Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.

$P$, $Q$, $R$ and $S$ are (distinct) points on a circle. $PS$ is a diameter and $QR$ is parallel to the diameter $PS$. $PR$ and $QS$ meet at $A$. Let $O$ be the centre of the circle and let $B$ be chosen so that the quadrilateral $POAB$ is a parallelogram. Prove that $BQ$ = $BP$ .

Let $a, b, c$ denote the lengths of the sides $BC,CA,AB$, respectively, of a triangle $ABC$. If $P$ is any point on the circumference of the circle inscribed in the triangle, show that $aPA^2+bPB^2+cPC^2$ is constant.

[b]1.[/b] Some proper partitions $P_1, \dots , P_n$ of a finite set $S$ (that is, partitions containing at least two parts) are called [i]independent[/i] if no matter how we choose one class from each partition, the intersection of the chosen classes is nonempty. Show that if the inequality $\frac{\left | S \right | }{2} < \left |P_1 \right | \dots \left |P_n \right |\qquad \quad (*)$ holds for some independent partitions, then $P_1, \dots , P_n$ is maximal in the sense that there is no partition $P$ such that $P,P_1, \dots , P_n$ are independent. On the other hand, show that inequality $(*)$ is not necessary for this maximality. ([b]C.20[/b]) [E. Gesztelyi]

In a hat are $1992$ notes with all numbers from $1$ to $1992$. At random way, two bills are drawn simultaneously from the hat; the difference between the numbers on the two notes are written on a new note, which is placed in the hat, while the two drawn notes thrown away. It continues in this way until there is only one note left in it the hat. Show that there is an even number on this slip of paper.

Consider a triangle $\triangle ABC$, let the incircle centered at $I$ touch the sides $BC,CA,AB$ at points $D,E,F$ respectively. Let the angle bisector of $\angle BIC$ meet $BC$ at $M$, and the angle bisector of $\angle EDF$ meet $EF$ at $N$. Prove that $A,M,N$ are collinear.

Let $\{a_n\}^\infty_{n=0}$ be the sequence of real numbers satisfying $a_0=0$, $a_1=1$ and $$a_{n+2}=a_{n+1}+\frac{a_n}{2^n}$$for every $n\ge0$. Prove that $$\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.$$

Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is [i]row-valid[/i] if the numbers in each row can be permuted to form an arithmetic progression, and [i]column-valid[/i] if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

Let $P$ be the number of ways to partition $2013$ into an ordered tuple of prime numbers. What is $\log_2 (P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor$ or zero, whichever is larger.

Let $\triangle ABC$ be given, it's $A-$mixtilinear incirlce, $\omega$, and it's excenter $I_A$. Let $H$ be the foot of altitude from $A$ to $BC$, $E$ midpoint of arc $\overarc{BAC}$ and denote by $M$ and $N$, midpoints of $BC$ and $AH$, respectively. Suposse that $MN\cap AE=\{ P \}$ and that line $I_AP$ meet $\omega$ at $S$ and $T$ in this order: $I_A-T-S-P$. Prove that circumcircle of $\triangle BSC$ and $\omega$ are tangent to each other. [hide=Diagram] [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(10.48006497171429cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -14.1599662489562, xmax = 15.320098722758091, ymin = -10.77766346689139, ymax = 5.5199497728160765; /* image dimensions */ /* draw figures */ draw(circle((0.025918528679950675,-0.7846460299371317), 4.597564438637676)); draw(circle((-0.9401249699191643,-2.0521899279943225), 3.003855690249927), red); draw((-2.4211259444978057,3.107599095143759)--(-4.242260757102907,-2.493518275554076), linewidth(1.2) + blue); draw((-4.242260757102907,-2.493518275554076)--(4.286443606492271,-2.5125131627798423), linewidth(1.2) + blue); draw((4.286443606492271,-2.5125131627798423)--(-2.4211259444978057,3.107599095143759), linewidth(1.2) + blue); draw((-2.4211259444978057,3.107599095143759)--(-2.433609564871039,-2.4975464519291233)); draw((-4.381282878515476,2.5449802910435655)--(1.435215864174395,-10.327847927108488), linewidth(1.2) + dotted); draw((-4.381282878515476,2.5449802910435655)--(0.022091424694681727,-2.5030157191669593), linetype("4 4")); draw(circle((0.0212183867796688,-2.8950097429721975), 4.282341626812516), red); draw((0.03615806773666919,3.8129070061099433)--(-2.4211259444978057,3.107599095143759)); draw((-2.4211259444978057,3.107599095143759)--(-4.381282878515476,2.5449802910435655), linetype("2 2") + green); /* dots and labels */ dot((-4.242260757102907,-2.493518275554076),linewidth(4.pt) + dotstyle); label("$B$", (-4.8714663963993114,-2.647851734263423), NE * labelscalefactor); dot((4.286443606492271,-2.5125131627798423),linewidth(4.pt) + dotstyle); label("$C$", (4.474018117829703,-2.5908670725861245), NE * labelscalefactor); dot((-2.4211259444978057,3.107599095143759),linewidth(4.pt) + dotstyle); label("$A$", (-2.5350952678420575,3.278553080175656), NE * labelscalefactor); dot((0.022091424694681727,-2.5030157191669593),linewidth(3.pt) + dotstyle); label("$M$", (-0.027770154268419445,-3.0657392532302814), NE * labelscalefactor); label("$\omega$", (-1.1294736132628969,0.5812790941168441), NE * labelscalefactor,red); dot((-2.433609564871039,-2.4975464519291233),linewidth(3.pt) + dotstyle); label("$H$", (-2.9529827867709972,-3.0657392532302814), NE * labelscalefactor); dot((-2.4273677546844223,0.3050263216073179),linewidth(3.pt) + dotstyle); label("$N$", (-2.2691668467054598,0.25836601127881736), NE * labelscalefactor); dot((1.435215864174395,-10.327847927108488),linewidth(3.pt) + dotstyle); label("$I_A$", (1.5678003725511684,-10.11284241398957), NE * labelscalefactor); dot((-4.381282878515476,2.5449802910435655),linewidth(3.pt) + dotstyle); label("$P$", (-4.643527749710799,2.708706463402667), NE * labelscalefactor); dot((-3.1988410259345286,-0.0719498450384039),linewidth(3.pt) + dotstyle); label("$S$", (-3.0859469973392963,-0.17851639491380702), NE * labelscalefactor); dot((-0.9468150550874253,-5.056038168270003),linewidth(3.pt) + dotstyle); label("$T$", (-0.8255554176782134,-4.908243314129611), NE * labelscalefactor); dot((0.03615806773666919,3.8129070061099433),linewidth(3.pt) + dotstyle); label("$E$", (-0.008775267044376731,3.962369020303242), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/hide]

To prepare a stay abroad, students meet at a workshop including an excursion. To promote interaction between the students, they are to be distributed to two busses such that not too many of the students in the same bus know each other. Every student knows all those who know her. The number of such pairwise acquaintances is $k$. Prove that it is possible to distribute the students such that the number of pairwise acquaintances in each bus is at most $\frac{k}{3}$.

Let $a,b,c,x,y,z,t$ be positive real numbers with $1\le x,y,z\le4$. Prove that $$\frac x{(2a)^t}+\frac y{(2b)^t}+\frac z{(2c)^t}\ge\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}.$$

The side of an equilateral triangle is $s$. A circle is inscribed in the triangle and a square is inscribed in the circle. The area of the square is: $ \text{(A)}\ \frac{s^2}{24}\qquad\text{(B)}\ \frac{s^2}{6}\qquad\text{(C)}\ \frac{s^2\sqrt{2}}{6}\qquad\text{(D)}\ \frac{s^2\sqrt{3}}{6}\qquad\text{(E)}\ \frac{s^2}{3} $

Let $ABC$ be a triangle with incircle touching $BC, CA, AB$ at $D, E, F,$ respectively. Let $O$ and $M$ be its circumcenter and midpoint of $BC.$ Suppose that circumcircles of $AEF$ and $ABC$ intersect at $X$ for the second time. Assume $Y \neq X$ is on the circumcircle of $ABC$ such that $OMXY$ is cyclic. Prove that circumcenter of $DXY$ lies on $BC.$ [i]Proposed by tenplusten.[/i]

Let $ Z$ and $ R$ denote the sets of integers and real numbers, respectively. Let $ f: Z \rightarrow R$ be a function satisfying: (i) $ f(n) \ge 0$ for all $ n \in Z$ (ii) $ f(mn)\equal{}f(m)f(n)$ for all $ m,n \in Z$ (iii) $ f(m\plus{}n) \le max(f(m),f(n))$ for all $ m,n \in Z$ (a) Prove that $ f(n) \le 1$ for all $ n \in Z$ (b) Find a function $ f: Z \rightarrow R$ satisfying (i), (ii),(iii) and $ 0<f(2)<1$ and $ f(2007) \equal{} 1$

Let $p$ be an odd prime. Show that for every integer $c$, there exists an integer $a$ such that $$a^{\frac{p+1}{2}} + (a+c)^{\frac{p+1}{2}} \equiv c\pmod p.$$

There is an equilateral triangle $ABC$. Let $E, F$ and $K$ be points such that $E{}$ lies on side $AB$, $F{}$ lies on the side $AC$, $K{}$ lies on the extension of side $AB$ and $AE = CF = BK$. Let $P{}$ be the midpoint of the segment $EF$. Prove that the angle $KPC$ is right. [i]Vladimir Rastorguev[/i]

Let $ABCDEF$ be a convex hexagon in which $AB=AF$, $BC=CD$, $DE=EF$ and $\angle ABC = \angle EFA = 90^{\circ}$. Prove that $AD\perp CE$.

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$. Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have. [i] (Andrei Bâra)[/i]

Find all solutions in integers of the equation $$x^2 + 2^2 = y^3 + 3^3.$$

Find the total number of solutions to the following system of equations: $ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\ b(a + d)\equiv b \pmod{37} \\ c(a + d)\equiv c \pmod{37} \\ bc + d^2\equiv d \pmod{37} \\ ad - bc\equiv 1 \pmod{37} \end{array}$

Evaluate the following expression: $2013^2 + 2011^2 + … + 5^2 + 3^2 -2012^2 -2010^2-…-4^2-2^2$

Prove that for any three positive real numbers $ a,b,c, \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \ge \frac{9}{2} \cdot \frac{1}{a\plus{}b\plus{}c}$.

Lucky starts doodling on a $5\times 5$ Bingo board. He puts his pencil at the center of the upper-left square (marked by ‘·’) and draws a continuous doodle ending on the Free Space, never going off the board or through a corner of a square. (See Figure 1.) (a) Is it possible for Lucky’s doodle to visit all squares exactly once? (The starting and ending squares are considered visited.) (b) Is it possible for Lucky’s doodle to visit all squares exactly twice?