For every integer $n$ not equal to $1$ or $-1$, define $S(n)$ as the smallest integer greater than $1$ that divides $n$. In particular, $S(0)=2$. We also define $S(1) = S(-1) = 1$. Let $f$ be a non-constant polynomial with integer coefficients such that $S(f(n)) \leq S(n)$ for every positive integer $n$. Prove that $f(0)=0$. [b]Note:[/b] A non-constant polynomial with integer coefficients is a function of the form $f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_k x^k$, where $k$ is a positive integer and $a_0,a_1,\ldots,a_k$ are integers such that $a_k \neq 0$. [i]Pitchayut Saengrungkongka, Thailand[/i]

Let $r_k$ denote the remainder when ${127 \choose k}$ is divided by $8$. Compute$ r_1 + 2r_2 + 3r_3 + · · · + 63r_{63}.$

Find the number of positive integers $x$ less than $100$ for which $$3^x + 5^x + 7^x + 11^x + 13^x + 17^x + 19^x$$ is prime.

Let $\left< F_n\right>$ be the Fibonacci sequence, that is, $F_0=0$, $F_1=1$, and $F_{n+2}=F_{n+1}+F_{n}$ holds for all nonnegative integers $n$. Find all pairs $(a,b)$ of positive integers with $a < b$ such that $F_n-2na^n$ is divisible by $b$ for all positive integers $n$.

(I'll skip over the whole "dressing" of the graph in cities and flights [color=#FF0000][Mod edit: Shu has posted the "dressed-up" version below][/color]) For an ordinary directed graph, show that there is a subset A of vertices such that: $1.$ There are no edges between the vertices of A. $2.$ For any vertex $v$, there is either a direct way from $v$ to a vertex in A, or a way passing through only one vertex and ending in A (like $v$ ->$v'$-> $a$, where $a$ is a vertex in A)

A cube with size $10\times 10\times 10$ consists of $1000$ unit cubes, all colored white. $A$ and $B$ play a game on this cube. $A$ chooses some pillars with size $1\times 10\times 10$ such that no two pillars share a vertex or side, and turns all chosen unit cubes to black. $B$ is allowed to choose some unit cubes and ask $A$ their colors. How many unit cubes, at least, that $B$ need to choose so that for any answer from $A$, $B$ can always determine the black unit cubes?

Hammie is in the $6^\text{th}$ grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds? $\textbf{(A)}\ \text{median, by 60} \qquad \textbf{(B)}\ \text{median, by 20} \qquad \textbf{(C)}\ \text{average, by 5} \qquad \textbf{(D)}\ \text{average, by 15}$ \\ $\textbf{(E)}\ \text{average, by 20}$

Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

Given any set of $9$ points in the plane such that there is no $3$ of them collinear, show that for each point $P$ of the set, the number of triangles with its vertices on the other $8$ points and that contain $P$ on its interior is even.

Find all pairs of positive integers $(a, b)$ such that $f(x)=x$ is the only function $f:\mathbb{R}\to \mathbb{R}$ that satisfies $$f^a(x)f^b(y)+f^b(x)f^a(y)=2xy$$ for all $x, y\in \mathbb{R}$.

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

Two permutations of $1,\ldots, n$ are written on the board: $a_1,\ldots,a_n$ $b_1,\ldots,b_n$ A move consists of one of the following two operations: 1) Change the first row to $b_{a_1},\ldots,b_{a_n}$ 2) Change the second row to $a_{b_1},\ldots,a_{b_n}$ The starting position is: $2134\ldots n$ $234\ldots n1$ Is it possible by finitely many moves to get: $2314\ldots n$ $234 \ldots n1$? [i]D. Zmiaikou[/i]

On a board the numbers $1,2,3,\dots,98,99$ are written. One has to mark $50$ of them, such that the sum of two marked numbers is never equal to $99$ or $100$. How many ways one can mark these numbers?

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ that satisfy the following two conditions: [list]$\bullet\ f(n)$ is a perfect square for all $n\in\mathbb{Z}_{>0}$ $\bullet\ f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}_{>0}$.[/list]

Given is a triangle $ABC$ with the property that $|AB| + |AC| = 3|BC|$. Let $T$ be the point on segment $AC$ such that $|AC| = 4|AT|$. Let $K$ and $L$ be points on the interior of line segments $AB$ and $AC$ respectively such that $KL \parallel BC$ and $KL$ is tangent to the inscribed circle of $\vartriangle ABC$. Let $S$ be the intersection of $BT$ and $KL$. Determine the ratio $\frac{|SL|}{|KL|}$

The median $AD$ is drawn in triangle $ABC$. Point $E$ is selected on segment $AC$, and on the ray $DE$ there is a point $F$, and $\angle ABC = \angle AED$ and $AF // BC$. Prove that from segments $BD, DF$ and $AF$, you can make a triangle, the area of ​​which is not less half the area of ​​triangle $ABC$.

Let $C',B',A'$ be points respectively on sides $AB,AC,BC$ of $\triangle ABC$ satisfying $ \tfrac{AB'}{B'C}= \tfrac{BC'}{C'A}=\tfrac{CA'}{A'B}=k$. Prove that the ratio of the area of the triangle formed by the lines $AA',BB',CC'$ over the area of $\triangle ABC$ is $\tfrac{(k-1)^2}{(k^2+k+1)}$.

Let $E$ be a common point of circles $\omega _1$ and $\omega _2$. Let $AB$ be a common tangent to these circles, and $CD$ be a line parallel to $AB$, such that $A$ and $C$ lie on $\omega _1$, $B$ and $D$ lie on $\omega _2$. The circles $ABE$ and $CDE$ meet for the second time at point $F$. Prove that $F$ bisects one of arcs $CD$ of circle $CDE$.

Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David? [i]Proposed by Vlad Spătaru[/i]

Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.

$(a_n)$ is a sequence with $a_1=1$ and $|a_n| = |a_{n-1}+2|$ for every positive integer $n\geq 2$. What is the minimum possible value of $\sum_{i = 1}^{2000}a_{i}$? $ \textbf{(A)}\ -4000 \qquad\textbf{(B)}\ -3000 \qquad\textbf{(C)}\ -2000 \qquad\textbf{(D)}\ -1000 \qquad\textbf{(E)}\ \text{None} $

Jeck and Lisa are playing a game on table dimensions $m \times n$ , where $m , n >2$. Lisa starts so that she puts knight figurine on arbitrary square of table.After that Jeck and Lisa put new figurine on table by the following rules: [b]1.[/b] Jeck puts queen figurine on any empty square of a table which is two squares vertically and one square horizontally distant, or one square vertically and two squares horizontally distant from last knight figurine which Lisa put on the table [b]2.[/b] Lisa puts knight figurine on any empty square of a table which is in the same row, column or diagonal as last queen figurine Jeck put on the table. Player which cannot put his figurine loses. For which pairs of $(m,n)$ Lisa has winning strategy? [i] Proposed by Stijn Cambie[/i]

The sequence $(a_n)$ is defined by $a_1=0$, $$ a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1. $$ Prove that $a_{2016}>{1\over 2}+a_{1000}$.

A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in [i]almost-order[/i] with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters. (b) How many different ways are there to line up $20$ people in [i]almost-order[/i] if their heights are $120, 125, 130,$ $135,$ $140,$ $145,$ $150,$ $155,$ $160,$ $164, 165, 170, 175, 180, 185, 190, 195, 200, 205$, and $210$ centimeters? (Note that there is someone of height $164$ centimeters.)