Let n be an integer such that if d | n then d + 1 | n + 1. Show that n is a prime number.

Given an undirected graph with $N$ vertices. For any set of $k$ vertices, where $1\le k\le N$, there are at most $2k-2$ edges, which join vertices of this set. Prove that the edges may be coloured in two colours so that each cycle contains edges of both colours. (Graph may contain multiple edges). [i]I. Bogdanov, G. Chelnokov[/i]

In the decimal expansion of a fraction $\frac{m}{n}$ with positive integers $m$ and $n$ you can find a string of numbers $7143$ after the comma. Show $n>1250$. [i]Example:[/i] I mean something like $0.7143$.

Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns. 1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$. 2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial $$P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)$$ 3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial $$P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)$$ The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.

Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.

$D$ is a closed disk radius $R$. Show that among any $8$ points of $D$ one can always find two whose distance apart is less than $R$.

Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BP D$?

There are two letters in a language. Every word consists of seven letters, and two different words always have different letters on at least three places. a. Show that such a language cannot have more than $16$ words. b. Can there be $16$ words in the language?

We say that a doubly infinite sequence $. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .$ is subaveraging if $s_n = (s_{n−1} + s_{n+1})/4$ for all integers n. (a) Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct. (b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such that $s_m = s_n$, then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .

Two children at a time can play pairball. For $90$ minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is $\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 36$

Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$. (Slovakia)

Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?

In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$. Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$. [i]Proposed by Ray Li[/i]

A team of geologists on a field expedition have taken with them $80$ tin cans of provisions. The $80$ cans have different weights, which are known (there is a list). After a while the names of the contents of the cans have become illegible. The cook knows what is in each can and claims that he can prove it without opening any can and only using the list and a balance which indicates the difference of weight of the objects placed on its two pans. Show that in order to do so, (a) four weight measurements will be enough, (b) three will not (AK Tolpygo)

Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

Two distinct lines pass through the center of three concentric circles of radii $3$, $2$, and $1$. The area of the shaded region in the diagram is $8/13$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.) [asy] defaultpen(linewidth(0.8)); pair O=origin; fill(O--Arc(O, 2, 20, 160)--cycle, mediumgray); fill(O--Arc(O, 1, 20, 160)--cycle, white); fill(O--Arc(O, 2, 200, 340)--cycle, mediumgray); fill(O--Arc(O, 1, 200, 340)--cycle, white); fill(O--Arc(O, 3, 160, 200)--cycle, mediumgray); fill(O--Arc(O, 2, 160, 200)--cycle, white); fill(O--Arc(O, 1, 160, 200)--cycle, mediumgray); fill(O--Arc(O, 3, -20, 20)--cycle, mediumgray); fill(O--Arc(O, 2, -20, 20)--cycle, white); fill(O--Arc(O, 1, -20, 20)--cycle, mediumgray); draw(Circle(origin, 1));draw(Circle(origin, 2));draw(Circle(origin, 3)); draw(5*dir(200)--5*dir(20)^^5*dir(160)--5*dir(-20));[/asy] $ \textbf{(A)} \frac{\pi}8\qquad \textbf{(B)}\frac{\pi}7\qquad \textbf{(C)}\frac{\pi}6\qquad \textbf{(D)}\frac{\pi}5\qquad \textbf{(E)}\frac{\pi}4 $

In rectangle $ABCD$, $AB=6$ and $BC=16$. Points $P, Q$ are chosen on the interior of side $AB$ such that $AP=PQ=QB$, and points $R, S$ are chosen on the interior of side $CD$ such that $CR=RS=SD$. Find the area of the region formed by the union of parallelograms $APCR$ and $QBSD$. [i]Proposed by Yannick Yao[/i]

In a plane $5$ points are given such that all triangles having vertices at these points are of area not greater than $1$. Show that there exists a trapezoid which contains all point in the interior (or on the sides) and having the area not exceeding $3$.

There are $ 2016 $ positions marked around a circle, with a token on one of them. A legitimate move is to move the token either 1 position or 4 positions from its location, clockwise. The restriction is that the token can not occupy the same position more than once. Players $ A $ and $ B $ take turns making moves. Player $ A $ has the first move. The first player who cannot make a legitimate move loses. Determine which of the two players has a winning strategy.

In a certain triangle $ABC$, there are points $K$ and $M$ on sides $AB$ and $AC$, respectively, such that if $L$ is the intersection of $MB$ and $KC$, then both $AKLM$ and $KBCM$ are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle $ABC$.

Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$. [asy] size(200); defaultpen(fontsize(10)); pair A=origin, B=(14,0), C=(9,12), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), P=centroid(A,B,C); draw(D--A--B--C--A^^B--E^^C--F); dot(A^^B^^C^^P); label("$a$", P--A, dir(-90)*dir(P--A)); label("$b$", P--B, dir(90)*dir(P--B)); label("$c$", P--C, dir(90)*dir(P--C)); label("$d$", P--D, dir(90)*dir(P--D)); label("$d$", P--E, dir(-90)*dir(P--E)); label("$d$", P--F, dir(-90)*dir(P--F)); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$P$", P, 1.8*dir(285));[/asy]

$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.

Find the sum of distinct residues of the number $2012^n+m^2$ on $\mod 11$ where $m$ and $n$ are positive integers. $ \textbf{(A)}\ 55 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 37$

Let $\alpha \geq 1$ be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be [i]flooded[/i]. Hephaestus is building a [i]levee[/i], which is a subset of unit edges of the grid (called [i]walls[/i]) forming a connected, non-self-intersecting path or loop*. The game then begins with Hephaestus moving first. On each of Hephaestus’s turns, he adds one or more walls to the levee, as long as the total length of the levee is at most $\alpha n$ after his $n$th turn. On each of Poseidon’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Hephaestus wins if the levee forms a closed loop such that all flooded cells are contained in the interior of the loop — hence stopping the flood and saving the world. For which $\alpha$ can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood? ----- [size=75]*More formally, there must exist lattice points $\mbox{\footnotesize \(A_0, A_1, \dotsc, A_k\)}$, pairwise distinct except possibly $\mbox{\footnotesize \(A_0 = A_k\)}$, such that the set of walls is exactly $\mbox{\footnotesize \(\{A_0A_1, A_1A_2, \dotsc , A_{k-1}A_k\}\)}$. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. $\mbox{\footnotesize \(A_0 = A_k\)}$) then Hephaestus cannot add more walls. Since each wall has length $\mbox{\footnotesize \(1\)}$, the length of the levee is $\mbox{\footnotesize \(k\)}$.[/size] [i]Nikolai Beluhov[/i]

Given regular $1976$-gon. The midpoints of all the sides and diagonals are marked. What is the greatest number of the marked points lying on one circumference?