Let $ABC$ be a triangle with $\angle C=90^{\circ}$, and $K, L $ be the midpoints of the minor arcs AC and BC of its circumcircle. Segment $KL$ meets $AC$,at point $N$. Find angle $NIC$ where $I$is the incenter of $ABC$.

As usual, let ${\mathbb Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : {\mathbb Z}[x] \to {\mathbb Z}$ such that for any polynomials $p,q \in {\mathbb Z}[x]$, [list] [*]$\theta(p+1) = \theta(p)+1$, and [*]if $\theta(p) \neq 0$ then $\theta(p)$ divides $\theta(p \cdot q)$. [/list] [i]Evan Chen and Yang Liu[/i]

If the base $8$ representation of a perfect square is $ab3c$, where $a\ne 0$, then $c$ equals $\textbf{(A) } 0\qquad \textbf{(B) }1 \qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } \text{not uniquely determined}$

Find all natural solutions of $3^{x}= 2^{x}y+1.$

Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.

We consider permutations $f$ of the set $\mathbb{N}$ of non-negative integers, i.e. bijective maps $f$ from $\mathbb{N}$ to $\mathbb{N}$, with the following additional properties: \[f(f(x)) = x \quad \mbox{and}\quad \left| f(x)-x\right| \leqslant 3\quad\mbox{for all } x \in\mathbb{N}\mbox{.}\] Further, for all integers $n > 42$, \[\left.M(n)=\frac{1}{n+1}\sum_{j=0}^n \left|f(j)-j\right|<2,011\mbox{.}\right.\] Show that there are infinitely many natural numbers $K$ such that $f$ maps the set \[\left\{ n\mid 0\leqslant n\leqslant K\right\}\] onto itself.

Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.

Let $r$, $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying \[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \] for all $n \ge 2023$ then the sum \[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \] is unbounded, i.e for all positive reals $M$ there is an $n$ such that $b_n > M$.

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

Let $ABC$ be a triangle and $M$ be the midpoint of $BC$, and let $AM$ meet the circumcircle of $ABC$ again at $R$. A line passing through $R$ and parallel to $BC$ meet the circumcircle of $ABC$ again at $S$. Let $U$ be the foot from $R$ to $BC$, and $T$ be the reflection of $U$ in $R$. $D$ lies in $BC$ such that $AD$ is an altitude. $N$ is the midpoint of $AD$. Finally let $AS$ and $MN$ meets at $K$. Prove that $AT$ bisector $MK$.

$\textbf{(Self-Isogonal Cubics)}$ Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, $BC = 4$. The $\emph{isogonal conjugate}$ of a point $P$, denoted $P^\ast$, is the point obtained by intersecting the reflection of lines $PA$, $PB$, $PC$ across the angle bisectors of $\angle A$, $\angle B$, and $\angle C$, respectively. Given a point $Q$, let $\mathfrak K(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $PP^\ast$ contains $Q$. Consider: [list] [*] the M'Cay cubic $\mathfrak K(O)$, where $O$ is the circumcenter of $\triangle ABC$, [*] the Thomson cubic $\mathfrak K(G)$, where $G$ is the centroid of $\triangle ABC$, [*] the Napoleon-Feurerbach cubic $\mathfrak K(N)$, where $N$ is the nine-point center of $\triangle ABC$, [*] the Darboux cubic $\mathfrak K(L)$, where $L$ is the de Longchamps point (the reflection of the orthocenter across point $O$), [*] the Neuberg cubic $\mathfrak K(X_{30})$, where $X_{30}$ is the point at infinity along line $OG$, [*] the nine-point circle of $\triangle ABC$, [*] the incircle of $\triangle ABC$, and [*] the circumcircle of $\triangle ABC$. [/list] Estimate $N$, the number of points lying on at least two of these eight curves. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|N-E|/6} \right\rfloor$ points.

Let $(a_1, a_2, ..., a_8)$ be a permutation of $(1, 2, ... , 8)$. Find, with proof, the maximum possible number of elements of the set $$\{a_1, a_1 + a_2, ... , a_1 + a_2 + ... + a_8\}$$ that can be perfect squares.

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Prove that for any integers $a,b$, the equation $2abx^4 - a^2x^2 - b^2 - 1 = 0$ has no integer roots. [i](Dan Schwarz)[/i]

Let $p>2$ be a prime number. For $k=1,2,\ldots,p-1$ we denote by $a_k$ the remainder when $k^p$ is divided by $p^2$. Prove that $$a_1+a_2+\ldots+a_{p-1}=\frac{p^3-p^2}2.$$

How many pairs $(a, b)$ for integers $1 \le a, b \le 2015$ which satisfy that $a$ is divisible by $b + 1$ and $2016 - a$ is divisible by $b$.

Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$. If $AB=4$ compute: a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$; b) the distance between the lines $AC$ and $BF$.

Polynomials $(P_n(X))_{n\in\mathbb{N}}$ are defined as: $$P_0(X)=0, \quad P_1(X)=X+2,$$ $$P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X), \quad (\forall) n\geq2.$$ Show that if $ k $ divides $m$ then $P_k(X)$ divides $P_m(X).$

Let $k$ and $m$, with $k > m$, be positive integers such that the number $km(k^2 - m^2)$ is divisible by $k^3 - m^3$. Prove that $(k - m)^3 > 3km$.

Solve the following system of equations in integers: $\begin{cases} x^2+2xy+8z=4z^2+4y+8\\ x^2+y+2z=156 \\ \end{cases}$

Let $n\geqslant 2$ be an integer and $A{}$ a set of $n$ points in the plane. Find all integers $1\leqslant k\leqslant n-1$ with the following property: any two circles $C_1$ and $C_2$ in the plane such that $A\cap\text{Int}(C_1)\neq A\cap\text{Int}(C_2)$ and $|A\cap\text{Int}(C_1)|=|A\cap\text{Int}(C_2)|=k$ have at least one common point. [i]Cristi Săvescu[/i]

Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.

Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let be a complex number $ z $ having the property that $ \Re \left( z^n \right) >\Im \left( z^n \right) , $ for any natural numbers $ n. $ Show that $ z $ is a positive real number. [i]Laurențiu Panaitopol[/i]

We say that a set of positive integers is [i]regular [/i] if, for any selection of numbers from the set, the sum of the chosen numbers is different from $1810$. Divide the set of integers from $452$ to $1809$ (inclusive) into the smallest possible number of regular sets.