Let $ [a]$ be the integer such that $ [a]\le a<[a]\plus{}1$. Find all real numbers $ (a,b,c)$ such that \[ \{a\}\plus{}[b]\plus{}\{c\}\equal{}2.9\\\{b\}\plus{}[c]\plus{}\{a\}\equal{}5.3\\\{c\}\plus{}[a]\plus{}\{b\}\equal{}4.0.\]

Andy and Eddie play a game in which they continuously flip a fair coin. They stop flipping when either they flip tails, heads, and tails consecutively in that order, or they flip three tails in a row. Then, if there has been an odd number of flips, Andy wins, and otherwise Eddie wins. Given that the probability that Andy wins is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Anderw Zhao and Zachary Perry[/i]

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

The numbers $1$ to $500$ are written on a board. Two pupils $A$ and $B$ play the following game: A player in turn deletes one of the numbers from the board. The game is over when only two numbers remain. Player $B$ wins if the sum of the two remaining numbers is divisible by $3,$ otherwise $A$ wins. If $A$ plays first, show that $B$ has a winning strategy.

Let $O$ be the center of an equilateral triangle $ABC$ of area $1/\pi$. As shown in the diagram below, a circle centered at $O$ meets the triangle at points $D$, $E$, $F$, $G$, $H$, and $I$, which trisect each of the triangle's sides. Compute the total area of all six shaded regions. [asy] unitsize(90); pair A = dir(0); pair B = dir(120); pair C = dir(240); draw(A -- B -- C -- cycle); pair D = (2*A + B)/3; pair E = (A + 2*B)/3; pair F = (2*B + C)/3; pair G = (B + 2*C)/3; pair H = (2*C + A)/3; pair I = (C + 2*A)/3; draw(E -- F); draw(G -- H); draw(I -- D); draw(D -- G); draw(E -- H); draw(F -- I); pair O = (0, 0); real r = 1/sqrt(3); draw(circle(O, r)); fill(O -- D -- E -- cycle, gray); fill(O -- F -- G -- cycle, gray); fill(O -- H -- I -- cycle, gray); fill(arc(O, r, -30, 30) -- cycle, gray); fill(arc(0, r, 90, 150) -- cycle, gray); fill(arc(0, r, 210, 270) -- cycle, gray); label("$A$", A, A); label("$B$", B, B); label("$C$", C, C); label("$D$", D, unit(D)); label("$E$", E, unit(E)); label("$F$", F, unit(F)); label("$G$", G, unit(G)); label("$H$", H, unit(H)); label("$I$", I, unit(I)); label("$O$", O, C); [/asy]

Determine the number of ordered quadruples of integers $(a,b,c,d)$ for which $0\leq a,b,c,d\leq 36$ and $37|a^2+b^2-c^3-d^3$.

Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer . (a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$. (b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$.

Compute the average of the integers $2, 3, 4, \dots, 2012$. [i]Proposed by Eugene Chen[/i]

In the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two numbers beneath it. Find the sum of all cells denoted with a question mark. [asy] unitsize(2cm); path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle; draw(box); label("$2$",(0,0)); draw(shift(1,0)*box); label("$?$",(1,0)); draw(shift(2,0)*box); label("$?$",(2,0)); draw(shift(3,0)*box); label("$?$",(3,0)); draw(shift(0.5,0.4)*box); label("$4$",(0.5,0.4)); draw(shift(1.5,0.4)*box); label("$?$",(1.5,0.4)); draw(shift(2.5,0.4)*box); label("$?$",(2.5,0.4)); draw(shift(1,0.8)*box); label("$5$",(1,0.8)); draw(shift(2,0.8)*box); label("$?$",(2,0.8)); draw(shift(1.5,1.2)*box); label("$9$",(1.5,1.2)); [/asy] $\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

Let $T$ denote the $15$-element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.

Prove that there infinitely many numbers of the form $18^{m}+45^{m}+50^{m}+125^{m}$, divisible by 2006. $m\in N$

In triangle $ABC$, the bisector of angle $B$ meets the opposite side $AC$ at $B'$. Similarly, the bisector of angle $C$ meets the opposite side $AB$ at $C'$ . Prove that $A=60^{\circ}$ if, and only if, $BC'+CB'=BC$.

Let $A_1,A_2,\ldots,A_{2017}$ be the vertices of a regular polygon with $2017$ sides.Prove that there exists a point $P$ in the plane of the polygon such that the vector $$\sum_{k=1}^{2017}k\frac{\overrightarrow{PA}_k}{\left\lVert\overrightarrow{PA}_k\right\rVert^5}$$ is the zero vector. (The notation $\left\lVert\overrightarrow{XY}\right\rVert$ represents the length of the vector $\overrightarrow{XY}$.)

A sequence consists of the digits $122333444455555\ldots$ such that the each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501$st and $4052$nd digits of this sequence.

Show that, for all positive real numbers $a, b, c$ such that $abc = 1$, the inequality $$\frac{1}{1 + a^2 + (b + 1)^2} +\frac{1}{1 + b^2 + (c + 1)^2} +\frac{1}{1 + c^2 + (a + 1)^2} \le \frac{1}{2}$$

Given $\triangle ABC$, whose all sides have different length. Point $P$ is chosen on altitude $AD$. Lines $BP$ and $CP$ intersect lines $AC, AB$ respectively and point $X, Y$.It is given that $AX=AY$. Prove that there is circle, whose centre lies on $BC$ and is tangent to sides $AC$ and $AB$ at points $X,Y$.

What is the largest number of cells that can be marked on a $100 \times 100$ board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.)

$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition: (1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$ (2) $d \mid (x_1+x_2+ \cdots x_n)$ Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.

In the diagram $ABCDEFG$ is a regular heptagon (a $7$ sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\dfrac{m}{n}$ where m and n are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); string lab[]={"A","B","C","D","E","F","G"}; real r = 360/7; pair A=dir(90-r),B=dir(90),C=dir(90+r),D=dir(90+2*r),E=dir(90+3*r),F=dir(90+4*r),G=dir(90+5*r); draw(A--E--B--F--C--G--D--cycle); for(int k = -1;k <= 5;++k) { label("$"+lab[k+1]+"$",dir(90+k*r),dir(90+k*r)); } [/asy]

Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether? [asy] picture corner; draw(corner,(5,0)--(35,0)); draw(corner,(0,-5)--(0,-35)); for (int i=0; i<3; ++i) { for (int j=0; j>-2; --j) { if ((i-j)<3) { add(corner,(50i,50j)); } } } draw((5,-100)--(45,-100)); draw((155,0)--(185,0),dotted+linewidth(2)); draw((105,-50)--(135,-50),dotted+linewidth(2)); draw((100,-55)--(100,-85),dotted+linewidth(2)); draw((55,-100)--(85,-100),dotted+linewidth(2)); draw((50,-105)--(50,-135),dotted+linewidth(2)); draw((0,-105)--(0,-135),dotted+linewidth(2));[/asy] $\textbf{(A)}\ 1920 \qquad \textbf{(B)}\ 1952 \qquad \textbf{(C)}\ 1980 \qquad \textbf{(D)}\ 2013 \qquad \textbf{(E)}\ 3932$

A traffic accident involved three cars: one blue, one green, and one red. Three witnesses spoke to the police and gave the following statements: **Person 1:** The red car was guilty, and either the green or the blue one was involved. **Person 2:** Either the green car or the red car was guilty, but not both. **Person 3:** Only one of the cars was guilty, but it was not the blue one. The police know that at least one car was guilty and that at least one car was not. However, the police do not know if any of the three witnesses lied. Which car(s) were responsible for the accident?

Cyclic quadrilateral $ABCD$ has $AC=AD=5, CD=6,$ and $AB=BC.$ If the length of $AB$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,c$ are relatively prime positive integers and $b$ is square-fre,e evaluate $a+b+c.$ [i]Proposed by Ada Tsui[/i]

Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[ \sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.