "5120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$ , where $a,\,b,\,c,\,d$ are positive integers. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | {
"answer": "15.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the length of side $x$ in the following diagram:
[asy]
import olympiad;
draw((0,0)--(2,0)--(0,2*sqrt(2))--cycle);
draw((0,0)--(-2,0)--(0,2*sqrt(2))--cycle);
label("10",(-1, sqrt(2)),NW);
label("$x$",(sqrt(2),sqrt(2)),NE);
draw("$30^{\circ}$",(2.2,0),NW);
draw("$45^{\circ}$",(-1.8,0),NE);
draw(rightanglemark((0,2*sqrt(2)),(0,0),(2,0),4));
[/asy] | {
"answer": "\\frac{20\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.