A beam of light strikes $\overline{BC}$ at point $C$ with angle of incidence $\alpha=19.94^\circ$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection. Given that $\beta=\alpha/10=1.994^\circ$ and $AB=AC,$ determine the number of times the light beam will bounce off the two line segments. Include the first reflection at $C$ in your count. [asy] size(250);defaultpen(linewidth(0.7)); real alpha=24, beta=32; pair B=origin, C=(1,0), A=dir(beta), D=C+0.5*dir(alpha); pair EE=2*dir(180-alpha), E=intersectionpoint(C--EE, A--B); pair EEE=reflect(B,A)*EE, EEEE=reflect(C,B)*EEE, F=intersectionpoint(E--EEE, B--C), G=intersectionpoint(F--EEEE, A--B); draw((1.4,0)--B--1.4*dir(beta)); draw(D--C, linetype("4 4"),EndArrow(5)); draw(C--E, linetype("4 4"),EndArrow(5)); draw(E--F, linetype("4 4"),EndArrow(5)); draw(F--G, linetype("4 4"),EndArrow(5)); markscalefactor=0.01; draw(anglemark(C,B,A)); draw(anglemark((1.4,0), C,D)); label("$\beta$", 0.07*dir(beta/2), dir(beta/2), fontsize(10)); label("$\alpha$", C+0.07*dir(alpha/2), dir(alpha/2), fontsize(10)); label("$A$", A, dir(90)*dir(A)); label("$B$", B, dir(beta/2+180)); label("$C$", C, S);[/asy]

Angle $A$ in triangle $ABC$ is equal to $a$. A circle passing through $A$ and $B$ and tangent to $BC$ intersects the median to side $BC$ (or its extension) at a point $M$ different from $A$. Find the angle $\angle BMC$.

Is $$\prod_{k=0}^\infty \left(1-\frac{1}{2022^{k!}}\right)$$ rational?

[b]a)[/b] Show that , if $ a,b>1 $ are two distinct real numbers, then $ \log_a\log_a b >\log_b\log_a b. $ [b]b)[/b] Show that if $ a_1>a_2>\cdots >a_n>1 $ are $ n\ge 2 $ real numbers, then $$ \log_{a_1}\log_{a_1} a_2 +\log_{a_2}\log_{a_2} a_3 +\cdots +\log_{a_{n-1}}\log_{a_{n-1}} a_n +\log_{a_n}\log_{a_n} a_1 >0. $$

Let $n$ be an odd positive integer and consider an $n \times n$ chessboard of $n^2$ unit squares. In some of the cells of the chessboard, we place a knight. A knight in a cell $c$ is said to [i]attack [/i] a cell $c'$ if the distance between the centers of $c$ and $c'$ is exactly $\sqrt{5}$ (in particular, a knight does not attack the cell which it occupies). Suppose each cell of the board is attacked by an even number of knights (possibly zero). Show that the configuration of knights is symmetric with respect to all four axes of symmetry of the board (i.e. the configuration of knights is both horizontally and vertically symmetric, and also unchanged by reflection along either diagonal of the chessboard). [i]NIkolai Beluhov[/i]

$\sqrt{\frac{8^{10}+4^{10}}{8^4+4^{11}}}=$ $ \textbf{(A)}\ \sqrt{2} \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 32 \qquad\textbf{(D)}\ 12^{\frac{2}{3}} \qquad\textbf{(E)}\ 512.5 $

There are $2003$ coins distributed in several bags. The bags are then distributed in several pockets. It is known that the total number of bags is greater than the number of coins in each of the pockets. Is it true that the total number of pockets is greater than the number of coins in some of the bags?

Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)[/i]

Let $ABC$ be an acute scalene triangle, and let $A_1, B_1, C_1$ be the feet of the altitudes from $A, B, C$. Let $A_2$ be the intersection of the tangents to the circle $ABC$ at $B, C$ and define $B_2, C_2$ similarly. Let $A_2A_1$ intersect the circle $A_2B_2C_2$ again at $A_3$ and define $B_3, C_3$ similarly. Show that the circles $AA_1A_3, BB_1B_3$, and $CC_1C_3$ all have two common points, $X_1$ and $X_2$ which both lie on the Euler line of the triangle $ABC$. [i]United Kingdom, Joe Benton[/i]

There are integers $v,w,x,y,z$ and real numbers $0\le \theta < \theta' \le \pi$ such that $$\cos 3\theta = \cos 3\theta' = v^{-1}, \qquad w+x\cos \theta + y\cos 2\theta = z\cos \theta'.$$ Given that $z\ne 0$ and $v$ is positive, find the sum of the $4$ smallest possible values of $v$. [i]Proposed by Vijay Srinivasan[/i]

The positive integer formed after writing $k$ consecutive positive integers from smallest to largest is called a $k-\text{continuous}$ number. For example $99100101$ is a $3-\text{continuous}$ number. Prove that: for $\forall N$, $k\in\mathbb Z^+$, there must be a $k-\text{continuous}$ number that can be divisible by $N$.

Let $ABCD$ be a square and $M,N$ points on sides $AB, BC$ respectively such that $\angle MDN = 45^{\circ}$. If $R$ is the midpoint of $MN$ show that $RP =RQ$ where $P,Q$ are points of intersection of $AC$ with the lines $MD, ND$.

Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.

Let $G$ be a graph, not containing $K_4$ as a subgraph and $|V(G)|=3k$ (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in $G$?

With the use of three different weights, namely 1 lb., 3 lb., and 9 lb., how many objects of different weights can be weighed, if the objects is to be weighed and the given weights may be placed in either pan of the scale? $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 7 $

The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.

A paper rectangle $ABCD$ has $AB = 8$ and $BC = 6$. After corner $B$ is folded over diagonal $AC$, what is $BD$?

The combination to open a safe is a five-digit number. different, randomly selected from $2$ to $9$. To open the box strong, you also need a key that is labeled with the number $410639104$, which is the sum of all combinations that do not open the box. What is the combination that opens the safe?

Find two consecutive integers with the property that the sums of their digits are each divisible by $2006$.

Let $H$ be an orhocenter of an acute triangle $ABC$ and $M$ midpoint of side $BC$. If $D$ and $E$ are foots of perpendicular of $H$ on internal and external angle bisector of angle $\angle BAC$, prove that $M$, $D$ and $E$ are collinear

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Let $n$ be a natural number divisible by $3$. We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ($m > 1$), the number of black squares is not more than white squares. Find the maximum number of black squares.

Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

4. Given is an integer $n\geq 1$. Find out the number of possible values of products $k \cdot m$, where $k,m$ are integers satisfying $n^{2}\leq k \leq m \leq (n+1)^{2}$.

Let $A'B'C'$ denote the reflection of scalene and acute triangle $ABC$ across its Euler-line. Let $P$ be an arbitrary point of the nine-point circle of $ABC$. For every point $X$, let $p(X)$ denote the reflection of $X$ across $P$. [b]a)[/b] Let $e_{AB}$ denote the line connecting the orthogonal projection of $A$ to line $BB'$ and the orthogonal projection of $B$ to line $AA'$. Lines $e_{BC}$ and $e_{CA}$ are defined analogously. Prove that these three lines are concurrent (and denote their intersection by $K$). [b]b)[/b] Prove that there are two choices of $P$ such that lines $Ap(A')$, $Bp(B')$ and $Cp(C')$ are concurrent, and the four points $p(A)p(A')\cap BC$, $p(B)p(B')\cap CA$, $p(C)p(C')\cap AB$, and $K$ are collinear. [i]Proposed by Áron Bán-Szabó, Budapest[/i]