19) Each cell of a $2$ × $5$ grid of unit squares is to be colored white or black. Compute the number of such colorings for which no $2$ × $2$ square is a single color. 20) Let $n$ be a three-digit integer with nonzero digits, not all of which are the same. Define $f(n)$ to be the greatest common divisor of the six integers formed by any permutation of $n$s digits. For example, $f(123) = 3$, because $gcd(123, 132, 213, 231, 312, 321) = 3$. Let the maximum possible value of $f(n)$ be $k$. Find the sum of all $n$ for which $f(n) = k$. 21) Consider a $2$ × $2$ grid of squares. Each of the squares will be colored with one of $10$ colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there? 22) Find all the roots of the polynomial $x^5 - 5x^4 + 11x^3 -13x^2+9x-3$ 23) Compute the smallest positive integer $n$ for which $0 < \sqrt[4]{n} - \left \lfloor{\sqrt[4]{n}}\right \rfloor < \dfrac{1}{2015}$. 24) Three ants begin on three different vertices of a tetrahedron. Every second, they choose one of the three edges connecting to the vertex they are on with equal probability and travel to the other vertex on that edge. They all stop when any two ants reach the same vertex at the same time. What is the probability that all three ants are at the same vertex when they stop? 25) Let $ABC$ be a triangle that satisfies $AB = 13$, $BC = 14$, $AC = 15$. Given a point $P$ in the plane, let $PA$, $PB$, $PC$ be the reflections of $A$, $B$, $C$ across $P$. Call $P$ [i]good[/i] if the circumcircle of $P_A P_B P_C$ intersects the circumcircle of $ABC$ at exactly 1 point. The locus of good points $P$ encloses a region $S$. Find the area of $S$. 26. Let $f : \mathbb{R}^+ \rightarrow \mathbb{R}$ be a continuous function satisfying $f(xy) = f(x) + f(y) + 1$ for all positive reals ${x,y}$. If $f(2) = 0$, compute $f(2015)$. 27) Let $ABCD$ be a quadrilateral with $A = (3,4)$, $B=(9,-40)$, $C = (-5,-12)$, $D = (-7,24)$. Let $P$ be a point in the plane (not necessarily inside the quadrilateral). Find the minimum possible value of $\overline{AP} + \overline{BP} + \overline{CP} + \overline{DP}$.

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.

Let $p$ be a prime number such that $p = 10^{d -1} + 10^{d-2} + ...+ 10 + 1$. Show that $d$ is a prime.

Let $ABCD$ be a square of side length $6$. Points $E$ and $F$ are selected on rays $AB$ and $AD$ such that segments $EF$ and $BC$ intersect at a point $L$, $D$ lies between $A$ and $F$, and the area of $\triangle AEF$ is 36. Clio constructs triangle $PQR$ with $PQ=BL$, $QR=CL$ and $RP=DF$, and notices that the area of $\triangle PQR$ is $\sqrt{6}$. If the sum of all possible values of $DF$ is $\sqrt{m} + \sqrt{n}$ for positive integers $m \ge n$, compute $100m+n$. [i]Based on a proposal by Calvin Lee[/i]

For geometrical sequence $(a_n)$, the first term $a_1=1536$, common ratio $q=-\frac{1}{2}$. Let $\pi_n=\prod_{i=1}^n a_i$, so the lagerest one in $(\pi_n)$ is $\text{(A)} \pi_9\qquad\text{(B)} \pi_{11}\qquad\text{(C)} \pi_{12}\qquad\text{(D)} \pi_{13}$

For $n\geq 3$ , determine all real solutions of the system of n equations : $x_1+x_2+...+x_{n-1}=\frac{1}{x_n}$ ....................... $x_1+x_2+...+x_{i-1}+x_{i+1}+...+x_n=\frac{1}{x_i}$ ....................... $x_2+...+x_{n-1}+x_n=\frac{1}{x_1}$

If $x$, $y$, and $z$ are real numbers such that $x^2+2y^2+3z^2=96$, what is the maximum possible value of $x+2y+3z$?

Eight all different sushis are placed evenly on the edge of a round table, whose surface can rotate around the center. Eight people also evenly sit around the table, each with one sushi in front. Each person has one favorite sushi among these eight, and they are all distinct. They find that no matter how they rotate the table, there are never more than three people who have their favorite sushis in front of them simultaneously. By this requirement, how many different possible arrangements of the eight sushis are there? Two arrangements that differ by a rotation are considered the same.

Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a function such that $f(ab)$ divides $\max \{f(a),b\}$ for any positive integers $a,b$. Must there exist infinitely many positive integers $k$ such that $f(k)=1$?

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $\mathcal{C}$ to the area of shaded region $\mathcal{B}$ is $11/5$. Find the ratio of shaded region $\mathcal{D}$ to the area of shaded region $\mathcal{A}$. [asy] defaultpen(linewidth(0.7)+fontsize(10)); for(int i=0; i<4; i=i+1) { fill((2*i,0)--(2*i+1,0)--(2*i+1,6)--(2*i,6)--cycle, mediumgray); } pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10); draw(B--A--C); fill((7.3,0)--(7.8,0)--(7.8,6)--(7.3,6)--cycle, white); clip(B--A--C--cycle); for(int i=0; i<9; i=i+1) { draw((i,1)--(i,6)); } label("$\mathcal{A}$", A+0.2*dir(-17), S); label("$\mathcal{B}$", A+2.3*dir(-17), S); label("$\mathcal{C}$", A+4.4*dir(-17), S); label("$\mathcal{D}$", A+6.5*dir(-17), S);[/asy]

Suppose a sequence of reals $\{a_n\}_{n\geq 0}$ satisfies $a_0 = 0$, $\frac{100}{101} <a_{100}<1$, and $$2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3$$ for every $n\geq 1$. (1) Define a sequence $b_n = a_n - \frac{n}{n+1}$. Prove that $b_n\leq b_{n+1}$ for any $n\geq 100$. (2) Determine whether infinite series $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converges or diverges.

Call a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0$ with integer coefficients primitive if and only if $\gcd(a_n,a_{n-1},\dots a_1,a_0) =1$. a)Let $P(x)$ be a primitive polynomial with degree less than $1398$ and $S$ be a set of primes greater than $1398$.Prove that there is a positive integer $n$ so that $P(n)$ is not divisible by any prime in $S$. b)Prove that there exist a primitive polynomial $P(x)$ with degree less than $1398$ so that for any set $S$ of primes less than $1398$ the polynomial $P(x)$ is always divisible by product of elements of $S$.

Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Let a,b,c be real numbers such that:$a\ge b\ge 1\ge c\ge 0$ and a+b+c=3. a)Prove that $2\le ab +bc+ca\le 3$ b)Prove that $\dfrac{24}{a^3+b^3+c^3}+\dfrac{25}{ab+bc+ca}\ge 14$. Find the equality cases

A square is divided into $N^2$ equal smaller non-overlapping squares, where $N \geq 3$. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). [list] [*] Show that it is possible to find a broken line composed of $4$ segments for $N = 3$. [*] Find the minimum number of segments in this broken line for arbitrary $N$. [/list]

Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$

Let $O$ be the center of the circle circumscribed around the the triangle $ABC$. The centers of the circles circumscribed around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.

Given a segment $AB$ of the length 1, define the set $M$ of points in the following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$

Triangle $ABC$ has $\angle A = 60^o$, $\angle B = 45$, and $AC = 6$. Let $D$ be on $AB$ such that $AD = 3$. There is exactly one point $E$ on $BC$ such that $\overline{DE}$ divides $ABC$ into two cyclic polygons. Compute $DE^2$.

A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.

Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations \[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]

a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$, then $$A^n=\operatorname{tr}(A)A^{n-1}.$$ b) Does the assertion from a) also hold for odd positive integers $k$?

Let $\Omega$ be the circumcircle of an acute triangle $ABC$. Points $D$, $E$, $F$ are the midpoints of the inferior arcs $BC$, $CA$, $AB$, respectively, on $\Omega$. Let $G$ be the antipode of $D$ in $\Omega$. Let $X$ be the intersection of lines $GE$ and $AB$, while $Y$ the intersection of lines $FG$ and $CA$. Let the circumcenters of triangles $BEX$ and $CFY$ be points $S$ and $T$, respectively. Prove that $D$, $S$, $T$ are collinear. [i]Proposed by kyou46 and Li4.[/i]

For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.