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?

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ $\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$

It is said that a positive integer is not GOOD, if there exists a permutation of the integers from 1 to n, $(a_1,a_2,...,a_n)$ such that $k + a_k$ is a perfect square for all $k$. For example $5$ is a GOOD number, since the permutation $(3,2,1,5,4)$ checks the condition: $1 + 3 = 2^2$, $2 + 2 = 2^2$, $3 + 1 = 2^2$; $4 + 5 = 3^2$ and $5 +4 = 3^2$. Find all GOOD numbers up to $12$.

In $\triangle ABC$, the incircle $\omega$ has center $I$ and is tangent to $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$ respectively. The circumcircle of $\triangle{BIC}$ meets $\omega$ at $P$ and $Q$. Lines $AI$ and $BC$ meet at $D$, and the circumcircle of $\triangle PDQ$ meets $\overline{BC}$ again at $X$. Suppose that $EF = PQ = 16$ and $PX + QX = 17$. Then $BC^2$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$. [i]Proposed by Ankan Bhattacharya and Michael Ren

a) How many distinct ways are there are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b)* How many distinct ways are there are there of painting the faces of a dodecahedron $12$ different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.)

Given are a positive integer $n$ and an infinite sequence of proper fractions $x_0 = \frac{a_0}{n}$, $\ldots$, $x_i=\frac{a_i}{n+i}$, with $a_i < n+i$. Prove that there exist a positive integer $k$ and integers $c_1$, $\ldots$, $c_k$ such that \[ c_1 x_1 + \ldots + c_k x_k = 1. \] [i]Proposed by M. Dubashinsky[/i]

Unit square $ABCD$ is divided into four rectangles by $EF$ and $GH$, with $BF = \frac14$. $EF$ is parallel to $AB$ and $GH$ parallel to $BC$. $EF$ and $GH$ meet at point $P$. Suppose $BF + DH = FH$, calculate the nearest integer to the degree of $\angle FAH$. [asy] size(100); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } // NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55. pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0]; draw(A--B--C--D--cycle); draw(F--A--H); draw(E--F); draw(G--H); label("$A$",D2(A),NW); label("$B$",D2(B),SW); label("$C$",D2(C),SE); label("$D$",D2(D),NE); label("$E$",D2(E),plain.N); label("$F$",D2(F),S); label("$G$",D2(G),W); label("$H$",D2(H),plain.E); label("$P$",D2(P),SE); [/asy]

In all tetrahedron $ABCD$ holds [list=1] [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [/list] for all $n\in \mathbb{N}^*$