Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$. Giorgi Arabidze, Georgia,

To commemorate the $43rd$ anniversary of the restoration of mathematics competitions, a mathematics enthusiast arranges the first $2021$ integers $1,2,\dots,2021$ into a sequence $\{a_n\}$ in a certain order, so that the sum of any consecutive $43$ items in the sequence is a multiple of $43$. (1) If the sequence of numbers is connected end to end into a circle, prove that the sum of any consecutive $43$ items on the circle is also a multiple of $43$; (2) Determine the number of sequences $\{a_n\}$ that meets the conditions of the question.

Let $f = f(x,y,z)$ be a polynomial in three variables $x$, $y$, $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$. Show that there exist three polynomials $A$, $B$, $C$ in these same three variables such that $A + B + C = 0$ and \[ f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). \] Is there any polynomial $f$ for which these $A$, $B$, $C$ are uniquely determined?

Let $p$ be an odd prime. Show that $\frac{1}{\pi}\cdot\cos^{-1}\left(\frac{1}{p}\right)$ is irrational. (Note: $\cos^{-1}\left(x\right)$ is defined to be the unique $y$ with $0\leq y\leq\pi$ such that $\cos\left(y\right)=x$.)

Let be a natural number $ n\ge 2 $ and two $ n\times n $ complex matrices $ A,B $ that satisfy $ (AB)^3=O_n. $ Does this imply that $ (BA)^3=O_n ? $

For each $a$, $1<a<2$, the graphs of functions $y=1-|x-1|$ and $y=|2x-a|$ determine a figure. Prove that the area of this figure is less than $\frac13$.

Let $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1)=91$ and $P(-1)=-121$, compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\] [i]Proposed by David Altizio[/i]

Let $P$ be an arbitrary point in the interior of triangle $\triangle ABC$. Lines$\overline{BP}$ and $\overline{CP}$ intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$, respectively. Let $K$ and $L$ be the midpoints of the segments $BF$ and $CE$, respectively. Let the lines through $L$ and $K$ parallel to $\overline{CF}$ and $\overline{BE}$ intersect $\overline{BC}$ at $S$ and $T$, respectively; moreover, denote by $M$ and $N$ the reflection of $S$ and $T$ over the points $L$ and $K$, respectively. Prove that as $P$ moves in the interior of triangle $\triangle ABC$, line $\overline{MN}$ passes through a fixed point. [i]Proposed by Ali Zamani[/i]

Let be a sequence of functions $ \left( f_n \right)_{n\ge 2}:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ defined, for each $ n\ge 2, $ as $$ f_n(x)=2nx^{2+n} -2(n+2)x^{1+n} +(2+n)x +1. $$ [b]a)[/b] Prove that $ f_n $ has an unique local maxima $ x_n, $ for any $ n\ge 2. $ [b]b)[/b] Show that $ 1=\lim_{n\to\infty } x_n. $ [i]Cătălin Zîrnă[/i]

Denote by $g_n$ the number of connected graphs of degree $n$ whose vertices are labeled with numbers $1,2,...,n$. Prove that $g_n \ge (\frac{1}{2}).2^{\frac{n(n-1)}{2}}$. [b][u]Note[/u][/b]:If you prove that for $c < \frac{1}{2}$, $g_n \ge c.2^{\frac{n(n-1)}{2}}$, you will earn some point! [i]proposed by Seyed Reza Hosseini and Mohammad Amin Ghiasi[/i]

$\xi, \xi'$ are iid random variables. let F have the distribution function $\xi+\xi'$, and G have the uniform distribution over the interval [-1,1]. Prove that $\max | F ( x ) - G ( x ) | \geq 10^{-1994}$ .

For any positive integer n, evaluate $$\sum_{i=0}^{\lfloor \frac{n+1}{2} \rfloor} {n-i+1 \choose i}$$ , where $\lfloor n \rfloor$ is the greatest integer less than or equal to $n$ .

Let $ABCD$ be a rhombus. $P$ is a point on side $ BC$ and $Q$ is a point on side $CD$ such that $BP = CQ$. Prove that centroid of triangle $APQ$ lies on the segment $BD.$ [i](6 points)[/i]

Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB$, $\angle OBA$, $\angle OBC$, $\angle OCB$, $\angle OAC$, $\angle OCA$ are acute.

a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers \[g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).\] b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers \[g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.\]

Suppose that $y$ is a positive integer written only with digit $1$, in base $9$ system. Prove that $y$ is a triangular number, that is, exists positive integer $n$ such that the number $y$ is the sum of the $n$ natural numbers from $1$ to $n$.

The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy $a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1}$ and $b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}$ for every integer $n = 1, 2, \dots, 2013$. If $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute \[ \sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right). \][i]Proposed by Evan Chen[/i]

How many different integers can be expressed as the sum of three distinct members of the set $ \{1, 4, 7, 10, 13, 16, 19\}$? $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$

Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones. Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible. (a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains. (b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.

As shown in figure , it is known that $M$ is the midpoint of side $BC$ of $\vartriangle ABC$. $\odot O$ passes through points $A, C$ and is tangent to $AM$. The extension of the segment $BA$ intersects $\odot O$ at point $D$. The lines $CD$ and $MA$ intersect at the point $P$. Prove that $PO \perp BC$. [img]https://cdn.artofproblemsolving.com/attachments/8/a/da3570ec7eb0833c7a396e22ffac2bd8902186.png[/img]

Consider an isosceles triangle $ABC$ ($AB = BC$). Let $D$ be on $BC$ such that $AD \perp BC$ and $O$ be a circle with diameter $BC$. Suppose that segment $AD$ intersects circle $O$ at $E$. If $CA = 2$ what is $CE$?

Natalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value $x\in [0,1]$ uniformly at random, and (a) If $x\leq\tfrac12$ she colors the interval $[x,x+\tfrac12]$ with her marker. (b) If $x>\tfrac12$ she colors the intervals $[x,1]$ and $[0,x-\tfrac12]$ with her marker. What is the expected value of the number of steps Natalie will need to color the entire interval black?

Let $n$ be a positive integer. Let $B_n$ be the set of all binary strings of length $n$. For a binary string $s_1\hdots s_n$, we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by $b$. Then, we replace $s_b$ with $1-s_b$. A string $a$ is said to be a [i]descendant[/i] of $b$ if $a$ can be obtained from $b$ through a finite number of twists. A subset of $B_n$ is called [i]divided[/i] if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of $B_n$. [i]Remark.[/i] Here is an example of a twist: $101100 \rightarrow 101000$ because $1\mid 0\mid 11\mid 00$ has $4$ blocks of consecutive digits. [i]Viktor Simjanoski[/i]

Two people A and B play a game in the $m \times n$ grid ($m,n \in \mathbb{N^*}$). Each person respectively (A plays first) draw a segment between two point of the grid such that this segment doesn't contain any point (except the 2 ends) and also the segment (except the 2 ends) doesn't intersect with any other segments. The last person who can't draw is the loser. Which one (of A and B) have the winning tactics?

Let $ABC$ be a triangle, and let $P$ be a distinct point on the plane. Moreover, let $A'B'C'$ be a homothety of $ABC$ with ratio $2$ and center $P$, and let $O$ and $O'$ be the circumcenters of $ABC$ and $A'B'C'$, respectively. The circumcircles of $AB'C'$, $A'BC'$, and $A'B'C$ meet at points $X$, $Y$, and $Z$, different from $A'$, $B'$, and $C'$. In a similar way, the circumcircles of $A'BC$, $AB'C$, and $ABC'$ meet at $X'$, $Y'$, and $Z'$, different from $A$, $B$, $C$. Let $W$ and $W'$ be the circumcenters of $XYZ$ and $X'Y'Z'$, respectively. Prove that $OW$ is parallel to $O'W'$. [i]Proposed by Mateus Thimóteo, Brazil.[/i]