Let $n\ge 2$ be a positive integer. The integers $1,2,\cdots,n$ are written on a board. In a move, Alice can pick two integers written on the board $a\neq b$ such that $a+b$ is an even number, erase both $a$ and $b$ from the board and write the number $\frac{a+b}{2}$ on the board instead. Find all $n$ for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board. [b]Note.[/b] When $n=3$, Alice changes $(1,2,3)$ to $(2,2)$ and can't make any further moves. [i]Proposed by Rohan Goyal[/i]

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$. Prove that the three circles which pass: a) through $O, A, B$, b) through $O, B, C$, and c) through $O, C, A$, are equal

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

A sequence $a_n$ satisfies $a_1 =2t-3$ ($t \ne 1,-1$), and $a_{n+1}=\dfrac{(2t^{n+1}-3)a_n+2(t-1)t^n-1}{a_n+2t^n-1}$. [list] [b][i]i)[/i][/b] Find $a_n$, [b][i]ii)[/i][/b] If $t>0$, compare $a_{n+1}$ with $a_n$.[/list]

Find $\log_2 3 * \log_3 4 * \log_4 5 * ... * \log_{62} 63 * \log_{63} 64$ .

Let $M$ be a set of real $n \times n$ matrices such that i) $I_{n} \in M$, where $I_n$ is the identity matrix. ii) If $A\in M$ and $B\in M$, then either $AB\in M$ or $-AB\in M$, but not both iii) If $A\in M$ and $B \in M$, then either $AB=BA$ or $AB=-BA$. iv) If $A\in M$ and $A \ne I_n$, there is at least one $B\in M$ such that $AB=-BA$. Prove that $M$ contains at most $n^2 $ matrices.

Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that \[\alpha < \frac{\phi(m)}{m} < \beta.\]

Let $p,q,r\in R $ with $pqr=1$. Prove that $$\left(\frac{1}{1-p}\right)^2+\left(\frac{1}{1-q}\right)^2+\left(\frac{1}{1-r}\right)^2\ge 1$$ [i](Real Matrik)[/i]

Let $S$ be a set of $100$ positive integer numbers having the following property: “Among every four numbers of $S$, there is a number which divides each of the other three or there is a number which is equal to the sum of the other three.” Prove that the set $S$ contains a number which divides all other $99$ numbers of $S$. [i]Proposed by Tajikistan[/i]

What is the largest power of 2 that is a divisor of $13^4-11^4$? $\textbf{(A) } 8\qquad\textbf{(B) } 16\qquad\textbf{(C) } 32\qquad\textbf{(D) } 64\qquad \textbf{(E) } 128$

Show that ${2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)$ for all positive integers $n$.

Let $n\geq 2$ is positive integer. Find the best constant $C(n)$ such that \[\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})\] is true for all real numbers $x_{i}\in(0,1),i=1,...,n$ for which $(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.$

Evaluate: $\sum^{\infty}_{k=1} \tfrac{k}{a^{k-1}}$ for all $|a|<1$.

Let $O, M, P$ and $D$ be distinct digits from each other, and different from zero, such that $O < M < P < D$, and the following equation is true: \[ \overline{\text{OMPD}} \times \left( \overline{\text{OM}} - \overline{\text{D}} \right) = \overline{\text{MDDMP}} - \overline{\text{OM}} \] (a) Using estimates, explain why it is impossible for the value of $O$ to be greater than or equal to $3$. (b) Explain why $O$ cannot be equal to $1$. (c) Is it possible for $M$ to be greater than or equal to $5$? Justify. (d) Determine the values of $M$, $P$, and $D$.

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.

Find all 4-digit numbers $\overline{abcd}$ that are multiples of $11$, such that the 2-digit number $\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$.

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]

Compute the maximum real value of $a$ for which there is an integer $b$ such that $\frac{ab^2}{a+2b} = 2019$. Compute the maximum possible value of $a$.

Find all triples $(x,y,z)$ of positive integers satisfying the system of equations \[\begin{cases} x^2=2(y+z)\\ x^6=y^6+z^6+31(y^2+z^2)\end{cases}\]

Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.

\[\int_0^{\frac \pi 4} \cot(x)\sqrt{\sin(x)}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Given positive integers $n$ and $k \leq n$. Consider an equilateral triangular board with side $n$, which consists of circles: in the first (top) row there is one circle, in the second row there are two circles, $\ldots$ , in the bottom row there are $n$ circles (see the figure below). Let us place checkers on this board so that any line parallel to a side of the triangle (there are $3n$ such lines) contains no more than $k$ checkers. Denote by $T(k, n)$ the largest possible number of checkers in such a placement. [img]https://i.ibb.co/bJjjK1M/Image2.jpg[/img] a) Prove that the following upper bound is true: $$T(k,n) \leq \lfloor \frac{k(2n+1)}{3} \rfloor$$ b) Find $T(1,n)$ and $T(2,n)$ [i]D. Zmiaikou[/i]

Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.

For any natural $n$ , define $n!!=(n!)!$ e.g. $3!!=(3!)!=6!=720$. Let $a_1,a_2,...,a_n$ be a positive integer Prove that $$\frac{(a_1+a_2+\cdots+a_n)!!}{a_1!!a_2!!\cdots a_n!!}$$ is an integer. [i](nooonuii)[/i]