Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game?

Set $A$ consists of natural numbers such that these numbers can be expressed as $2x^2+3y^2,$ where $x$ and $y$ are integers. $(x^2+y^2\not=0)$ $a)$ Prove that there is no perfect square in the set $A.$ $b)$ Prove that multiple of odd number of elements of the set $A$ cannot be a perfect square.

An integer number $n\geq 2$ and real numbers $x_1<x_2<\cdots < x_n$ are given. $f: \mathbb R \to \mathbb R$ is a function defined as $$ f(x) = \left | \dfrac{(x-x_2)(x-x_3)\cdots (x-x_n)}{(x_1-x_2)(x_1-x_3)\cdots (x_1-x_n)} \right | + \cdots + \left | \dfrac{(x-x_1)(x-x_2)\cdots (x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\cdots (x_n-x_{n-1})} \right |. $$ Prove that there exists $i\in \{1,2,\cdots,n-1\}$ such that for all $x\in (x_i,x_{i+1})$ one has $f(x)< \sqrt n$. Proposed by [i]Navid Safaei[/i]

Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions: [b]I.[/b] $|A_i| = 7, i = 1, 2, \ldots, n$; [b]II.[/b] $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$ [b]III.[/b] For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$. Find the smallest possible value of $n$.

Let $y = mx + b$ be the image when the line $x - 3y + 11 = 0$ is reflected across the x-axis. The value of $m + b$ is $ \textbf{(A)}\ -6\qquad\textbf{(B)}\ -5\qquad\textbf{(C)}\ -4\qquad\textbf{(D)}\ -3\qquad\textbf{(E)}\ -2 $

A triangle has side lengths $18$, $24$, and $30$. Find the area of the triangle whose vertices are the incenter, circumcenter, and centroid of the original triangle.

Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.

Show that:\[\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2 (k\in \mathbb{Z}) \Longleftrightarrow\ |{\tan \alpha}| + |{\cot \alpha}| = 4\]

Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at $\text{(A)}\ \text{3:40 P.M.} \qquad \text{(B)}\ \text{3:55 P.M.} \qquad \text{(C)}\ \text{4:10 P.M.} \qquad \text{(D)}\ \text{4:25 P.M.} \qquad \text{(E)}\ \text{4:40 P.M.}$

(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite. (b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.

Let $f \colon [0,1] \to \mathbb{R}$ be a continuous function which is differentiable on $(0,1)$. Prove that either $f(x) = ax + b$ for all $x \in [0,1]$ for some constants $a,b \in \mathbb{R}$ or there exists $t \in (0,1)$ such that $|f(1) - f(0)| < |f'(t)|$.

A coin is tossed 9 times. Hence $2^{9}$ different outcomes are possible. In how many cases 2 consecutive heads does not appear? [list=1] [*] 34 [*] 55 [*] 89 [*] None of these [/list]

On a blackboard a positive integer $n_0$ is written. Two players, $A$ and $B$ are playing a game, which respects the following rules: $-$ acting alternatively per turn, each player deletes the number written on the blackboard $n_k$ and writes instead one number denoted with $n_{k+1}$ from the set $\left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\}$; $-$ player $A$ starts first deleting $n_0$ and replacing it with $n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\}$; $-$ the game ends when the number on the table is 0 - and the player who wrote it is the winner. Find which player has a winning strategy in each of the following cases: a) $n_0=120$; b) $n_0=\dsp \frac {3^{2002}-1}2$; c) $n_0=\dsp \frac{3^{2002}+1}2$.

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

Let $ABC$ be an acute angled triangle. The arc between $A$ and $B$ of the circumcircle of $ABC$ is reflected through the line $AB$, and the arc between $A$ and $C$ of the circumcircle of $ABC$ is reflected over the line $AC$. Obviously these two reflected arcs intersect at the point $A$. Prove that they also intersect at another point inside the triangle $ABC$.

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.

[i](5 pts)[/i] Let $n$ be a positive integer, $K = \{1, 2, \dots, n\}$, and $\sigma : K \rightarrow K$ be a function with the property that $\sigma(i) = \sigma(j)$ if and only if $i = j$ (in other words, $\sigma$ is a \textit{bijection}). Show that there is a positive integer $m$ such that \[ \underbrace{\sigma(\sigma( \dots \sigma(i) \dots ))}_\textrm{$m$ times} = i \] for all $i \in K$.

Construct a holomorphic function $f(z) = \sum \limits_{n = 0} ^ \infty a_n z^n$ ( | z | <1 ) in the unit circle that can be analytically continued to all points of the unit circle except one point, and for which the sequence $\{a_n\}$ has two limit points, $\infty$ and a finite value.

Solve in the real numbers the equation $ \cos \left( \pi\log_3 (x+6) \right)\cdot \cos \left( \pi \log_3 (x-2) \right) =1. $

Find the sum of all distinct four-digit numbers that contain only the digits $1, 2, 3, 4,5$, each at most once.

Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$