Let $n$ be a fixed odd positive integer. For each odd prime $p$, define $$a_p=\frac{1}{p-1}\sum_{k=1}^{\frac{p-1}{2}}\bigg\{\frac{k^{2n}}{p}\bigg\}.$$ Prove that there is a real number $c$ such that $a_p = c$ for infinitely many primes $p$. [i]Note: $\left\{x\right\} = x - \left\lfloor x\right\rfloor$ is the fractional part of $x$.[/i]

A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$. Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.

A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.

It is known that there is a number $S$ such that if $ a+b+c+d = S$ and $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d} = S$ $(a, b, c, d$ are different from zero and one$)$, then $\frac{1}{a- 1} ++ \frac{1}{b- 1} + \frac{1}{c- 1} + \frac{1}{d -1} = S.$ Find $S$.

If $ f$ is a nonnegative, continuous, concave function on the closed interval $ [0,1]$ such that $ f(0)=1$, then \[ \int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command. \int_0^1 f(x)dx \right]^2.\] [i]Z. Daroczy[/i]

Denote by $\mathbb{Z}^{*}$ the set of all nonzero integers and denote by $\mathbb{N}_{0}$ the set of all nonnegative integers. Find all functions $f:\mathbb{Z}^{*} \rightarrow \mathbb{N}_{0}$ such that: $(1)$ For all $a,b \in \mathbb{Z}^{*}$ such that $a+b \in \mathbb{Z}^{*}$ we have $f(a+b) \geq $ [b]min[/b] $\left \{ f(a),f(b) \right \}$. $(2)$ For all $a, b \in \mathbb{Z}^{*}$ we have $f(ab) = f(a)+f(b)$.

Bunbury the bunny is hopping on the positive integers. First, he is told a positive integer $n$. Then Bunbury chooses positive integers $a,d$ and hops on all of the spaces $a,a+d,a+2d,\dots,a+2013d$. However, Bunbury must make these choices so that the number of every space that he hops on is less than $n$ and relatively prime to $n$. A positive integer $n$ is called [i]bunny-unfriendly[/i] if, when given that $n$, Bunbury is unable to find positive integers $a,d$ that allow him to perform the hops he wants. Find the maximum bunny-unfriendly integer, or prove that no such maximum exists.

As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(real x, real y) { pair P = (x,y); dot(P,linewidth(3)); return P; } int big = 30, small = 14; filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5)); draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare); label("$A$",D2(0,big),NW); label("$B$",D2(0,0),SW); label("$C$",D2(big,0),SW); label("$D$",D2(big,big),N); label("$E$",D2(big+small,0),SE); label("$F$",D2(big+small,small),NE); label("$G$",D2(big,small),NE); [/asy]

Let $ABC$ be a triangle, $I$ its incenter, $\omega$ its incircle, $P$ a point such that $PI\perp BC$ and $PA\parallel BC$, $Q\in (AB), R\in (AC)$ such that $QR\parallel BC$ and $QR$ tangent to $\omega$. Show that $\angle QPB = \angle CPR$.

If $x$ is a real number satisfying the equation $$9\log_3 x - 10\log_9 x =18 \log_{27} 45,$$ then the value of $x$ is equal to $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$. [i]Proposed by pog[/i]

Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$ [i](Macedonia)[/i]

In a country, some cities are connected by two-way airlines, and one can get from any city to any other city in no more than $n{}$ flights. Prove that all airlines can be distributed among $n{}$ companies so that a route can be built between any two cities in which no more than two flights of each company would meet. [i]From the folklore[/i]

In acute triangle $\triangle {ABC}$, $\angle A > \angle B > \angle C$. $\triangle {AC_1B}$ and $\triangle {CB_1A}$ are isosceles triangles such that $\triangle {AC_1B} \stackrel{+}{\sim} \triangle {CB_1A}$. Let lines $BB_1, CC_1$ intersects at ${T}$. Prove that if all points mentioned above are distinct, $\angle ATC$ isn't a right angle.

Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$ holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$. [i]Proposed by Ivan Chan Kai Chin[/i]

A robot is placed at point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$. Player $B$'s aim is to stop $A$ from achieving his aim. For which $P$ can $A$ win?

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. [i]Proposed by Mongolia[/i]

Given is a parallelogram $ABCD$ with $\angle A < 90^o$ and $|AB| < |BC|$. The angular bisector of angle $A$ intersects side $BC$ in $M$ and intersects the extension of $DC$ in $N$. Point $O$ is the centre of the circle through $M, C$, and $N$. Prove that $\angle OBC = \angle ODC$. [asy] unitsize (1.2 cm); pair A, B, C, D, M, N, O; A = (0,0); B = (2,0); D = (1,3); C = B + D - A; M = extension(A, incenter(A,B,D), B, C); N = extension(A, incenter(A,B,D), D, C); O = circumcenter(C,M,N); draw(D--A--B--C); draw(interp(D,N,-0.1)--interp(D,N,1.1)); draw(A--interp(A,N,1.1)); draw(circumcircle(M,C,N)); label("$\circ$", A + (0.45,0.15)); label("$\circ$", A + (0.25,0.35)); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, dir(90)); dot("$D$", D, dir(90)); dot("$M$", M, SE); dot("$N$", N, dir(90)); dot("$O$", O, SE); [/asy]

In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

We have a set of 2n positive integers whose sum is a multiple of n. One operation consists of choosing n of them and adding the same positive integer to all of them. Show that, starting from the initial 2n numbers, we can get all are equal, performing a maximum of 2n - 1 operations.

Yu Semo and Yu Sejmo have created sequences of symbols $\mathcal{U} = (\text{U}_1, \ldots, \text{U}_6)$ and $\mathcal{J} = (\text{J}_1, \ldots, \text{J}_6)$. These sequences satisfy the following properties. [list] [*] Each of the twelve symbols must be $\Sigma$, $\#$, $\triangle$, or $\mathbb{Z}$. [*] In each of the sets $\{\text{U}_1, \text{U}_2, \text{U}_4, \text{U}_5\}$, $\{\text{J}_1, \text{J}_2, \text{J}_4, \text{J}_5\}$, $\{\text{U}_1, \text{U}_2, \text{U}_3\}$, $\{\text{U}_4, \text{U}_5, \text{U}_6\}$, $\{\text{J}_1, \text{J}_2, \text{J}_3\}$, $\{\text{J}_4, \text{J}_5, \text{J}_6\}$, no two symbols may be the same. [*] If integers $d \in \{0, 1\}$ and $i, j \in \{1, 2, 3\}$ satisfy $\text{U}_{i + 3d} = \text{J}_{j + 3d}$, then $i < j$. [/list] How many possible values are there for the pair $(\mathcal{U}, \mathcal{J})$?

Show that there are infinitely many triangles with side lengths $a$, $b$, $c$, where $a$ is a prime, $b$ is a power of $2$ and $c$ is the square of an odd integer.

We call a nonempty set $M$ good if its elements are positive integers, each having exactly $4$ divisors. If the good set $M$ has $n$ elements, we denote by $S_{M}$ the sum of all $4n$ divisors of its members (the sum may contain repeating terms). a) Prove that $A=\{2\cdot37,19\cdot37,29\cdot37\}$ is good and $S_{A}=2014$. b) Prove that if the set $B$ is good and $8\in B$, then $S_{B}\neq2014$.

Let $E$ be the set of all continuously differentiable real valued functions $f$ on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. Define $$J(f)=\int^1_0(1+x^2)f'(x)^2\text dx.$$ a) Show that $J$ achieves its minimum value at some element of $E$. b) Calculate $\min_{f\in E}J(f)$.

On a circle there are $99$ natural numbers. If $a,b$ are any two neighbouring numbers on the circle, then $a-b$ is equal to $1$ or $2$ or $ \frac{a}{b}=2 $. Prove that there exists a natural number on the circle that is divisible by $3$. [i]S. Berlov[/i]