Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$, $n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.

Find all $n$ such that \[n = d (n) ^ 4\] Where $d (n)$ is the number of divisors of $n$, for example $n = 2 \cdot 3\cdot 5\implies d (n) = 2 \cdot 2\cdot 2$.

A closed line on a plane is such that any quadrangle inscribed in it has the sum of opposite angles equal to $180^o$. Prove that this line is a circle.

$ABCBE$ is a regular pentagon. Point $B'$ is symmetric to point $B$ wrt line $AC$ (see figure). Is it possible to pave the plane with pentagons equal to $AB'CBE$? (S. Markelov) [img]https://cdn.artofproblemsolving.com/attachments/9/2/cbb5756517e85e56c4a931e761a6b4da8fe547.png[/img]

A computer outputs the values of the expression $ (n\plus{}1) . 2^n$ for $ n \equal{} 1, n \equal{} 2, n \equal{} 3$, etc. What is the largest number of consecutive values that are perfect squares?

Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$ \[ a\plus{}b\plus{}c|f(a)\plus{}f(b)\plus{}f(c)\]

Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$

Let $k$ be a natural number.Find all the couples of natural numbers $(n,m)$ such that : $(2^k)!=2^n*m$

The integers between $1$ and $9$ inclusive are distributed in the units of a $3$ x $3$ table. You sum six numbers of three digits: three that are read in the rows from left to right, and three that are read in the columns from top to bottom. Is there any such distribution for which the value of this sum is equal to $2001$?

In $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1$ and $A_7A_2$ are congruent to each other and diagonals $A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2$ and $A_7A_3$ are also congruent to each other. Is the polygon necessarily regular?

Prove that in a topological space X , if all discrete subspaces have compact closure , then X is compact.

Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.

Ivan is playing Lego with $4n^2$ $1 \times 2$ blocks. First, he places $2n^2$ $1 \times 2$ blocks to fit a $2n \times 2n$ square as the bottom layer. Then he builds the top layer on top of the bottom layer using the remaining $2n^2$ $1 \times 2$ blocks. Note that the blocks in the bottom layer are connected to the blocks above it in the top layer, just like real Lego blocks. He wants the whole two-layered building to be connected and not in seperate pieces. Prove that if he can do so, then the four $1\times 2$ blocks connecting the four corners of the bottom layer, must be all placed horizontally or all vertically. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that \[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\] Prove that \[5m+12n\le 581.\]

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?

There are $15$ boys and $15$ girls in the class. The first girl is friends with $4$ boys, the second with $5$, the third with $6$, . . . , the $11$th with $14$, and each of the other four girls is friends with all the boys. It turned out that there are exactly $3 \cdot 2^{25}$ ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too. [i]G.M.Sharafetdinova[/i]

Let $\tau(n)$ denote the number of positive integer divisors of a positive integer $n$ (for example, $\tau(2022) = 8$). Given a polynomial $P(X)$ with integer coefficients, we define a sequence $a_1, a_2,\ldots$ of nonnegative integers by setting \[a_n =\begin{cases}\gcd(P(n), \tau (P(n)))&\text{if }P(n) > 0\\0 &\text{if }P(n) \leq0\end{cases}\] for each positive integer $n$. We then say the sequence [i]has limit infinity[/i] if every integer occurs in this sequence only finitely many times (possibly not at all). Does there exist a choice of $P(X)$ for which the sequence $a_1$, $a_2$, . . . has limit infinity? [i]Jovan Vuković[/i]

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). [Hereby, $a=BC,$ $b=CA,$ $c=AB, $ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$ Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

In the triangle $ABC$ holds $|AB| = |AC|$ and the inscribed circle touches the sides $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ at the points $D$, $E$ and $F$ respectively . The perpendicular from the point $D$ to the line $EF$ intersects the side $\overline{AB}$ at the point $G$, and the circles circumscribed around the triangles $AEF$ and $ABC$ intersect at the points $A $and $T$. Prove that the lines $T G$ and $T F$ are perpendicular.

Let $\mathbb N$ denote the set of positive integers. A function $f\colon\mathbb N\to\mathbb N$ has the property that for all positive integers $m$ and $n$, exactly one of the $f(n)$ numbers \[f(m+1),f(m+2),\ldots,f(m+f(n))\] is divisible by $n$. Prove that $f(n)=n$ for infinitely many positive integers $n$.

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]

[u]Round 1[/u] [b]p1.[/b] What is the remainder when $2021$ is divided by $102$? [b]p2.[/b] Brian has $2$ left shoes and $2$ right shoes. Given that he randomly picks $2$ of the $4$ shoes, the probability he will get a left shoe and a right shoe is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p3.[/b] In how many ways can $59$ be written as a sum of two perfect squares? (The order of the two perfect squares does not matter.) [u]Round 2 [/u] [b]p4.[/b] Two positive integers have a sum of $60$. Their least common multiple is $273$. What is the positive diffeerence between the two numbers? [b]p5.[/b] How many ways are there to distribute $13$ identical apples among $4$ identical boxes so that no two boxes receive the same number of apples? A box may receive zero apples. [b]p6.[/b] In square $ABCD$ with side length $5$, $P$ lies on segment $AB$ so that $AP = 3$ and $Q$ lies on segment $AD$ so that $AQ = 4$. Given that the area of triangle $CPQ$ is $x$, compute $2x$. [u]Round 3 [/u] [b]p7.[/b] Find the number of ordered triples $(a, b, c)$ of nonnegative integers such that $2a+3b+5c = 15$. [b]p8.[/b] What is the greatest integer $n \le 15$ such that $n + 1$ and $n^2 + 3$ are both prime? [b]p9.[/b] For positive integers $a, b$, and $c$, suppose that $gcd \,\,(a, b) = 21$, $gcd \,\,(a, c) = 10$, and $gcd \,\,(b,c) = 11$. Find $\frac{abc}{lcm \,\,(a,b,c)}$ . (Note: $gcd$ is the greatest common divisor function and $lcm$ is the least common multiple function.) [u]Round 4[/u] [b]p10.[/b] The vertices of a square in the coordinate plane are at $(0, 0)$, $(0, 6)$, $(6, 0)$, and $(6, 6)$. Line $\ell$ intersects the square at exactly two lattice points (that is, points with integer coordinates). How many such lines $\ell$ are there that divide the square into two regions, one of them having an area of $12$? [b]p11.[/b] Let $f(n)$ be defined as follows for positive integers $n$: $f(1) = 0$, $f(n) = 1$ if $n$ is prime, and $f(n) = f(n - 1) + 1$ otherwise. What is the maximum value of $f(n)$ for $n \le 120$? [b]p12.[/b] The graph of the equation $y = x^3 + ax^2 + bx + c$ passes through the points $(2,4)$, $(3, 9)$, and $(4, 16)$. What is $b$? PS. You should use hide for answers. Rounds 5- 8 have been posted [url=https://artofproblemsolving.com/community/c3h2949415p26408227]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].