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In the election for the class president, Petya and Vasya competed. Over three hours, 27 students voted for one of the two candidates. During the first two hours, Petya received 9 more votes than Vasya. In the last two hours, Vasya received 9 more votes than Petya. In the end, Petya won. By the largest possible margin, how many more votes could Petya have received than Vasya? | 9 | 2/8 |
Define polynomials $f_{n}(x)$ for $n \geq 0$ by $f_{0}(x)=1, f_{n}(0)=0$ for $n \geq 1,$ and $$ \frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1) $$ for $n \geq 0 .$ Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes. | 101^{99} | 7/8 |
A cycle of three conferences had a constant attendance, meaning that each session had the same number of participants. However, half of those who attended the first session did not return; a third of those who attended the second conference only attended that one, and a quarter of those who attended the third conference did not attend either the first or the second. Knowing that there were 300 registered participants and that each one attended at least one conference, determine:
a) How many people attended each conference?
b) How many people attended all three conferences? | 37 | 5/8 |
A right cylinder with a height of 8 inches is enclosed inside another cylindrical shell of the same height but with a radius 1 inch greater than the inner cylinder. The radius of the inner cylinder is 3 inches. What is the total surface area of the space between the two cylinders, in square inches? Express your answer in terms of $\pi$. | 16\pi | 1/8 |
Let $n$ be a positive integer. Compute, in terms of $n$ , the number of sequences $(x_1,\ldots,x_{2n})$ with each $x_i\in\{0,1,2,3,4\}$ such that $x_1^2+\dots+x_{2n}^2$ is divisible by $5$ .
*2020 CCA Math Bonanza Individual Round #13* | 5^{2n-1}+4\cdot5^{n-1} | 4/8 |
Let \( T \) be the set of positive real numbers. Let \( g : T \to \mathbb{R} \) be a function such that
\[ g(x) g(y) = g(xy) + 2006 \left( \frac{1}{x} + \frac{1}{y} + 2005 \right) \] for all \( x, y > 0 \).
Let \( m \) be the number of possible values of \( g(3) \), and let \( t \) be the sum of all possible values of \( g(3) \). Find \( m \times t \). | \frac{6019}{3} | 3/8 |
Consider the game of "tic-tac-toe" on a 3D cubic grid $8 \times 8 \times 8$. How many straight lines can be indicated on which there are 8 symbols in a row? | 244 | 4/8 |
At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.
[color=#BF0000]Rewording of the last line for clarification:[/color]
Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other. | \left\lfloor \frac{n}{2} \right\rfloor +1 | 4/8 |
\(A, B, C, D\) are consecutive vertices of a parallelogram. Points \(E, F, P, H\) lie on sides \(AB\), \(BC\), \(CD\), and \(AD\) respectively. Segment \(AE\) is \(\frac{1}{3}\) of side \(AB\), segment \(BF\) is \(\frac{1}{3}\) of side \(BC\), and points \(P\) and \(H\) bisect the sides they lie on. Find the ratio of the area of quadrilateral \(EFPH\) to the area of parallelogram \(ABCD\). | 37/72 | 7/8 |
Consider a sequence $\{a_n\}$ with property P: if $a_p = a_q$ for $p, q \in \mathbb{N}^{*}$, then it must hold that $a_{p+1} = a_{q+1}$. Suppose the sequence $\{a_n\}$ has property P, and it is given that $a_1=1$, $a_2=2$, $a_3=3$, $a_5=2$, and $a_6+a_7+a_8=21$. Determine the value of $a_{2017}$. | 16 | 7/8 |
On the extension of side \(AC\) of the equilateral triangle \(ABC\) beyond point \(A\), a point \(M\) is taken, and circles are circumscribed around the triangles \(ABM\) and \(MBC\). Point \(A\) divides the arc \(MAB\) in the ratio \(MA:AB = n\). In what ratio does point \(C\) divide the arc \(MCB\)? | 2n+1 | 1/8 |
Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider $0$ to be a natural number. | 3 | 1/8 |
Find all real \( x \) such that \( 0 < x < \pi \) and \(\frac{8}{3 \sin x - \sin 3x} + 3 \sin^2 x \le 5\). | \frac{\pi}{2} | 7/8 |
There are 11 quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this
$$
\star x^{2}+\star x+\star=0 \text {. }
$$
Two players are playing a game making alternating moves. In one move each of them replaces one star with a real nonzero number.
The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible.
What is the maximal number of equations without roots that the first player can achieve if the second player plays to her best? Describe the strategies of both players. | 6 | 1/8 |
Let \(ABCD\) be a rectangle inscribed in circle \(\Gamma\), and let \(P\) be a point on minor arc \(AB\) of \(\Gamma\). Suppose that \(PA \cdot PB = 2\), \(PC \cdot PD = 18\), and \(PB \cdot PC = 9\). The area of rectangle \(ABCD\) can be expressed as \(\frac{a \sqrt{b}}{c}\), where \(a\) and \(c\) are relatively prime positive integers and \(b\) is a squarefree positive integer. Compute \(100a + 10b + c\). | 21055 | 1/8 |
What is the base $2$ representation of $84_{10}$? | 1010100_2 | 1/8 |
Given an equilateral triangle ABC. Point D is chosen on the extension of side AB beyond point A, point E on the extension of BC beyond point C, and point F on the extension of AC beyond point C such that CF = AD and AC + EF = DE. Find the angle BDE. | 60 | 7/8 |
Three circles $\mathcal{K}_1$ , $\mathcal{K}_2$ , $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$ . The point $O$ lies inside the triangle $ABC$ .
Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$ . Let $ \alpha = \frac {OA_1}{AA_1} $ , $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$ . Prove that
\[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \] | \alphaR_1+\betaR_2+\gammaR_3\geR | 1/8 |
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. | \sqrt{2} - 1 | 1/8 |
Prove that if \( A, B, C, \) and \( D \) are arbitrary points on a plane, then four circles, each passing through three points: the midpoints of the segments \( AB, AC \) and \( AD; BA, BC \) and \( BD; CA, CB \) and \( CD; DA, DB \) and \( DC \), have a common point.
| 4 | 1/8 |
A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \frac{\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
$\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)} \ 2+2\sqrt{3} \qquad\textbf{(C)} \ 6 \qquad\textbf{(D)} \ 3 + 2\sqrt{3} \qquad\textbf{(E)} \ 6 + \frac{\sqrt{3}}{3}$ | \textbf{(D)}\3+2\sqrt{3} | 1/8 |
$ABCD$ - quadrilateral inscribed in circle, and $AB=BC,AD=3DC$ . Point $R$ is on the $BD$ and $DR=2RB$ . Point $Q$ is on $AR$ and $\angle ADQ = \angle BDQ$ . Also $\angle ABQ + \angle CBD = \angle QBD$ . $AB$ intersect line $DQ$ in point $P$ .
Find $\angle APD$ | 90 | 1/8 |
The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George and Alex? | 3795 | 5/8 |
How many 8-digit numbers begin with 1 , end with 3 , and have the property that each successive digit is either one more or two more than the previous digit, considering 0 to be one more than 9 ? | 21 | 7/8 |
A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7 \times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell). | 43 | 1/8 |
In an \( n \times n \) grid, each cell is filled with one of the numbers from \( 1 \) to \( n^2 \). If, no matter how the grid is filled, there are always two adjacent cells whose numbers differ by at least 1011, find the smallest possible value of \( n \). | 2020 | 2/8 |
Let points \( C \) and \( D \) be the trisection points of \( AB \). At 8:00, person \( A \) starts walking from \( A \) to \( B \) at a constant speed. At 8:12, person \( B \) starts walking from \( B \) to \( A \) at a constant speed. After a few more minutes, person \( C \) starts walking from \( B \) to \( A \) at a constant speed. When \( A \) and \( B \) meet at point \( C \), person \( C \) reaches point \( D \). When \( A \) and \( C \) meet at 8:30, person \( B \) reaches \( A \) exactly at that time. At what time did person \( C \) start walking? | 8:16 | 7/8 |
A box contains seven cards, each with a different integer from 1 to 7 written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh "I know the sum of the numbers on your cards is even." What is the sum of the numbers on Avani's cards?
A 6
B 9
C 10
D 11
E 12 | 12 | 1/8 |
Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$ ? | 3267 | 1/8 |
In triangle $ABC$ , $AB = 28$ , $AC = 36$ , and $BC = 32$ . Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$ , and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$ . Find the length of segment $AE$ .
*Ray Li* | 18 | 7/8 |
When point P moves on the circle $C: x^2 - 4x + y^2 = 0$, there exist two fixed points $A(1, 0)$ and $B(a, 0)$, such that $|PB| = 2|PA|$, then $a = \ $. | -2 | 7/8 |
It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{1}{3}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{3}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^2 y \). | \frac{4}{5} | 6/8 |
Interior numbers begin in the third row of Pascal's Triangle. Calculate the sum of the squares of the interior numbers in the eighth row. | 3430 | 2/8 |
Ben "One Hunna Dolla" Franklin is flying a kite KITE such that $I E$ is the perpendicular bisector of $K T$. Let $I E$ meet $K T$ at $R$. The midpoints of $K I, I T, T E, E K$ are $A, N, M, D$, respectively. Given that $[M A K E]=18, I T=10,[R A I N]=4$, find $[D I M E]$. | 16 | 2/8 |
What is the smallest base-10 integer that can be represented as $XX_6$ and $YY_8$, where $X$ and $Y$ are valid digits in their respective bases? | 63 | 1/8 |
Determine the largest natural number \( n \) such that
\[ 4^{995} + 4^{1500} + 4^{n} \]
is a square number. | 2004 | 6/8 |
Given that the domain of the function \( f \) is
$$
\{(x, y) \mid x, y \in \mathbf{R}, xy \neq 0\},
$$
and it takes values in the set of positive real numbers, satisfying the following conditions:
(i) For any \( x, y \neq 0 \), we have
$$
f(xy, z) = f(x, z) f(y, z);
$$
(ii) For any \( x, y \neq 0 \), we have
$$
f(x, yz) = f(x, y) f(x, z);
$$
(iii) For any \( x \neq 0, 1 \), we have
$$
f(x, 1-x) = 1.
$$
Prove:
(1) For all \( x \neq 0 \), we have
$$
f(x, x) = f(x, -x) = 1;
$$
(2) For all \( x, y \neq 0 \), we have
$$
f(x, y) f(y, x) = 1.
$$ | f(x,y)f(y,x)=1 | 1/8 |
In quadrilateral $ABCD$, there exists a point $E$ on segment $AD$ such that $\frac{AE}{ED}=\frac{1}{9}$ and $\angle BEC$ is a right angle. Additionally, the area of triangle $CED$ is 27 times more than the area of triangle $AEB$. If $\angle EBC=\angle EAB, \angle ECB=\angle EDC$, and $BC=6$, compute the value of $AD^{2}$. | 320 | 1/8 |
For each integer $n\geqslant2$ , determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\]
*(4th Middle European Mathematical Olympiad, Team Competition, Problem 2)* | \frac{1}{2n} | 3/8 |
How many non-negative integers with not more than 1993 decimal digits have non-decreasing digits? [For example, 55677 is acceptable, 54 is not.] | \binom{2002}{9} | 1/8 |
Four roads branch off a highway to four villages A, B, C, and D sequentially. It is known that the distance along the road/highway/road from A to B is 9 km, from A to C is 13 km, from B to C is 8 km, and from B to D is 14 km. Find the length of the path along the road/highway/road from A to D. Explain your answer. | 19\, | 1/8 |
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^n a_n = 2n-1$. Find the sum of the first $80$ terms of $\{a_n\}$. | 3240 | 4/8 |
The product of several distinct positive integers is divisible by ${2006}^{2}$ . Determine the minimum value the sum of such numbers can take. | 228 | 1/8 |
Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$. | 295 | 1/8 |
A computer program evaluates expressions without parentheses in the following way:
1) First, it performs multiplications and divisions from left to right one by one.
2) Then, it performs additions and subtractions from left to right.
For example, the value of the expression $1-2 / 3-4$ is $-3 \frac{2}{3}$. How many different results can we get if in the following expression each $*$ is independently replaced by one of the operators $+$, $-$, $/$, $\times$?
$1 * 1 * 1 * 1 * 1 * 1 * 1 * 1$ | 15 | 3/8 |
Given that $\alpha$ and $\beta$ are acute angles, and the equation $\frac{\sin ^{4} \alpha}{\cos ^{2} \beta}+\frac{\cos ^{4} \alpha}{\sin ^{2} \beta}=1$ holds, prove that $\alpha + \beta = \frac{\pi}{2}$. | \alpha+\beta=\frac{\pi}{2} | 1/8 |
The length of a crocodile from head to tail is three times smaller than ten ken, and from tail to head it is three ken and two shaku. It is known that one shaku is equal to 30 cm. Find the length of the crocodile in meters. (Ken and shaku are Japanese units of length.) | 6\, | 1/8 |
\( p(x) \) is a polynomial with integer coefficients. The sequence of integers \( a_1, a_2, \ldots, a_n \) (where \( n > 2 \)) satisfies \( a_2 = p(a_1) \), \( a_3 = p(a_2) \), \(\ldots\), \( a_n = p(a_{n-1}) \), \( a_1 = p(a_n) \). Show that \( a_1 = a_3 \). | a_1=a_3 | 4/8 |
Angry reviews about the work of an online store are left by $80\%$ of dissatisfied customers (those who were poorly served in the store). Of the satisfied customers, only $15\%$ leave a positive review. A certain online store earned 60 angry and 20 positive reviews. Using this statistic, estimate the probability that the next customer will be satisfied with the service in this online store. | 0.64 | 7/8 |
Given the function $g(x)=\ln x+\frac{1}{2}x^{2}-(b-1)x$.
(1) If the function $g(x)$ has a monotonically decreasing interval, find the range of values for the real number $b$;
(2) Let $x_{1}$ and $x_{2}$ ($x_{1} < x_{2}$) be the two extreme points of the function $g(x)$. If $b\geqslant \frac{7}{2}$, find the minimum value of $g(x_{1})-g(x_{2})$. | \frac{15}{8}-2\ln 2 | 3/8 |
In the rhombus \(ABCD\), the heights \(BP\) and \(BQ\) intersect the diagonal \(AC\) at points \(M\) and \(N\) (point \(M\) lies between \(A\) and \(N\)), \(AM = p\) and \(MN = q\). Find \(PQ\). | \frac{q(2p+q)}{p+q} | 2/8 |
A regular triangle is constructed on the diameter of a semicircle such that its sides are equal to the diameter. What is the ratio of the areas of the parts of the triangle lying outside and inside the semicircle? | \frac{3\sqrt{3}-\pi}{3\sqrt{3}+\pi} | 1/8 |
The equality $\frac{1}{x-1}=\frac{2}{x-2}$ is satisfied by:
$\textbf{(A)}\ \text{no real values of }x\qquad\textbf{(B)}\ \text{either }x=1\text{ or }x=2\qquad\textbf{(C)}\ \text{only }x=1\\ \textbf{(D)}\ \text{only }x=2\qquad\textbf{(E)}\ \text{only }x=0$ | \textbf{(E)}\onlyx=0 | 1/8 |
Given $|\vec{a}|=1$, $|\vec{b}|=6$, and $\vec{a}\cdot(\vec{b}-\vec{a})=2$, calculate the angle between $\vec{a}$ and $\vec{b}$. | \dfrac{\pi}{3} | 2/8 |
If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount:
$\textbf{(A)}\ {\pm 2ax + a^2}\qquad \textbf{(B)}\ {2ax \pm a^2}\qquad \textbf{(C)}\ {\pm a^2 - 3} \qquad \textbf{(D)}\ {(x + a)^2 - 3}\qquad\\ \textbf{(E)}\ {(x - a)^2 - 3}$ | \textbf{(A)}\{\2ax+^2} | 1/8 |
Complex numbers \( p, q, r \) form an equilateral triangle with a side length of 24 in the complex plane. If \( |p + q + r| = 48 \), find \( |pq + pr + qr| \). | 768 | 5/8 |
Four points \( A, O, B, O' \) are aligned in this order on a line. Let \( C \) be the circle centered at \( O \) with radius 2015, and \( C' \) be the circle centered at \( O' \) with radius 2016. Suppose that \( A \) and \( B \) are the intersection points of two common tangents to the two circles. Calculate \( AB \) given that \( AB \) is an integer \( < 10^7 \) and that \( AO \) and \( AO' \) are integers. | 8124480 | 2/8 |
Find all roots of the polynomial $x^3+x^2-4x-4$. Enter your answer as a list of numbers separated by commas. | -1,2,-2 | 3/8 |
Calculate the land tax on a plot of land with an area of 15 acres, if the cadastral value of one acre is 100,000 rubles, and the tax rate is $0.3 \%$. | 4500 | 5/8 |
How many squares of side at least $8$ have their four vertices in the set $H$, where $H$ is defined by the points $(x,y)$ with integer coordinates, $-8 \le x \le 8$ and $-8 \le y \le 8$? | 285 | 1/8 |
If $x$ is positive and $\log{x} \ge \log{2} + \frac{1}{2}\log{x}$, then:
$\textbf{(A)}\ {x}\text{ has no minimum or maximum value}\qquad \\ \textbf{(B)}\ \text{the maximum value of }{x}\text{ is }{1}\qquad \\ \textbf{(C)}\ \text{the minimum value of }{x}\text{ is }{1}\qquad \\ \textbf{(D)}\ \text{the maximum value of }{x}\text{ is }{4}\qquad \\ \textbf{(E)}\ \text{the minimum value of }{x}\text{ is }{4}$ | \textbf{(E)}\ | 1/8 |
Let $\Delta ABC$ be an equilateral triangle. How many squares in the same plane as $\Delta ABC$ share two vertices with the triangle? | 9 | 2/8 |
There are two engineering teams, Team A and Team B, each with a certain number of people. If 90 people are transferred from Team A to Team B, then the total number of people in Team B becomes twice the number of people in Team A. If a certain number of people are transferred from Team B to Team A, then the total number of people in Team A becomes six times the number of people in Team B. What is the minimum number of people originally in Team A? | 153 | 7/8 |
Let's define "addichiffrer" as the process of adding all the digits of a number. For example, if we addichiffrer 124, we get $1+2+4=7$.
What do we obtain when we addichiffrer $1998^{1998}$, then addichiffrer the result, and continue this process three times in total? | 9 | 7/8 |
Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. Determine how many positive integers $b_1 \le 3000$ are such that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$. | 2000 | 1/8 |
In an orthocentric tetrahedron \(ABCD\), the angle \(ADC\) is a right angle. Prove that \(\frac{1}{h^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\), where \(h\) is the length of the altitude of the tetrahedron drawn from vertex \(D\), \(a=DA\), \(b=DB\), and \(c=DC\). | \frac{1}{^2}=\frac{1}{^2}+\frac{1}{b^2}+\frac{1}{^2} | 4/8 |
Arrange $\frac{n(n+1)}{2}$ different real numbers into a triangular number table, with $k$ numbers in the $k$th row ($k=1,2,\cdots,n$). Let the maximum number in the $k$th row be $M_{k}$. Find the probability that $M_{1}<M_{2}<\cdots<M_{n}$. | \frac{2^n}{(n+1)!} | 1/8 |
Given the lateral area of a cylinder with a square cross-section is $4\pi$, calculate the volume of the cylinder. | 2\pi | 7/8 |
Solve in the set of real numbers the equation \[ 3x^3 \minus{} [x] \equal{} 3,\] where $ [x]$ denotes the integer part of $ x.$ | x = \sqrt [3]{\frac {4}{3}} | 7/8 |
If $\angle A = 60^\circ$, $\angle E = 40^\circ$ and $\angle C = 30^\circ$, then $\angle BDC =$ | 50^\circ | 1/8 |
Given that the simplest quadratic radical $\sqrt{m+1}$ and $\sqrt{8}$ are of the same type of quadratic radical, the value of $m$ is ______. | m = 1 | 7/8 |
In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating $2$ or $3$ new chains. For example, if she cuts the $1111$ th link out of her chain first, then she will have $3$ chains, of lengths $1110$ , $1$ , and $907$ . What is the least number of links she needs to remove in order to be able to pay for anything costing from $1$ to $2018$ links using some combination of her chains?
*2018 CCA Math Bonanza Individual Round #10* | 10 | 1/8 |
In a math competition, 5 problems were assigned. There were no two contestants who solved exactly the same problems. However, for any problem that is disregarded, for each contestant there is another contestant who solved the same set of the remaining 4 problems. How many contestants participated in the competition? | 32 | 5/8 |
If \( AC = 1.5 \, \text{cm} \) and \( AD = 4 \, \text{cm} \), what is the relationship between the areas of triangles \( \triangle ABC \) and \( \triangle DBC \)? | 3/5 | 5/8 |
Suppose that \(a, b, c,\) and \(d\) are positive integers which are not necessarily distinct. If \(a^{2}+b^{2}+c^{2}+d^{2}=70\), what is the largest possible value of \(a+b+c+d?\) | 16 | 5/8 |
What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another? | 50 | 5/8 |
A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?
$\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$ | \textbf{(B)}76 | 1/8 |
On the side \(AB\) of an equilateral triangle \(\mathrm{ABC}\), a right triangle \(\mathrm{AHB}\) is constructed (\(\mathrm{H}\) is the vertex of the right angle) such that \(\angle \mathrm{HBA}=60^{\circ}\). Let the point \(K\) lie on the ray \(\mathrm{BC}\) beyond the point \(\mathrm{C}\), and \(\angle \mathrm{CAK}=15^{\circ}\). Find the angle between the line \(\mathrm{HK}\) and the median of the triangle \(\mathrm{AHB}\) drawn from the vertex \(\mathrm{H}\). | 15 | 1/8 |
The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$ , find all real numbers $x$ satisfying the equation $P_{n}(x)=0$ . | 0 | 2/8 |
For the set \( \mathrm{T} = \{1, 2, \ldots, 999\} \), find the maximum number \( \mathrm{k} \) of different subsets \( \mathrm{A}_1, \mathrm{A}_2, \ldots, \mathrm{A}_\mathrm{k} \) such that for any \( \mathrm{i}, \mathrm{j} \) with \( 1 \leq \mathrm{i} < \mathrm{j} \leq \mathrm{k} \), we have \( A_i \cup A_j = \mathrm{T} \). | 1000 | 4/8 |
Originally, there were 5 books on the bookshelf. If 2 more books are added, but the relative order of the original books must remain unchanged, then there are $\boxed{\text{different ways}}$ to place the books. | 42 | 7/8 |
The determinant
$$
D=\left|\begin{array}{rrr}
3 & 1 & 2 \\
-1 & 2 & 5 \\
0 & -4 & 2
\end{array}\right|
$$
to be expanded: a) by the elements of the 1st row; b) by the elements of the 2nd column. | 82 | 6/8 |
Currently, the exchange rates for the dollar and euro are as follows: $D = 6$ yuan and $E = 7$ yuan. The People's Bank of China determines these exchange rates regardless of market conditions and follows a strategy of approximate currency parity. One bank employee proposed the following scheme for changing the exchange rates. Each year, the exchange rates can be adjusted according to the following four rules: Either change $D$ and $E$ to the pair $(D + E, 2D \pm 1)$, or to the pair $(D + E, 2E \pm 1)$. Moreover, it is prohibited for the dollar and euro rates to be equal at the same time.
For example: From the pair $(6, 7)$, after one year the following pairs are possible: $(13, 11)$, $(11, 13)$, $(13, 15)$, or $(15, 13)$. What is the smallest possible value of the difference between the higher and lower of the simultaneously resulting exchange rates after 101 years? | 2 | 4/8 |
A circle centered at $O$ has radius 1 and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\angle
AOB=\theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, then express $OC$ in terms of $s$ and $c,$ where $s = \sin \theta$ and $c = \cos \theta.$
[asy]
pair A,B,C,O;
O=(0,0);
A=(1,0);
C=(0.6,0);
B=(1,2);
label("$\theta$",(0.1,0),NE);
label("$O$",O,S);
label("$C$",C,S);
label("$A$",A,E);
label("$B$",B,E);
draw(A--O--B--cycle,linewidth(0.7));
draw(C--B,linewidth(0.7));
draw(Circle(O,1),linewidth(0.7));
[/asy] | \frac{1}{1 + s} | 5/8 |
A country has a total of 1985 airports. From each airport, a plane takes off and lands at the farthest airport from its origin. Is it possible for all 1985 planes to land at just 50 specific airports? (Assume the ground is a plane, flights are in straight lines, and the distance between any two airports is unique.) | Yes | 2/8 |
Given point $P(2,-1)$,
(1) Find the general equation of the line that passes through point $P$ and has a distance of 2 units from the origin.
(2) Find the general equation of the line that passes through point $P$ and has the maximum distance from the origin. Calculate the maximum distance. | \sqrt{5} | 7/8 |
A circle with radius \(\frac{2}{\sqrt{3}}\) is inscribed in an isosceles trapezoid. The angle between the diagonals of the trapezoid, which intersects at the base, is \(2 \operatorname{arctg} \frac{2}{\sqrt{3}}\). Find the segment connecting the points of tangency of the circle with the larger base of the trapezoid and one of its lateral sides. | 2 | 1/8 |
Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime. | (2,14) | 5/8 |
Let \( ABC \) be a right triangle with the hypotenuse \( BC \) measuring \( 4 \) cm. The tangent at \( A \) to the circumcircle of \( ABC \) meets the line \( BC \) at point \( D \). Suppose \( BA = BD \). Let \( S \) be the area of triangle \( ACD \), expressed in square centimeters. Calculate \( S^2 \). | 27 | 7/8 |
Given that \(\alpha, \beta \in \left(\frac{3\pi}{4}, \pi \right)\), \(\cos (\alpha + \beta) = \frac{4}{5}\), and \(\sin \left(\alpha - \frac{\pi}{4}\right) = \frac{12}{13}\), find \(\cos \left(\beta + \frac{\pi}{4}\right)\). | -\frac{56}{65} | 7/8 |
Find the smallest positive integer $n$ such that, if there are initially $2n$ townspeople and 1 goon, then the probability the townspeople win is greater than $50\%$. | 3 | 1/8 |
Among 50 school teams participating in the HKMO, no team answered all four questions correctly. The first question was solved by 45 teams, the second by 40 teams, the third by 35 teams, and the fourth by 30 teams. How many teams solved both the third and the fourth questions? | 15 | 5/8 |
Today is January 30th, so we first write down 130. The rule for writing subsequent numbers is: if the last number written is even, divide it by 2 and then add 2; if the last number written is odd, multiply it by 2 and then subtract 2. Thus the sequence obtained is: $130, 67, 132, 68, \cdots$. What is the 2016th number in this sequence? | 6 | 7/8 |
In $\triangle PQR$, points $X$ and $Y$ lie on $\overline{QR}$ and $\overline{PR}$, respectively. If $\overline{PX}$ and $\overline{QY}$ intersect at $Z$ such that $PZ/ZX = 2$ and $QZ/ZY = 5$, what is $RX/RY$? | \frac{5}{4} | 1/8 |
If the fractional equation $\frac{3}{{x-2}}+1=\frac{m}{{4-2x}}$ has a root, then the value of $m$ is ______. | -6 | 3/8 |
Find all positive integers \( k \) such that for any positive numbers \( a, b, c \) satisfying \( k(ab + bc + ca) > 5(a^2 + b^2 + c^2) \), there exists a triangle with side lengths \( a, b, c \). | 6 | 1/8 |
It is given an acute triangle $ABC$ , $AB \neq AC$ where the feet of altitude from $A$ its $H$ . In the extensions of the sides $AB$ and $AC$ (in the direction of $B$ and $C$ ) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.
Find the ratio $\tfrac{HP}{HA}$ . | 1 | 1/8 |
Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a}(\overrightarrow{a}+ \overrightarrow{b})=5$, and $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{\pi}{3} | 4/8 |
Given the parabola $y=x^2$ and the moving line $y=(2t-1)x-c$ have common points $(x_1, y_1)$, $(x_2, y_2)$, and $x_1^2+x_2^2=t^2+2t-3$.
(1) Find the range of the real number $t$;
(2) When does $t$ take the minimum value of $c$, and what is the minimum value of $c$? | \frac{11-6\sqrt{2}}{4} | 5/8 |
Equilateral triangle \( \triangle ABC \) and square \( ABDE \) have a common side \( AB \). The cosine of the dihedral angle \( C-ABD \) is \(\frac{\sqrt{3}}{3}\). If \( M \) and \( N \) are the midpoints of \( AC \) and \( BC \) respectively, then the cosine of the angle between \( EM \) and \( AN \) is \(\qquad\). | \frac{1}{6} | 4/8 |
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