One hundred sages play the following game. They are waiting in some fixed order in front of a room. The sages enter the room one after another. When a sage enters the room, the following happens - the guard in the room chooses two arbitrary distinct numbers from the set {$1,2,3$}, and announces them to the sage in the room. Then the sage chooses one of those numbers, tells it to the guard, and leaves the room, and the next enters, and so on. During the game, before a sage chooses a number, he can ask the guard what were the chosen numbers of the previous two sages. During the game, the sages cannot talk to each other. At the end, when everyone has finished, the game is considered as a failure if the sum of the 100 chosen numbers is exactly $200$; else it is successful. Prove that the sages can create a strategy, by which they can win the game.

Let $q$ be a real number. Suppose there are three distinct positive integers $a, b,c$ such that $q + a$, $q + b$,$q + c$ is a geometric progression. Show that $q$ is rational.

Points $A, B, C$ and $D$ lie on a circle, in that order clockwise, such that there is a point $E$ on segment $CD$ with the property that $AD = DE$ and $BC = EC$. Prove that the intersection point of the bisectors of the angles $\angle DAB$ and $\angle ABC$ is on the line $CD$.

The sequence $(a_n)$ is determined by $a_1 = 0$ and $(n+1)^3a_{n+1} = 2n^2(2n+1)a_n+2(3n+1)$ for $n \geq 1$. Prove that infinitely many terms of the sequence are positive integers.

$G$ is a group that order of each element of it Commutator group is finite. Prove that subset of all elemets of $G$ which have finite order is a subgroup og $G$.

What is the correct ordering of the three numbers $\frac5{19}$, $\frac7{21}$, and $\frac9{23}$, in increasing order? $\textbf{(A)}\hspace{.05in}\dfrac9{23} < \dfrac7{21} < \dfrac5{19}$ $\textbf{(B)}\hspace{.05in}\dfrac5{19} < \dfrac7{21} < \dfrac9{23}$ $\textbf{(C)}\hspace{.05in}\dfrac9{23} < \dfrac5{19} < \dfrac7{21}$ $\textbf{(D)}\hspace{.05in}\dfrac5{19} < \dfrac9{23} < \dfrac7{21}$ $\textbf{(E)}\hspace{.05in}\dfrac7{21} < \dfrac5{19} < \dfrac9{23}$

A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.

Let $ a,b,c $ be three real numbers such that $ \cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0. $ Prove that [b]i)[/b] $ \cos 6a+\cos 6b+\cos 6c=3\cos (2a+2b+2c) $ [b]ii)[/b] $ \sin 6a+\sin 6b+\sin 6c=3\sin (2a+2b+2c) $ [i]Vasile Pop[/i]

Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with every possible height-weight combination. The USAMTS wants to field a competitive team, so there are some strict requirements. [list] [*] If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team. [*] If person $P$ is on the team, then no one whose weight is the same as $P$’s height can also be on the team. [/list] Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]

Let $A$, $B$, $C$, $D$, $E$, $F$ be $6$ points on a circle in that order. Let $X$ be the intersection of $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is the intersection of $CF$ and $BE$. $X$ lies on segments $BZ$ and $AY$ and $Y$ lies on segment $CZ$. Given that $AX = 3$, $BX = 2$, $CY = 4$, $DY = 10$, $EZ = 16$, and $FZ = 12$, find the perimeter of triangle $XYZ$.

The point $M$ , inside $\vartriangle ABC$, satisfies the conditions that $\angle BMC = 90^o +\frac12 \angle BAC$ and that the line $AM$ contains the centre of the circumscribed circle of $\vartriangle BMC$. Prove that $M$ is the centre of the inscribed circle of $\vartriangle ABC$.

Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide. by Evangelos Psychas, Greece

How many ways are there to express $2548$ as a sum of at least two positive integers, where two sums that differ in order are considered different?

Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$? $\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }22$

Consider the set $E$ consisting of pairs of integers $(a, b)$, with $a \geq 1$ and $b \geq 1$, that satisfy in the decimal system the following properties: [b](i)[/b] $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$, $\alpha_2 \neq 0$; [b](ii)[/b] $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$; [b](iii)[/b] $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$ Find the elements of $E$.

Let $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$. Determine $\displaystyle n$ sets $\displaystyle A_i$, $\displaystyle 1 \leq i \leq n$, from the plane, pairwise disjoint, such that: (a) for every circle $\displaystyle \mathcal C$ from the plane and for every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$ we have $\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi$; (b) for all lines $\displaystyle d$ from the plane and every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$, the projection of $\displaystyle A_i$ on $\displaystyle d$ is not $\displaystyle d$.

Prove that for positive real numbers $a,b,c$ satisfying $abc=1$ the following inequality holds: $$ \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \ge \frac{a^2+1}{2a}+\frac{b^2+1}{2b}+\frac{c^2+1}{2c}.$$

A certain type of Bessel function has the form $I(x) = \frac{1}{\pi} \int_0^{\pi}e^{x \cos \theta} d\theta$ for all real $x$. Evaluate $\int_0^{\infty} x I(2x) e^{-x^2}dx$.

Let $x$ and $y$ be positive integers that are at least $2$. Suppose Johnny hits $1$ out of every $10$ free throws, Abigail hits $1$ out of every $x$ free throws, and Demar hits $2$ out of every $y$ free throws. It turns out that the mean of Abigail's and Demar's individual free throw percentages are the same as Johnny's free throw percentage. Find the sum of all possible values of $x$. [i]Individual #4[/i]

A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.

A trapezoid is given in which one base is twice as large as the other. Use one ruler (no divisions) to draw the midline of this trapezoid.

What is the smallest amount that may be invested at interest rate $i$, compounded annually, in order that one may withdraw $1$ dollar at the end of the first year, $4$ dollars at the end of the second year, $\ldots$ , $n^2$ dollars at the end of the $n$-th year, in perpetuity?

The sum of all integers between 50 and 350 which end in 1 is $ \textbf{(A)}\ 5880 \qquad \textbf{(B)}\ 5539 \qquad \textbf{(C)}\ 5208 \qquad \textbf{(D)}\ 4877 \qquad \textbf{(E)}\ 4566$