In the Game of Life, each square in an infinite grid of squares is either shaded or blank. Every day, if a square shares an edge with exactly zero or four shaded squares, it becomes blank the next day. If a square shares an edge with exactly two or three shaded squares, it becomes shaded the next day. Otherwise, it does not change. On day $1$, each square is randomly shaded or blank with equal probability. If the probability that a given square is shaded on day 2 is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, find $a + b.$

A large pond contains infinitely many lily pads labelled $1$, $2$, $3$,$ ... $, placed in a line, where for each $k$, lily pad $k + 1$ is one unit to the right of lily pad $k$. A frog starts at lily pad $100$. Each minute, if the frog is at lily pad $n$, it hops to lily pad $n + 1$ with probability $\frac{n-1}{n}$ , and hops all the way back to lily pad $1$ with probability $\frac{1}{n}$. Let $N$ be the position of the frog after $1000$ minutes. What is the expected value of $N$?

The $64$ whole numbers from $1$ through $64$ are written, one per square, on a checkerboard (an $8$ by $8$ array of $64$ squares). The first $8$ numbers are written in order across the first row, the next $8$ across the second row, and so on. After all $64$ numbers are written, the sum of the numbers in the four corners will be $\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134$

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

$$\int\left(\frac{x-1}{x^2+1}\right)^2e^x\,dx$$ [i]Proposed by Connor Gordon[/i]

If A,B are invertible and the set {A^k - B^k | k is a natural number} is finite , then there exists a natural number m such that A^m = B^m.

Let $a$, $b$ be such integers that $gcd(a + n,b + n) > 1$ for every integer $n \geq 1$. Prove that $a = b$.

Prove that for each $ a\in\mathbb N$, there are infinitely many natural $ n$, such that \[ n\mid a^{n \minus{} a \plus{} 1} \minus{} 1. \]

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

If the function $f:[0,1]\to (0.+\infty)$ is increasing and continuous, then for every $a\geq 0$ the following inequality holds: $$\int \limits_0^1 \frac{x^{a+1}}{f(x)}dx \leq \frac{a+1}{a+2} \int \limits_0^1 \frac{x^{a}}{f(x)}dx$$

Prove the following inequality holds if $\{a_n\}$ is a deceasing sequence of positive reals, and $0<\theta<\frac{\pi}{2}$. $$\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}$$

Let $A,B\in \mathcal{M}_n(\mathbb{R})$. Prove that $\text{rank}\ A+\text{rank}\ B\le n$ if and only if there exists an invertible matrix $X\in \mathcal{M}_n(\mathbb{R})$ such that $AXB=O_n$.

The number $2020!$ can be expressed as $7^k \cdot m$, where $k, m$ are integers and $m$ is not divisible by $7$. Find the remainder when $m$ is divided by $49$.

Prove that for each positive integer $ n$, there are pairwise relatively prime integers $ k_0,k_1,\ldots,k_n$, all strictly greater than $ 1$, such that $ k_0k_1\ldots k_n\minus{}1$ is the product of two consecutive integers.

Prove that for any $\lambda > 3$ there is a number $x$ for which $$\sin x + \sin (\lambda x) \ge 1.8.$$

$ \diamondsuit $ and $ \Delta $ are whole numbers and $ \diamondsuit\times\Delta =36 $. The largest possible value of $ \diamondsuit+\Delta $ is $ \text{(A)}\ 12\qquad\text{(B)}\ 13\qquad\text{(C)}\ 15\qquad\text{(D)}\ 20\ \qquad\text{(E)}\ 37 $

For every set $S$ with $n(\ge3)$ distinct integers, show that there exists a function $f:\{1,2,\dots,n\}\rightarrow S$ satisfying the following two conditions. (i) $\{ f(1),f(2),\dots,f(n)\} = S$ (ii) $2f(j)\neq f(i)+f(k)$ for all $1\le i<j<k\le n$.

Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes.

What figure can the central projection of a triangle be? (The center of the projection does not lie on the plane of the triangle.)

$P$ is a point in the interior of acute triangle $ABC$. $D,E,F$ are the reflections of $P$ across $BC,CA,AB$ respectively. Rays $AP,BP,CP$ meet the circumcircle of $\triangle ABC$ at $L,M,N$ respectively. Prove that the circumcircles of $\triangle PDL,\triangle PEM,\triangle PFN$ meet at a point $T$ different from $P$.

Find $m$ and $n$ such that the set of points whose coordinates $x$ and $y$ satisfy the equation $|y-2x|=x$, coincides with the set of points specified by the equation $|mx + ny| = y$.

We have $n\ge3$ points on the plane such that no three are collinear. Prove that it's possible to name them $P_1,P_2,\ldots,P_n$ such that for all $1<i<n$, the angle $\angle P_{i-1}P_iP_{i+1}$ is acute.

Determine all sequences $a_1,a_2,a_3,\dots$ of positive integers that satisfy the equation $$(n^2+1)a_{n+1} - a_n = n^3+n^2+1$$ for all positive integers $n$.

A uniform pool ball of radius $r$ and mass $m$ begins at rest on a pool table. The ball is given a horizontal impulse $J$ of fixed magnitude at a distance $\beta r$ above its center, where $-1 \le \beta \le 1$. The coefficient of kinetic friction between the ball and the pool table is $\mu$. You may assume the ball and the table are perfectly rigid. Ignore effects due to deformation. (The moment of inertia about the center of mass of a solid sphere of mass $m$ and radius $r$ is $I_{cm} = \frac{2}{5}mr^2$.) [asy] size(250); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.8)); draw((-3,-1)--(3,-1)); draw((-2.4,0.1)--(-2.4,0.6),EndArrow); draw((-2.5,0)--(2.5,0),dashed); draw((-2.75,0.7)--(-0.8,0.7),EndArrow); label("$J$",(-2.8,0.7),W); label("$\beta r$",(-2.3,0.35),E); draw((0,-1.5)--(0,1.5),dashed); draw((1.7,-0.1)--(1.7,-0.9),BeginArrow,EndArrow); label("$r$",(1.75,-0.5),E); [/asy] (a) Find an expression for the final speed of the ball as a function of $J$, $m$, and $\beta$. (b) For what value of $\beta$ does the ball immediately begin to roll without slipping, regardless of the value of $\mu$?

Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?