2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

Call a positive integer [i]challenging[/i] if it can be expressed as $2^a(2^b+1)$, where $a,b$ are positive integers. Prove that if $X$ is a set of challenging numbers smaller than $2^n (n$ is a given positive integer) and $|X|\ge \frac{4}{3}(n-1)$, there exist two disjoint subsets $A,B\subset X$ such that $|A|=|B|$ and $\sum_{a\in A}a=\sum_{b \in B}b$.

Find the smallest positive integer $x$ such that $2^{254}$ divides $x^{2005} + 1$.

In a village $X_0$ there are $80$ tourists who are about to visit $5$ nearby villages $X_1,X_2,X_3,X_4,X_5$.Each of them has chosen to visit only one of them.However,there are cases when the visit in a village forces the visitor to visit other villages among $X_1,X_2,X_3,X_4,X_5$.Each tourist visits only the village he has chosen and the villages he is forced to.If $X_1,X_2,X_3,X_4,X_5$ are totally visited by $40,60,65,70,75$ tourists respectively,then find how many tourists had chosen each one of them and determine all the ordered pairs $(X_i,X_j):i,j\in \{1,2,3,4,5\}$ which are such that,the visit in $X_i$ forces the visitor to visit $X_j$ as well.

Let $n$ be integer, $n>1.$ An element of the set $M=\{ 1,2,3,\ldots,n^2-1\}$ is called [i]good[/i] if there exists some element $b$ of $M$ such that $ab-b$ is divisible by $n^2.$ Furthermore, an element $a$ is called [i]very good[/i] if $a^2-a$ is divisible by $n^2.$ Let $g$ denote the number of [i]good[/i] elements in $M$ and $v$ denote the number of [i]very good[/i] elements in $M.$ Prove that \[v^2+v \leq g \leq n^2-n.\]

$\vartriangle ABC$ satisfies $AB = 16$, $BC = 30$, and $\angle ABC = 90^o$. On the circumcircle of $\vartriangle ABC$, let $P$ be the midpoint of arc $AC$ not containing $B$, and let $X$ and $Y$ lie on lines $AB$ and $BC$, respectively, with $PX \perp AB$ and $PY \perp BC$. Find $XY^2$.

Let $A = \{ 1,2, 3 , \ldots, 100 \}$ and $B$ be a subset of $A$ having $53$ elements. Show that $B$ has 2 distinct elements $x$ and $y$ whose sum is divisible by $11$.

Given two angles $ACT$ and $TCB$, where $A$, $C$ and $B$ lie on a line in that order. A circle $\alpha$ is inscribed in the first angle, and $\beta$ in the second. $\alpha$ is tangent to $AB$ and $CT$ at points $A$ and $E$, and $\beta$ is tangent to $AE$ and $BF$ at $B$ and $F \neq E$. Lines $AE$ and $BF$ intersect at $P$. Circumcircle $\omega$ of triangle $PEF$ intersects $\alpha$ and $\beta$ at $X$ and $Y$ respectively. Prove that $AX$ and $BY$ intersect on $\omega$. [i]Matsvei Zorka[/i]

For which finite group G does there exist natural number s with the following property: for any subgroup H of a finite direct power of G, each subgroup of H is produced as an intersection of subgroups of H with index at most s. not sure of translation.

A function $f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}$ is called [i]good[/i] if the function $g(n)=|f(n)-n|$ is injective. Furthermore, a good function $f$ is called [i]excellent[/i] if there exists another good function $f'$ such that $f(n)-f'(n)$ is nonzero for exactly one value of $n$. Let $N$ be the number of good functions that are not excellent. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Nathan Ramesh

Let $AL$ and $BK$ be the angle bisectors in a non-isosceles triangle $ABC,$ where $L$ lies on $BC$ and $K$ lies on $AC.$ The perpendicular bisector of $BK$ intersects the line $AL$ at $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK.$ Prove that $LN=NA.$

The point $ O$ is the center of the circle circumscribed about $ \triangle ABC$, with $ \angle BOC \equal{} 120^\circ$ and $ \angle AOB \equal{} 140^\circ$, as shown. What is the degree measure of $ \angle ABC$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair B=dir(80), A=dir(220), C=dir(320), O=(0,0); draw(unitcircle); draw(A--B--C--O--A--C); draw(O--B); draw(anglemark(C,O,A,2)); label("$A$",A,SW); label("$B$",B,NNE); label("$C$",C,SE); label("$O$",O,S); label("$140^{\circ}$",O,NW,fontsize(8pt)); label("$120^{\circ}$",O,ENE,fontsize(8pt));[/asy]$ \textbf{(A)}\ 35 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$

Is there a tetrahedron $ABCD$ such that $AB+BC+CD+DA=12\text{ cm}$ with volume $\mathrm V\ge2\sqrt3\text{ cm}^3?$

Let n be a set of integers. $S(n)$ is defined as the sum of the elements of n. $T=\{1,2,3,4,5,6,7,8,9\}$ and A and B are subsets of T such that A $\cup$ $B=T$ and A $\cap$ $B=\varnothing$. The probability that $S(A)\geq4S(B)$ can be expressed as $\frac{p}{q}$. Compute $p+q$. [i]2022 CCA Math Bonanza Team Round #8[/i]

A paper triangle with the angles $20^o$, $20^o$ and $140^o$ is cut into two triangles by the bisector of one of its angles. Then one of these triangles is cut into two by its bisector, and so on. Prove that it is impossible to get a triangle similar to the initial one. (AI Galochkin)

Determine all pairs $\{a, b\}$ of positive integers with the property that, in whatever manner you color the positive integers with two colors $A$ and $B$, there always exist two positive integers of color $A$ having their difference equal to $a$ [b]or[/b] of color $B$ having their difference equal to $b$.

Let {$a_1, a_2,\cdots,a_n$} be a set of $n$ real numbers whos sym equals S. It is known that each number in the set is less than $\frac{S}{n-1}$. Prove that for any three numbers $a_i$, $a_j$ and $a_k$ in the set, $a_i+a_j>a_k$.

Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$, $(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$. Find $\lim_{n\to \infty}(x_n,y_n)$.

On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does [b] not [/b] touch the outer edge of the board? [asy] unitsize(10); draw((0,0)--(8,0)--(8,8)--(0,8)--cycle); draw((1,8)--(1,0)); draw((7,8)--(7,0)); draw((6,8)--(6,0)); draw((5,8)--(5,0)); draw((4,8)--(4,0)); draw((3,8)--(3,0)); draw((2,8)--(2,0)); draw((0,1)--(8,1)); draw((0,2)--(8,2)); draw((0,3)--(8,3)); draw((0,4)--(8,4)); draw((0,5)--(8,5)); draw((0,6)--(8,6)); draw((0,7)--(8,7)); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,black); fill((6,0)--(7,0)--(7,1)--(6,1)--cycle,black); fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,black); fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black); fill((4,2)--(5,2)--(5,3)--(4,3)--cycle,black); fill((6,2)--(7,2)--(7,3)--(6,3)--cycle,black); fill((0,4)--(1,4)--(1,5)--(0,5)--cycle,black); fill((2,4)--(3,4)--(3,5)--(2,5)--cycle,black); fill((4,4)--(5,4)--(5,5)--(4,5)--cycle,black); fill((6,4)--(7,4)--(7,5)--(6,5)--cycle,black); fill((0,6)--(1,6)--(1,7)--(0,7)--cycle,black); fill((2,6)--(3,6)--(3,7)--(2,7)--cycle,black); fill((4,6)--(5,6)--(5,7)--(4,7)--cycle,black); fill((6,6)--(7,6)--(7,7)--(6,7)--cycle,black); fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,black); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,black); fill((5,1)--(6,1)--(6,2)--(5,2)--cycle,black); fill((7,1)--(8,1)--(8,2)--(7,2)--cycle,black); fill((1,3)--(2,3)--(2,4)--(1,4)--cycle,black); fill((3,3)--(4,3)--(4,4)--(3,4)--cycle,black); fill((5,3)--(6,3)--(6,4)--(5,4)--cycle,black); fill((7,3)--(8,3)--(8,4)--(7,4)--cycle,black); fill((1,5)--(2,5)--(2,6)--(1,6)--cycle,black); fill((3,5)--(4,5)--(4,6)--(3,6)--cycle,black); fill((5,5)--(6,5)--(6,6)--(5,6)--cycle,black); fill((7,5)--(8,5)--(8,6)--(7,6)--cycle,black); fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black); fill((3,7)--(4,7)--(4,8)--(3,8)--cycle,black); fill((5,7)--(6,7)--(6,8)--(5,8)--cycle,black); fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black);[/asy] $ \textbf{(A)}\frac{1}{16}\qquad\textbf{(B)}\frac{7}{16}\qquad\textbf{(C)}\frac12\qquad\textbf{(D)}\frac{9}{16}\qquad\textbf{(E)}\frac{49}{64} $

Let $C$ be a circle of radius $1$ and $O$ its center. Let $\overline{AB}$ be a chord of the circle and $D$ a point on $\overline{AB}$ such that $OD =\frac{\sqrt2}{2}$ such that $D$ is closer to $ A$ than it is to $ B$, and if the perpendicular line at $D$ with respect to $\overline{AB}$ intersects the circle at $E $and $F$, $AD = DE$. The area of the region of the circle enclosed by $\overline{AD}$, $\overline{DE}$, and the minor arc $AE$ may be expressed as $\frac{a + b\sqrt{c} + d\pi}{e}$ where $a, b, c, d, e$ are integers, gcd $(a, b, d, e) = 1$, and $c$ is squarefree. Find $a + b + c + d + e$

a) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{14}x_{15} = 0 \\ 1 - x_{15}x_1 = 0 \end{cases}$ b) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{n-1}x_{n} = 0 \\ 1 - x_{n}x_1 = 0 \end{cases}$ How does the solution vary for distinct values of $n$?

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

Ryan uses $91$ puzzle pieces to make a rectangle. Each of them is identical to one of the tiles shown. Given that pieces can be flipped or rotated, find the number of pieces that are red in the puzzle. (He is not allowed to join two ``flat sides'' together.)

Let $\Gamma$ is a circle with diameter $AB$. Let $\ell$ be the tangent of $\Gamma$ at $A$, and $m$ be the tangent of $\Gamma$ through $B$. Let $C$ be a point on $\ell$, $C \ne A$, and let $q_1$ and $q_2$ be two lines that passes through $C$. If $q_i$ cuts $\Gamma$ at $D_i$ and $E_i$ ($D_i$ is located between $C$ and $E_i$) for $i = 1, 2$. The lines $AD_1, AD_2, AE_1, AE_2$ intersects $m$ at $M_1, M_2, N_1, N_2$ respectively. Prove that $M_1M_2 = N_1N_2$.