Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots. Determine the maximum number of elements in $S(a, b, c)$.

Does there exist any integer $a,b,c$ such that $a^2bc+2,ab^2c+2,abc^2+2$ are perfect squares?

Five airlines operate in a country consisting of $36$ cities. Between any pair of cities exactly one airline operates two way flights. If some airlines operates between cities $A,B$ and $B,C$ we say that the ordered triple $A,B,C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.

[asy] draw(ellipse((0,-5),10,3)); fill((-10,-5)--(10,-5)--(10,5)--(-10,5)--cycle,white); draw(ellipse((0,0),10,3)); draw((10,0)--(10,-5)); draw((-10,0)--(-10,-5)); draw((0,0)--(7,-3*sqrt(51)/10)); label("10",(7/2,-3*sqrt(51)/20),NE); label("5",(-10,-3),E); [/asy] Which cylinder has twice the volume of the cylinder shown above? [asy] unitsize(4); draw(ellipse((0,-5),20,6)); fill((-20,-5)--(20,-5)--(20,5)--(-20,5)--cycle,white); draw(ellipse((0,0),20,6)); draw((20,0)--(20,-5)); draw((-20,0)--(-20,-5)); draw((0,0)--(14,-3*sqrt(51)/5)); label("20",(7,-3*sqrt(51)/10),NE); label("5",(-20,-4),E); label("(A)",(0,6),N); draw(ellipse((31,-7),10,3)); fill((21,-7)--(41,-7)--(41,7)--(21,7)--cycle,white); draw(ellipse((31,3),10,3)); draw((41,3)--(41,-7)); draw((21,3)--(21,-7)); draw((31,3)--(38,3-3*sqrt(51)/10)); label("10",(34.5,3-3*sqrt(51)/20),NE); label("10",(21,-4),E); label("(B)",(31,6),N); draw(ellipse((47,-15.5),5,3/2)); fill((42,-15.5)--(42,-15.5)--(42,15.5)--(42,15.5)--cycle,white); draw(ellipse((47,4.5),5,3/2)); draw((42,4.5)--(42,-15.5)); draw((52,4.5)--(52,-15.5)); draw((47,4.5)--(50.5,4.5-3*sqrt(51)/20)); label("5",(48.75,4.5-3*sqrt(51)/40),NE); label("10",(42,-6),E); label("(C)",(47,6),N); draw(ellipse((73,-10),20,6)); fill((53,-10)--(93,-10)--(93,5)--(53,5)--cycle,white); draw(ellipse((73,0),20,6)); draw((53,0)--(53,-10)); draw((93,0)--(93,-10)); draw((73,0)--(87,-3*sqrt(51)/5)); label("20",(80,-3*sqrt(51)/10),NE); label("10",(53,-6),E); label("(D)",(73,6),N); [/asy] $\text{(E)}\ \text{None of the above}$

Let $ABC$ be an acute triangle with circumcenter $O$, on the sides $BC, CA$ and $AB$ they take the points $D, E$ and $F$, respectively, in such a way that $BDEF$ is a parallelogram. Supposing that $DF^2 = AE\cdot EC <\frac{AC^2}{4}$ show that the circles circumscribed to the triangles $FBD$ and $AOC$ are tangent.

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

All non-zero coefficients of the polynomial $f(x)$ equal $1$, while the sum of the coefficients is $20$. Is it possible that thirteen coefficients of $f^2(x)$ equal $9$? [i](S. Ivanov, K. Kokhas)[/i]

As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length $2$ so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles? [asy] size(140); fill((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--cycle,gray(0.4)); fill(arc((2,0),1,180,0)--(2,0)--cycle,white); fill(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle,white); fill(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle,white); fill(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle,white); fill(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle,white); fill(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle,white); draw((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--(1,0)); draw(arc((2,0),1,180,0)--(2,0)--cycle); draw(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle); draw(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle); draw(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle); draw(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle); draw(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle); label("$2$",(3.5,3sqrt(3)/2),NE); [/asy] $\textbf{(A)}\ 6\sqrt3-3\pi \qquad\textbf{(B)}\ \frac{9\sqrt3}{2}-2\pi \qquad\textbf{(C)}\ \frac{3\sqrt3}{2}-\frac{\pi}{3} \qquad\textbf{(D)}\ 3\sqrt3-\pi \\ \qquad\textbf{(E)}\ \frac{9\sqrt3}{2}-\pi$

In the diagram below, square $ABCD$ has side length $4$. Two congruent square $EGIK$ and $FHJL$ are drawn such that $AE=FB=BG=HC=CI=JD=DK=LA=1$ and $EF=GH=IJ=KL=2$. Compute the area of the region that lies in both $EGIK$ and $FHJL$.

Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

Let $c>1$ be a real constant. For the sequence $a_1,a_2,...$ we have: $a_1=1$, $a_2=2$, $a_{mn}=a_m a_n$, and $a_{m+n}\leq c(a_m+a_n)$. Prove that $a_n=n$.

Let $ ABC$ be a triangle with $ \angle BAC \equal{} 90^\circ$. A circle is tangent to the sides $ AB$ and $ AC$ at $ X$ and $ Y$ respectively, such that the points on the circle diametrically opposite $ X$ and $ Y$ both lie on the side $ BC$. Given that $ AB \equal{} 6$, find the area of the portion of the circle that lies outside the triangle. [asy]import olympiad; import math; import graph; unitsize(20mm); defaultpen(fontsize(8pt)); pair A = (0,0); pair B = A + right; pair C = A + up; pair O = (1/3, 1/3); pair Xprime = (1/3,2/3); pair Yprime = (2/3,1/3); fill(Arc(O,1/3,0,90)--Xprime--Yprime--cycle,0.7*white); draw(A--B--C--cycle); draw(Circle(O, 1/3)); draw((0,1/3)--(2/3,1/3)); draw((1/3,0)--(1/3,2/3)); label("$A$",A, SW); label("$B$",B, down); label("$C$",C, left); label("$X$",(1/3,0), down); label("$Y$",(0,1/3), left);[/asy]

How many subsets of the set $\{1, 2, \ldots, 11\}$ have median 6? [i]Proposed by Michael Tang

A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 5$

In the quadrilateral $ABCD$ it is known that $\angle A = 90^o$, $\angle C = 45^o$ . Diagonals $AC$ and $BD$ intersect at point $F$, and $BC = CF$, and the diagonal $AC$ is the bisector of angle $A$. Determine the other two angles of the quadrilateral $ABCD$. (Maria Rozhkova)

Given that $5^3+5^3 + 5^3 + 5^3 + 5^3 = 5^J$ and $3^2 + 3^2 + 3^2 = 3^N$ , what is the value of $J^ N$ ?

Find all positive integers $n\geqslant 2$ such that $d_{i+1}/d_i$ is an integer for all $1\leqslant i < k$, where $1=d_1<d_2<\dots <d_k=n$ are all the positive divisors of $n$. [i]Proposed by Pavel Ciurea[/i]

Let $x_0, x_1, . . . , x_{2014}$ be a sequence of real numbers, which for all $i < j$ satisfy $x_i + x_j \le 2j$. Determine the largest possible value of the sum $x_0 + x_1 + · · · + x_{2014}$.

The position [i]vs.[/i] time graph for an object moving in a straight line is shown below. What is the instantaneous velocity at $\text{t = 2 s}$? [asy] size(300); // x-axis draw((0,0)--(12,0)); // x-axis tick marks draw((4,0)--(4,-0.2)); draw((8,0)--(8,-0.2)); draw((12,0)--(12,-0.2)); // x-axis labels label("1",(4,0),2*S); label("2",(8,0),2*S); label("3",(12,0),2*S); //y-axis draw((0,-3)--(0,5)); //y-axis tick marks draw((-0.2,-2)--(0,-2)); draw((-0.2,0)--(0,0)); draw((-0.2,2)--(0,2)); draw((-0.2,4)--(0,4)); // y-axis labels label("-2",(0,-2),2*W); label("0",(0,0),2*W); label("2",(0,2),2*W); label("4",(0,4),2*W); // Axis Labels label("Time (s)",(10,0),N); label(rotate(90)*"Position (m)",(0,1),6*W); //Line draw((0,4)--(12,-2)); [/asy] (a) $-\text{2 m/s}$ (b) $-\frac{1}{2} \ \text{m/s}$ (c) $\text{0 m/s}$ (d) $\text{2 m/s}$ (e) $\text{4 m/s}$

$ A$ takes $ m$ times as long to do a piece of work as $ B$ and $ C$ together; $ B$ takes $ n$ times as long as $ C$ and $ A$ together; and $ C$ takes $ x$ times as long as $ A$ and $ B$ together. Then $ x$, in terms of $ m$ and $ n$, is: $ \textbf{(A)}\ \frac {2mn}{m \plus{} n} \qquad \textbf{(B)}\ \frac {1}{2(m \plus{} n)} \qquad \textbf{(C)}\ \frac {1}{m \plus{} n \minus{} mn} \qquad \textbf{(D)}\ \frac {1 \minus{} mn}{m \plus{} n \plus{} 2mn} \qquad \textbf{(E)}\ \frac {m \plus{} n \plus{} 2}{mn \minus{} 1}$

Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.

When $126$ is added to its reversal, $621,$ the sum is $126 + 621 = 747.$ Find the greatest integer which when added to its reversal yields $1211.$

Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.

$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$