Let $n$ be a positive integer and $p$ be a prime number such that $np+1$ is a perfect square. Prove that $n+1$ can be written as the sum of $p$ perfect squares.

$32$ teams, ranked $1$ through $32$, enter a basketball tournament that works as follows: the teams are randomly paired and in each pair, the team that loses is out of the competition. The remaining $16$ teams are randomly paired, and so on, until there is a winner. A higher ranked team always wins against a lower-ranked team. If the probability that the team ranked $3$ (the third-best team) is one of the last four teams remaining can be written in simplest form as $\dfrac{m}{n}$, compute $m+n$.

Find all positive integers $n$ that have precisely $\sqrt{n + 1}$ natural divisors.

Let $ABC$ be an acute-angled triangle and $D$ a point on the altitude through $C$. Let $E$, $F$, $G$ and $H$ be the midpoints of the segments $AD$, $BD$, $BC$ and $AC$. Show that $E$, $F$, $G$, and $H$ form a rectangle. (G. Anegg, Innsbruck)

Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.

Let $ABC$ be a triangle with incenter $I $, the line $AI$ intersects $BC$ at $ L$ and the circumcircle of $ABC$ at $L'$. Show that the triangles $BLI$ and $L'IB$ are similar if and only if $AC = AB + BL$.

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$: (a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$, (b) $f(2n) \ge f(n)$ where $Z_{\ge 0}$ is the set of nonnegative integers. Solve the equation $f(n) = 1000$

Prove the inequality \[ {x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1, \] where $2 < x, y, z < 4.$ [i]Proposed by A. Golovanov[/i]

Let $ABC$ be a triangle with $AB<AC$ and $I$ be your incenter. Let $M$ and $N$ be the midpoints of the sides $BC$ and $AC$, respectively. If the lines $AI$ and $IN$ are perpendicular, prove that the line $AI$ is tangent to the circumcircle of $\triangle IMC$.

Given a triangle $ABC$ with circumcircle $\Gamma$, and two arbitrary points $X, Y$ on $\Gamma$. Let $D$, $E$, $F$ be points on lines $BC$, $CA$, $AB$, respectively, such that $AD$, $BE$, and $CF$ concur at a point $P$. Let $U$ be a point on line $BC$ such that $X$, $Y$, $D$, $U$ are concyclic. Similarly, let $V$ be a point on line $CA$ such that $X$, $Y$, $E$, $V$ are concyclic, and let $W$ be a point on line $AB$ such that $X$, $Y$, $F$, $W$ are concyclic. Prove that $AU$, $BV$, $CW$ concur at a single point. [i]Proposed by chengbilly[/i]

Find the smallest positive integer $n$ such that $n^4 + (n+1)^4$ is composite.

Let $ABCD$ be a trapezoid with $AB \parallel CD.$ Point $X$ is placed on segment $BC$ such that $\angle{BAX} = \angle{XDC}.$ Given that $AB = 5, BX =3, CX =4,$ and $CD =12,$ compute $AX.$

Maria was a brilliant mathematician who found the following property about her year of birth: if $f$ is a function defined in the set of natural numbers $N = \{0, 1, 2, 3, 4, 5,...\}$ such that $f(1) = 1335$ and $f(n+1) = f(n)-2n+43$ for all $n \in N$, then his year of birth is the maximum value that $f(n)$ can reach when $n$ takes values in $N$. Determine the year of birth of Mary.

Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$.

Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.

In a certain forest there are at least three crossroads, and for any three crossroads of roads A, B and C there is a road from A to B without passing through C. A deer and a hunter are standing at crossroads of different paths. Is it possible that they can exchange positions without their paths crossing at other points, that are not their initial positions?

Alex thinks of a two-digit integer (any integer between $10$ and $99$). Greg is trying to guess it. If the number Greg names is correct, or if one of its digits is equal to the corresponding digit of Alex’s number and the other digit differs by one from the corresponding digit of Alex’s number, then Alex says “hot”; otherwise, he says “cold”. (For example, if Alex’s number was $65$, then by naming any of $64, 65, 66, 55$ or $75$ Greg will be answered “hot”, otherwise he will be answered “cold”.) [list][b](a)[/b] Prove that there is no strategy which guarantees that Greg will guess Alex’s number in no more than 18 attempts. [b](b)[/b] Find a strategy for Greg to find out Alex’s number (regardless of what the chosen number was) using no more than $24$ attempts. [b](c)[/b] Is there a $22$ attempt winning strategy for Greg?[/list]

When $a_0, a_1, \dots , a_{1000}$ denote digits, can the sum of the $1001$-digit numbers $a_0a_1\cdots a_{1000}$ and $a_{1000}a_{999}\cdots a_0$ have odd digits only?

Given a triangle $ABC$, let the bisector of $\angle BAC$ meets the side $BC$ and circumcircle of triangle $ABC$ at $D$ and $E$, respectively. Let $M$ and $N$ be the midpoints of $BD$ and $CE$, respectively. Circumcircle of triangle $ABD$ meets $AN$ at $Q$. Circle passing through $A$ that is tangent to $BC$ at $D$ meets line $AM$ and side $AC$ respectively at $P$ and $R$. Show that the four points $B,P,Q,R$ lie on the same line. [i]Proposer: Fajar Yuliawan[/i]

Let $ABC$ be a scalene, non-right triangle. Let $\omega$ be the incircle and let $\gamma$ be the nine-point circle (the circle through the feet of the altitudes) of $\triangle ABC$, with centers $I$ and $N$, respectively. Suppose $\omega$ and $\gamma$ are tangent at a point $F$. Let $D$ be the foot of the perpendicular from $A$ to line $BC$ and let $M$ be the midpoint of side $\overline{BC}$. The common tangent to $\omega$ and $\gamma$ at $F$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let lines $DP$ and $DQ$ intersect $\gamma$ at points $P_1 \neq D$ and $Q_1 \neq D$, respectively. Suppose that point $Z$ lies on line $P_1Q_1$ such that $\angle MFZ=90^{\circ}$ and $MZ \perp DF$. Suppose that $\gamma$ has radius 11 and $\omega$ has radius 5. Then $DI=\frac{k\sqrt{m}}{n}$, where $k,m,$ and $n$ are positive integers such that $\gcd(k,n)=1$ and $m$ is not divisible by the square of any prime. Compute $100k+10m+n$. [i]Proposed by Luke Robitaille[/i]

Let \(ABC\) be a triangle and \(D\) be the mid-point of \(BC\). Suppose the angle bisector of \(\angle ADC\) is tangent to the circumcircle of triangle \(ABD\) at \(D\). Prove that \(\angle A=90^{\circ}\).

Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

For every natural number $ t$, $ f(t)$ is the probability that if a fair coin is tossed $ t$ times, the number of times we get heads is 2008 more than the number of tails. What is the value of $ t$ for which $ f(t)$ attains its maximum? (if there is more than one, describe all of them)

Determine the number of triangles, of any size and shape, in the following figure: [asy] size(4cm); draw(2*dir(0)--dir(120)--dir(240)--cycle); draw(dir(60)--2*dir(180)--dir(300)--cycle); [/asy] [i]Proposed by William Yue[/i]

We say that a positive integer is [i]irregular [/i] if said number is not a multiple of none of its digits. For example, $203$ is irregular because $ 203$ is not a multiple of $2$, it is not multiple of $0$ and is not a multiple of $3$. Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$.