Let triangle $ABC$ be an acute triangle with $AB<AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. Let $\Gamma$ be the circle $BOC$. The line $AO$ and the circle of radius $AO$ centered at $A$ cross $\Gamma$ at $A’$ and $F$, respectively. Prove that $\Gamma$ , the circle on diameter $AA’$ and circle $AFH$ are concurrent. [i]Proposed by Romania, Radu-Andrew Lecoiu[/i]

Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$

Given triangle $ABC$. Lines $AL_a$ and $AM_a$ are the internal and the external bisectrix of angle $A$. Let $\omega_a$ be the reflection of the circumcircle of $\triangle AL_aM_a$ in the midpoint of $BC$. Circle $\omega_b$ is defined similarly. Prove that $\omega_a$ and $\omega_b$ touch if and only if $\triangle ABC$ is right-angled.

Peter writes a positive integer on the whiteboard. Each minute Basil multiplies the last written number by 2 or by 3 and writes the product on the whiteboard too. Can Peter choose the starting integer such that, irrespective of Basil's strategy, at any given moment the number of integers on the whiteboard starting with 1 or 2 would exceed the number of the ones starting with 7, 8 or 9 ? Maxim Didin

Let $\alpha$, $\beta$ and $\gamma$ be the roots of the polynomial $2023x^3-2023x^2-1$. Find $$\frac{1}{\alpha^3}+\frac{1}{\beta^3}+\frac{1}{\gamma^3}$$. [i]Proposed by Andy Xu[/i]

In a tennis tournament of $10$ players, everyone played against everyone once. In this tournament, if player $i$ won the match against player $j$, then the total number of matches $i$ lost plus the total number of matches $j$ won is greater than or equal to $8$. We will say that three players $i$, $j$, $k$ form an [i]atypical tri[/i]o if $i$ beat $j$, $j$ beat $k$ and $k$ beat $i$. Prove that in the tournament there were exactly $40$ atypical trios.

Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

Consider the expansion \[(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.\] [b](a)[/b] Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$ [b](b)[/b] Prove that $10^{340 }< a_{992} < 10^{347}.$

In a triangle $ABC$, $D$ and $E$ lies on $AB$ and $AC$ such that $DE$ is parallel to $BC$. There exists point $P$ in the interior of $BDEC$ such that \[ \angle BPD = \angle CPE = 90^{\circ} \]Prove that the line $AP$ passes through the circumcenter of triangles $EPD$ and $BPC$.

Let $A$ be the set of odd integers $\leq 2n-1.$ For a positive integer $m$, let $B=\{a+m\,|\, a\in A \}.$ Determine for which positive integers $n$ there exists a positive integer $m$ such that the product of all elements in $A$ and $B$ is a square.

Say that a positive integer is [i]red[/i] if it is of the form $n^{2020}$, where $n$ is a positive integer. Say that a positive integer is [i]blue[/i] if it is not red and is of the form $n^{2019}$, where $n$ is a positive integer. True or false: Between every two different red positive integers greater than $10^{100{,}000{,}000}$, there are at least 2019 blue positive integers. Prove that your answer is correct.

If $a,b \ge 0$ are real numbers, prove the inequality $$\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}$$ For each of the inequalities, find the cases of equality.

Let $\left( A,+, \cdot \right)$ be a ring that verifies the following properties: (i) it has a unit, $1$, and its order is $p$, a prime number; (ii) there is $B \subset A, \, |B| = p$, such that: for all $x,y \in A$, there is $b \in B$ such that $xy = byx$. Prove that $A$ is commutative. [i]Ion Savu[/i]

If a number $N$, $N\neq 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is: $\textbf{(A) }\dfrac1R\qquad \textbf{(B) }R\qquad \textbf{(C) }4\qquad \textbf{(D) }\dfrac14\qquad \textbf{(E) }-R$

Three schools have $200$ students each. Every student has at least one friend in each school (if the student $a$ is a friend of the student $b$ then $b$ is a friend of $a$). It is known that there exists a set $E$ of $300$ students (among the $600$) such that for any school $S$ and any two students $x,y\in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different. Show that one can find a student in each school such that they are friends with each other.

Let $A_1,B_1,C_1$ be the midpoints of the sides $BC,CA,AB$ respectively of a triangle $ABC$. Points $K$ on segment $C_1A_1$ and $L$ on segment $A_1B_1$ are taken such that $$\frac{C_1K}{KA_1}=\frac{BC+AC}{AC+AB}\enspace\enspace\text{and}\enspace\enspace\frac{A_1L}{LB_1}=\frac{AC+AB}{BC+AB}.$$If $BK$ and $CL$ meet at $S$, prove that $\angle C_1A_1S=\angle B_1A_1S$.

Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then \[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \] must hold.

From a point $A$ not on line $\varepsilon$, we drop the perpendicular $AB$ on $\varepsilon$ and three other not perpendicular lines $AC$, $AD$,$AE $ which lie on the same semiplane defines by $AB$, such that $(AD )>\frac{1}{2}((AC)+(AE))$. Prove that $(CD )>(DE).$ (Points $B,C,D,,E$ lie on line $\varepsilon$ ) .

Let $n$ be a fixed positive integer. Ben is playing a computer game. The computer picks a tree $T$ such that no vertex of $T$ has degree $2$ and such that $T$ has exactly $n$ leaves, labeled $v_1,\ldots, v_n$. The computer then puts an integer weight on each edge of $T$, and shows Ben neither the tree $T$ nor the weights. Ben can ask queries by specifying two integers $1\leq i < j \leq n$, and the computer will return the sum of the weights on the path from $v_i$ to $v_j$. At any point, Ben can guess whether the tree's weights are all zero. He wins the game if he is correct, and loses if he is incorrect. (a) Show that if Ben asks all $\binom n2$ possible queries, then he can guarantee victory. (b) Does Ben have a strategy to guarantee victory in less than $\binom n2$ queries? [i]Brandon Wang[/i]

Someone wrote six letters to six people and addressed six envelopes to them. How many ways can the letters be put into the envelopes so that none of the letters end up in the correct envelope?

Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?

In a triangle $ABC$, $AB>AC$. The foot of the altitude from $A$ to $BC$ is $D$, the intersection of bisector of $B$ and $AD$ is $K$, the foot of the altitude from $B$ to $CK$ is $M$ and let $BM$ and $AK$ intersect at point $N$. The line through $N$ parallel to $DM$ intersects $AC$ at $T$. Prove that $BM$ is the bisector of angle $\widehat{TBC}$.

The triangles $BCD$ and $ACE$ are externally constructed to sides $BC$ and $CA$ of a triangle $ABC$ such that $AE = BD$ and $\angle BDC+\angle AEC = 180^\circ$. Let $F$ be a point on segment $AB$ such that ${AF\over FB}={CD\over CE}$. Prove that ${DE\over CD+CE}={EF\over BC}={FD\over AC}$.

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

Find the index of the singular point $0$ of the vector field with components \[(x^4+y^4+z^4,x^3y-xy^3,xyz^2)\]