problem stringlengths 10 5.15k | answer dict |
|---|---|
**polyhedral**
we call a $12$ -gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points $(0,1)$ , $(1,0)$ , $(-1,0)$ and $(0,-1)$ .
find the faces of the <u>massivest</u> polyhedral that it's image on every three plane $xy$ , $yz$ and $zx$ is a good $12$ -gon.
(it's obvios that centers of these three $12$ -gons are the origin of coordinates for three dimensions.)
time allowed for this question is 1 hour. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB= 400$, $BC=480$, and $AC=560$. An interior point $P$ is identified, and segments are drawn through $P$ parallel to the sides of the triangle. These three segments are of equal length $d$. Determine $d$. | {
"answer": "218\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $a$, $b$, $c$ form an arithmetic sequence in that order.
$(1)$ If the vectors $\overrightarrow{m}=(3,\sin B)$ and $\overrightarrow{n}=(2,\sin C)$ are collinear, find the value of $\cos A$;
$(2)$ If $ac=8$, find the maximum value of the area $S$ of $\triangle ABC$. | {
"answer": "2 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the students are numbered from 01 to 70, determine the 7th individual selected by reading rightward starting from the number in the 9th row and the 9th column of the random number table. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$|2+i^{2}+2i^{2}|=$ | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $Q(x) = 0$ be the polynomial equation of the least possible degree, with rational coefficients, having $\sqrt[4]{13} + \sqrt[4]{169}$ as a root. Compute the product of all of the roots of $Q(x) = 0.$ | {
"answer": "-13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[\frac{(10^4+400)(26^4+400)(42^4+400)(58^4+400)}{(2^4+400)(18^4+400)(34^4+400)(50^4+400)}.\] | {
"answer": "962",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest $2$-digit prime factor of the integer $n = {300 \choose 150}$? | {
"answer": "89",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five distinct points are arranged on a plane such that the segments connecting them form lengths $a$, $a$, $a$, $a$, $a$, $b$, $b$, $2a$, and $c$. The shape formed by these points is no longer restricted to a simple polygon but could include one bend (not perfectly planar). What is the ratio of $c$ to $a$?
**A)** $\sqrt{3}$
**B)** $2$
**C)** $2\sqrt{3}$
**D)** $3$
**E)** $\pi$ | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that both $m$ and $n$ are non-negative integers, find the number of "simple" ordered pairs $(m, n)$ with a value of 2019. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M$ be the number of ways to write $3050$ in the form $3050 = b_3 \cdot 10^3 + b_2 \cdot 10^2 + b_1 \cdot 10 + b_0$, where the $b_i$'s are integers, and $0 \le b_i \le 99$. Find $M$. | {
"answer": "306",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parallelogram \(ABCD\) with \(\angle B = 60^\circ\). Point \(O\) is the center of the circumcircle of triangle \(ABC\). Line \(BO\) intersects the bisector of the exterior angle \(\angle D\) at point \(E\). Find the ratio \(\frac{BO}{OE}\). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \subseteq \{0, 1, 2, \cdots, 29\} \) such that for any integers \( k \) and any numbers \( a \) and \( b \) (possibly \( a = b \)), the expression \( a + b + 30k \) is not equal to the product of two consecutive integers. Determine the maximum possible number of elements in \( A \). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A, B, and C start from the same point on a circular track with a circumference of 360 meters: A starts first and runs in the counterclockwise direction; before A completes a lap, B and C start simultaneously and run in the clockwise direction; when A and B meet for the first time, C is exactly half a lap behind them; after some time, when A and C meet for the first time, B is also exactly half a lap behind them. If B’s speed is 4 times A’s speed, then how many meters has A run when B and C start? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polynomial \( x^{2n} + 1 + (x+1)^{2n} \) cannot be divided by \( x^2 + x + 1 \) under the condition that \( n \) is equal to: | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area of the Crescent Gemini. | {
"answer": "\\frac{17\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an acute-angled $\triangle ABC$, find the minimum value of $3 \tan B \tan C + 2 \tan A \tan C + \tan A \tan B$. | {
"answer": "6 + 2\\sqrt{3} + 2\\sqrt{2} + 2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bookshelf, there are four volumes of Astrid Lindgren's collected works in order, each containing 200 pages. A little worm living in these volumes burrowed a path from the first page of the first volume to the last page of the fourth volume. How many pages did the worm burrow through? | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $DEF$, $\angle E = 45^\circ$, $DE = 100$, and $DF = 100 \sqrt{2}$. Find the sum of all possible values of $EF$. | {
"answer": "\\sqrt{30000 + 5000(\\sqrt{6} - \\sqrt{2})}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $\triangle ABC$ with $\angle BAC = 90^\circ$, medians $\overline{AD}$ and $\overline{BE}$ are given such that $AD = 18$ and $BE = 24$. If $\overline{AD}$ is the altitude from $A$, find the area of $\triangle ABC$. | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that real numbers x and y satisfy x + y = 5 and x * y = -3, find the value of x + x^4 / y^3 + y^4 / x^3 + y. | {
"answer": "5 + \\frac{2829}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x,$ $y,$ $z$ be real numbers such that $0 \leq x, y, z \leq 1$. Find the maximum value of
\[
\frac{1}{(2 - x)(2 - y)(2 - z)} + \frac{1}{(2 + x)(2 + y)(2 + z)}.
\] | {
"answer": "\\frac{12}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the probability of bus No. 3 arriving at the bus stop within 5 minutes is 0.20 and the probability of bus No. 6 arriving within 5 minutes is 0.60, calculate the probability that the passenger can catch the bus he needs within 5 minutes. | {
"answer": "0.80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The expression $\frac{\sqrt{3}\tan 12^{\circ} - 3}{(4\cos^2 12^{\circ} - 2)\sin 12^{\circ}}$ equals \_\_\_\_\_\_. | {
"answer": "-4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2 - 96x + 2210}{x^2 + 65x - 3510}$. What is $\alpha + \beta$? | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\triangle ABC$ satisfies that $\cot A, \cot B, \cot C$ form an arithmetic sequence, then the maximum value of $\angle B$ is ______. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Give an example of a function that, when \( x \) is equal to a known number, takes the form \( \frac{0}{0} \), but as \( x \) approaches this number, tends to a certain limit. | {
"answer": "\\frac{8}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram below shows part of a city map. The small rectangles represent houses, and the spaces between them represent streets. A student walks daily from point $A$ to point $B$ on the streets shown in the diagram, and can only walk east or south. At each intersection, the student has an equal probability ($\frac{1}{2}$) of choosing to walk east or south (each choice is independent of others). What is the probability that the student will walk through point $C$? | {
"answer": "$\\frac{21}{32}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers? | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For an arithmetic sequence $\{a_n\}$ with a non-zero common difference, some terms $a_{k_1}$, $a_{k_2}$, $a_{k_3}$, ... form a geometric sequence $\{a_{k_n}\}$, and it is given that $k_1 \neq 1$, $k_2 \neq 2$, $k_3 \neq 6$. Find the value of $k_4$. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each twin from the first 4 sets shakes hands with all twins except his/her sibling and with one-third of the triplets; the remaining 8 sets of twins shake hands with all twins except his/her sibling but does not shake hands with any triplet; and each triplet shakes hands with all triplets except his/her siblings and with one-fourth of all twins from the first 4 sets only. | {
"answer": "394",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We draw diagonals in some of the squares on a chessboard in such a way that no two diagonals intersect at a common point. What is the maximum number of diagonals that can be drawn this way? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right square pyramid with base edges of length $12$ units each and slant edges of length $15$ units each is cut by a plane that is parallel to its base and $4$ units above its base. What is the volume, in cubic units, of the top pyramid section that is cut off by this plane? | {
"answer": "\\frac{1}{3} \\times \\left(\\frac{(144 \\cdot (153 - 8\\sqrt{153}))}{153}\\right) \\times (\\sqrt{153} - 4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A wooden cube with edges of length $3$ meters has square holes, of side one meter, centered in each face, cut through to the opposite face. Find the entire surface area, including the inside, of this cube in square meters. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of permutations of \( n \) distinct elements \( a_1, a_2, \cdots, a_n \) (where \( n \geqslant 2 \)) such that \( a_1 \) is not in the first position and \( a_2 \) is not in the second position. | {
"answer": "32,527,596",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangular box is to be cut from an equilateral triangle of length 30 cm. Find the largest possible volume of the box (in cm³). | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two distinguishable flagpoles, and there are $17$ flags, of which $11$ are identical red flags, and $6$ are identical white flags. Determine the number of distinguishable arrangements using all of the flags such that each flagpole has at least one flag and no two white flags on either pole are adjacent. Compute the remainder when this number is divided by $1000$. | {
"answer": "164",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the center of circle $M$ lies on the $y$-axis, the radius is $1$, and the chord intercepted by line $l: y = 2x + 2$ on circle $M$ has a length of $\frac{4\sqrt{5}}{5}$. Additionally, the circle center $M$ is located below line $l$.
(1) Find the equation of circle $M$;
(2) Let $A(t, 0), B(t + 5, 0) \, (-4 \leqslant t \leqslant -1)$, if $AC, BC$ are tangent lines to circle $M$, find the minimum value of the area of $\triangle ABC$. | {
"answer": "\\frac{125}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all real numbers \( p \) such that the cubic equation \( 5x^{3} - 5(p+1)x^{2} + (71p - 1)x + 1 = 66p \) has three roots, all of which are positive integers. | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.
Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$ . Find the expected value of $E$ .
(Note: Negative numbers are permitted, so 13-22 gives $E = -9$ . Any excess operators are parsed as signs, so -2-+3 gives $E=-5$ and -+-31 gives $E = 31$ . Trailing operators are discarded, so 2++-+ gives $E=2$ . A string consisting only of operators, such as -++-+, gives $E=0$ .)
*Proposed by Lewis Chen* | {
"answer": "1866",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the largest four-digit number $M$ has digits with a product of $72$, find the sum of its digits. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three of the four endpoints of the axes of an ellipse are, in some order, \[(10, -3), \; (15, 7), \; (25, -3).\] Find the distance between the foci of the ellipse. | {
"answer": "11.18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The maximum number that can be formed by the digits 0, 3, 4, 5, 6, 7, 8, 9, given that there are only thirty 2's and twenty-five 1's. | {
"answer": "199",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a whole number, $M$, such that $\frac{M}{5}$ is strictly between 9.5 and 10.5. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the difference between each number in a row and the number immediately to its left in the given diagram is the same, and the quotient of each number in a column divided by the number immediately above it is the same, then $a + b \times c =\quad$ | {
"answer": "540",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor, the difference between the highest and lowest ranking (from different judges) is at most 3. What is the highest score the winner could have obtained? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p = 2017,$ a prime number. Let $N$ be the number of ordered triples $(a,b,c)$ of integers such that $1 \le a,b \le p(p-1)$ and $a^b-b^a=p \cdot c$ . Find the remainder when $N$ is divided by $1000000.$ *Proposed by Evan Chen and Ashwin Sah*
*Remark:* The problem was initially proposed for $p = 3,$ and $1 \le a, b \le 30.$ | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{P}$ be a convex polygon with $50$ vertices. A set $\mathcal{F}$ of diagonals of $\mathcal{P}$ is said to be *$minimally friendly$* if any diagonal $d \in \mathcal{F}$ intersects at most one other diagonal in $\mathcal{F}$ at a point interior to $\mathcal{P}.$ Find the largest possible number of elements in a $\text{minimally friendly}$ set $\mathcal{F}$ . | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of cells required to mark on a chessboard so that each cell of the board (marked or unmarked) is adjacent by side to at least one marked cell? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $f(x) = x(x - m)^2$ reaches its maximum value at $x = -2$. Determine the value of $m$. | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cindy wants to arrange her coins into $X$ piles, each consisting of the same number of coins, $Y$. Each pile will have more than one coin and no pile will have all the coins. If there are 16 possible values for $Y$ given all of the restrictions, what is the smallest number of coins she could have? | {
"answer": "131072",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of \(\log _{2}\left[2^{3} 4^{4} 8^{5} \cdots\left(2^{20}\right)^{22}\right]\).
Choose one of the following options:
(a) 3290
(b) 3500
(c) 3710
(d) 4172 | {
"answer": "5950",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An assembly line produces an average of $85\%$ first-grade products. How many products need to be selected so that with a probability of 0.997, the deviation of the frequency of first-grade products from the probability $p=0.85$ does not exceed $0.01$ in absolute value? | {
"answer": "11171",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On graph paper, large and small triangles are drawn (all cells are square and of the same size). How many small triangles can be cut out from the large triangle? Triangles cannot be rotated or flipped (the large triangle has a right angle in the bottom left corner, the small triangle has a right angle in the top right corner). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A flag in the shape of a square has a symmetrical cross of uniform width, featuring a center decorated with a green square, and set against a yellow background. The entire cross (comprising both the arms and the green center) occupies 49% of the total area of the flag. What percentage of the flag's area is occupied by the green square? | {
"answer": "25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A light flashes green every 3 seconds. Determine the number of times the light has flashed green after 671 seconds. | {
"answer": "154",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ be a real number such that
\[ x^2 + 8 \left( \frac{x}{x-3} \right)^2 = 53. \]
Find all possible values of $y = \frac{(x - 3)^3 (x + 4)}{2x - 5}.$ | {
"answer": "\\frac{17000}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that angles $\angle A, \angle B, \angle C$ are the interior angles of triangle $ABC$, and vector $\alpha=\left(\cos \frac{A-B}{2}, \sqrt{3} \sin \frac{A+B}{2}\right)$ with $|\alpha|=\sqrt{2}$. If when $\angle C$ is maximized, there exists a moving point $M$ such that $|MA|, |AB|, |MB|$ form an arithmetic sequence, the maximum value of $|AB|$ is ____. | {
"answer": "\\frac{2\\sqrt{3} + \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $p$, $q$, $r$, $s$, $t$, and $u$ are integers for which $512x^3 + 64 = (px^2 + qx +r)(sx^2 + tx + u)$ for all x, find the value of $p^2 + q^2 + r^2 + s^2 + t^2 + u^2$. | {
"answer": "5472",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(1 \leqslant a_{1} \leqslant a_{2} \leqslant a_{3} \leqslant a_{4} \leqslant a_{5} \leqslant a_{6} \leqslant 64\), find the minimum value of \(Q = \frac{a_{1}}{a_{2}} + \frac{a_{3}}{a_{4}} + \frac{a_{5}}{a_{6}}\). | {
"answer": "3/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a, b, c$, and $d$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2 + b^2 &=& c^2 + d^2 &=& 1458, \\
ac &=& bd &=& 1156.
\end{array}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$. | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area bounded by the graph of $y = \arcsin(\cos x)$ and the $x$-axis on the interval $0 \le x \le 2\pi.$ | {
"answer": "\\frac{\\pi^2}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the series:
\[
\sum_{n=1}^\infty \frac{3^n}{1 + 3^n + 3^{n+1} + 3^{2n+1}}.
\] | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a park, 10,000 trees are planted in a square grid pattern (100 rows of 100 trees). What is the maximum number of trees that can be cut down such that if one stands on any stump, no other stumps are visible? (Trees can be considered thin enough for this condition.)
| {
"answer": "2500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x - \frac{1}{x} = \sqrt{3} \), find \( x^{2048} - \frac{1}{x^{2048}} \). | {
"answer": "277526",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A calculator has digits from 0 to 9 and signs of two operations. Initially, the display shows the number 0. You can press any keys. The calculator performs operations in the sequence of keystrokes. If an operation sign is pressed several times in a row, the calculator remembers only the last press. A distracted Scientist pressed many buttons in a random sequence. Find approximately the probability that the result of the resulting sequence of actions is an odd number? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$, find the value of $a$ such that the lines are parallel. | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the volume of the parallelepiped formed by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 4, find the volume of the parallelepiped formed by the vectors $\mathbf{2a} + \mathbf{b}$, $\mathbf{b} + 4\mathbf{c}$, and $\mathbf{c} - 5\mathbf{a}$. | {
"answer": "232",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express the quotient and remainder of $3232_5 \div 21_5$ in base $5$. | {
"answer": "130_5 \\, R2_5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $\dfrac{13! - 12! - 2 \times 11!}{10!}$. | {
"answer": "1430",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=3\sin \left(2x-\frac{\pi }{8}\right)$, determine the horizontal shift required to transform the graph of the function $y=3\sin 2x$. | {
"answer": "\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sam and Lee run at equal and constant rates. They also cycle and skateboard at equal and constant rates. Sam covers $120$ kilometers after running for $4$ hours, cycling for $5$ hours, and skateboarding for $3$ hours while Lee covers $138$ kilometers after running for $5$ hours, skateboarding for $4$ hours, and cycling for $3$ hours. Their running, cycling, and skateboarding rates are all whole numbers of kilometers per hour. Find the sum of the squares of Sam's running, cycling, and skateboarding rates. | {
"answer": "436",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the focus of the parabola $y^{2}=4x$, and a perpendicular line to the directrix is drawn from a point $M$ on the parabola, with the foot of the perpendicular being $N$. If $|MF|= \frac{4}{3}$, then $\angle NMF=$ . | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle \( A B C \) (with right angle at \( C \)), the medians \( A M \) and \( B N \) are drawn with lengths 19 and 22, respectively. Find the length of the hypotenuse of this triangle. | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assume that $y_1, y_2, ... , y_8$ are real numbers such that
\[
\begin{aligned}
y_1 + 4y_2 + 9y_3 + 16y_4 + 25y_5 + 36y_6 + 49y_7 + 64y_8 &= 3, \\
4y_1 + 9y_2 + 16y_3 + 25y_4 + 36y_5 + 49y_6 + 64y_7 + 81y_8 &= 15, \\
9y_1 + 16y_2 + 25y_3 + 36y_4 + 49y_5 + 64y_6 + 81y_7 + 100y_8 &= 140.
\end{aligned}
\]
Find the value of $16y_1 + 25y_2 + 36y_3 + 49y_4 + 64y_5 + 81y_6 + 100y_7 + 121y_8$. | {
"answer": "472",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle \(ABC\). On the extension of the hypotenuse \(BC\), a point \(D\) is chosen such that the line \(AD\) is tangent to the circumscribed circle \(\omega\) of triangle \(ABC\). The line \(AC\) intersects the circumscribed circle of triangle \(ABD\) at point \(E\). It turns out that the angle bisector of \(\angle ADE\) is tangent to the circle \(\omega\). In what ratio does point \(C\) divide the segment \(AE\)? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Zeus starts at the origin \((0,0)\) and can make repeated moves of one unit either up, down, left or right, but cannot make a move in the same direction twice in a row. What is the smallest number of moves that he can make to get to the point \((1056,1007)\)? | {
"answer": "2111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The side lengths \(a, b, c\) of triangle \(\triangle ABC\) satisfy the conditions:
1. \(a, b, c\) are all integers;
2. \(a, b, c\) form a geometric sequence;
3. At least one of \(a\) or \(c\) is equal to 100.
Find all possible sets of the triplet \((a, b, c)\). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest common divisor of $8!$ and $(6!)^3.$ | {
"answer": "11520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the circle described by the equation $x^2 - 4x + y^2 - 8y + 12 = 0$ that lies above the line $y = 3$. | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than 10,000 have at most three different digits? | {
"answer": "4119",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the grid made up of $1 \times 1$ squares, four digits of 2015 are written in the shaded areas. The edges are either horizontal or vertical line segments, line segments connecting the midpoints of adjacent sides of $1 \times 1$ squares, or the diagonals of $1 \times 1$ squares. What is the area of the shaded portion containing the digits 2015? | {
"answer": "$47 \\frac{1}{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is now 3:00:00 PM, as read on a 12-hour digital clock. In 315 hours, 58 minutes, and 16 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$? | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g(x) = |3\{x\} - 1.5|$ where $\{x\}$ denotes the fractional part of $x$. Determine the smallest positive integer $m$ such that the equation \[m g(x g(x)) = x\] has at least $3000$ real solutions. | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( c \) be the constant term in the expansion of \( \left(2 x+\frac{b}{\sqrt{x}}\right)^{3} \). Find the value of \( c \). | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price of a stock increased by $25\%$ during January, fell by $15\%$ during February, rose by $20\%$ during March, and fell by $x\%$ during April. The price of the stock at the end of April was the same as it had been at the beginning of January. Calculate the value of $x$. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence ${a_{n}}$ with the sum of its first $n$ terms denoted as $S_{n}$, if $S_{5}$, $S_{4}$, and $S_{6}$ form an arithmetic sequence, then determine the common ratio of the sequence ${a_{n}}$, denoted as $q$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of four numbers $p, q, r$, and $s$ is 100. If we increase $p$ by 10, we get the value $M$. If we decrease $q$ by 5, we get the value $M$. If we multiply $r$ by 10, we also get the value $M$. Lastly, if $s$ is divided by 2, it equals $M$. Determine the value of $M$. | {
"answer": "25.610",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the convex quadrilateral \(ABCD\), \(\angle ABC=60^\circ\), \(\angle BAD=\angle BCD=90^\circ\), \(AB=2\), \(CD=1\), and the diagonals \(AC\) and \(BD\) intersect at point \(O\). Find \(\sin \angle AOB\). | {
"answer": "\\frac{15 + 6\\sqrt{3}}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of
\[9y + \frac{1}{y^6}\] for \(y > 0\). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arithmetic sequence $\{a\_n\}$, the common difference $d=\frac{1}{2}$, and the sum of the first $100$ terms $S\_{100}=45$. Find the value of $a\_1+a\_3+a\_5+...+a\_{99}$. | {
"answer": "-69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + 2x + 15\] has three distinct roots, which are also roots of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 75x + c.\] Determine the value of $f(-1)$. | {
"answer": "-2773",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. After the recruitment and departure, 7 new leaders are elected from outside the organization. How many people total will be in the organization after four years? | {
"answer": "172",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain school organized a Chinese traditional culture activity week to promote traditional Chinese culture. During the activity period, a Chinese traditional culture knowledge competition was held, with classes participating in the competition. Each class selected 5 representatives through a Chinese traditional culture knowledge quiz to participate in the grade-level competition. The grade-level competition was divided into two stages: preliminary and final. In the preliminary stage, the questions of Chinese traditional culture were placed in two boxes labeled A and B. Box A contained 5 multiple-choice questions and 3 fill-in-the-blank questions, while box B contained 4 multiple-choice questions and 3 fill-in-the-blank questions. Each class representative team was required to randomly draw two questions to answer from either box A or box B. Each class representative team first drew one question to answer, and after answering, the question was not returned to the box. Then they drew a second question to answer. After answering the two questions, the questions were returned to their original boxes.
$(1)$ If the representative team of Class 1 drew 2 questions from box A, mistakenly placed the questions in box B after answering, and then the representative team of Class 2 answered the questions, with the first question drawn from box B. It is known that the representative team of Class 2 drew a multiple-choice question from box B. Find the probability that the representative team of Class 1 drew 2 multiple-choice questions from box A.
$(2)$ After the preliminary round, the top 6 representative teams from Class 6 and Class 18 entered the final round. The final round was conducted in the form of an idiom solitaire game, using a best-of-five format, meaning the team that won three rounds first won the match and the game ended. It is known that the probability of Class 6 winning the first round is $\frac{3}{5}$, the probability of Class 18 winning is $\frac{2}{5}$, and the probability of the winner of each round winning the next round is $\frac{2}{5}$, with each round having a definite winner. Let the random variable $X$ represent the number of rounds when the game ends. Find the expected value of the random variable $X$, denoted as $E(X)$. | {
"answer": "\\frac{537}{125}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with $AB=5$ , $AC=12$ and incenter $I$ . Let $P$ be the intersection of $AI$ and $BC$ . Define $\omega_B$ and $\omega_C$ to be the circumcircles of $ABP$ and $ACP$ , respectively, with centers $O_B$ and $O_C$ . If the reflection of $BC$ over $AI$ intersects $\omega_B$ and $\omega_C$ at $X$ and $Y$ , respectively, then $\frac{O_BO_C}{XY}=\frac{PI}{IA}$ . Compute $BC$ .
*2016 CCA Math Bonanza Individual #15* | {
"answer": "\\sqrt{109}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three different integers are randomly chosen from the set $$\{ -6, -3, 0, 2, 5, 7 \}$$. What is the probability that their sum is even? Express your answer as a common fraction. | {
"answer": "\\frac{19}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $a$ passes through point P(4, -3), find:
1. The value of $2\sin{a} - \cos{a}$
2. The coordinates of point P where the terminal side of angle $a$ intersects the unit circle. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square pyramid \(M-ABCD\) with a square base such that \(MA = MD\), \(MA \perp AB\), and the area of \(\triangle AMD\) is 1, find the radius of the largest sphere that can fit into this square pyramid. | {
"answer": "\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. | {
"answer": "619",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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