Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$. Prove that $M_1$ lies on the segment $BH_1$.

The midpoints of four sides of a cyclic pentagon were marked, after this the pentagon was erased. Restore it.

An L-shape is one of the following four pieces, each consisting of three unit squares: [asy] size(300); defaultpen(linewidth(0.8)); path P=(1,2)--(0,2)--origin--(1,0)--(1,2)--(2,2)--(2,1)--(0,1); draw(P); draw(shift((2.7,0))*rotate(90,(1,1))*P); draw(shift((5.4,0))*rotate(180,(1,1))*P); draw(shift((8.1,0))*rotate(270,(1,1))*P); [/asy] A $5\times 5$ board, consisting of $25$ unit squares, a positive integer $k\leq 25$ and an unlimited supply of L-shapes are given. Two players A and B, play the following game: starting with A they play alternatively mark a previously unmarked unit square until they marked a total of $k$ unit squares. We say that a placement of L-shapes on unmarked unit squares is called $\textit{good}$ if the L-shapes do not overlap and each of them covers exactly three unmarked unit squares of the board. B wins if every $\textit{good}$ placement of L-shapes leaves uncovered at least three unmarked unit squares. Determine the minimum value of $k$ for which B has a winning strategy.

Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent: a) $f$ is differentiable, with continuous first derivative. b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.

The root(s) of $ \frac {15}{x^2 \minus{} 4} \minus{} \frac {2}{x \minus{} 2} \equal{} 1$ is (are): $ \textbf{(A)}\ \minus{} 5\text{ and }3 \qquad\textbf{(B)}\ \pm 2 \qquad\textbf{(C)}\ 2\text{ only} \qquad\textbf{(D)}\ \minus{} 3\text{ and }5 \qquad\textbf{(E)}\ 3\text{ only}$

Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing \right\}$. Find the number of elements of $K(2004,2004)$.

[b]a)[/b] Determine all sequences of real numbers $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ that satisfy $ x_{n+2}+x_{n+1}=x_n, $ for any nonnegative integer $ n. $ [b]b)[/b] If $ y_k>0 $ and $ y_k^k=y_k+k, $ for all naturals $ k, $ calculate $ \lim_{n\to\infty }\frac{\ln n}{n\left( x_n-1\right)} . $

Sally rolls an $8$-sided die with faces numbered $1$ through $8$. Compute the probability that she gets a power of $2$.

Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b, c \in M$, the number $a^2 + bc$ is rational. Prove that there is a positive integer $n$ such that $a\sqrt n$ is rational for all $a \in M.$

In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then: [asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); real e=350,c=55; pair O=origin,E=dir(e),C=dir(c),B=dir(180+c),D=dir(180+e), rot=rotate(90,B)*O,A=extension(E,D,B,rot); path tangent=A--B; pair P=waypoint(tangent,abs(A-D)/abs(A-B)); draw(unitcircle^^C--B--A--E); dot(A^^B^^C^^D^^E^^P,linewidth(2)); label("$O$",O,dir(290)); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,NE); label("$D$",D,dir(120)); label("$E$",E,SE); label("$P$",P,SW);[/asy] $ \textbf{(A)} AP^2 = PB \times AB\qquad$ $\textbf{(B)}\ AP \times DO = PB \times AD\qquad$ $\textbf{(C)}\ AB^2 = AD \times DE\qquad$ $\textbf{(D)}\ AB \times AD = OB \times AO\qquad$ $\textbf{(E)}\ \text{none of these} $

Given are two distinct sequences of positive integers $(a_n)$ and $(b_n)$, such that their first two members are coprime and smaller than $1000$, and each of the next members is the sum of the previous two. 8-9 grade Prove that if $a_n$ is divisible by $b_n$, then $n<50$ 10-11 grade Prove that if $a_n^{100}$ is divisible by $b_n$ then $n<5000$

Let $ P_1,P_2,\ldots,P_8$ be $ 8$ distinct points on a circle. Determine the number of possible configurations made by drawing a set of line segments connecting pairs of these $ 8$ points, such that: $ (1)$ each $ P_i$ is the endpoint of at most one segment and $ (2)$ no two segments intersect. (The configuration with no edges drawn is allowed. An example of a valid configuration is shown below.) [asy]unitsize(1cm); pair[] P = new pair[8]; align[] A = {E, NE, N, NW, W, SW, S, SE}; for (int i = 0; i < 8; ++i) { P[i] = dir(45*i); dot(P[i]); label("$P_"+((string)i)+"$", P[i], A[i],fontsize(8pt)); } draw(unitcircle); draw(P[0]--P[1]); draw(P[2]--P[4]); draw(P[5]--P[6]);[/asy]

Two lines intersect inside a unit square, splitting it into four regions. Find the maximum product of the areas of the four regions. [i]Proposed by Nathan Ramesh

Solve for all complex numbers $z$ such that $z^4+4z^2+6=z$.

There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle. Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labellings of the $2022$ points. [i]Kevin Cong[/i]

Through the vertices $A, B$ of the parallelogram $ABCD$ passes a circle that intersects for the second time diagonals $BD$ and $AC$ at points $X$ and $Y$, respectively. The circumsccribed circle of $\vartriangle ADX$ intersects diagonal $AC$ for the second time at the point $Z$. Prove that $AY = CZ$.

Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials $f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$ have integer coefficients. Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$. Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j)\right|=1$

In a deck of cards, there are $kn$ cards numbered from $1$ to $n$ and there are $k$ cards of each number. Now, divide this deck into $k$ sub-decks with equal sizes. Prove that if $\gcd(k,n)=1$, then one could always pick $n$ cards, one from each sub-deck, such that the sum of those cards is divisible by $n$.

Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is [asy] size(180); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10); pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3; D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));[/asy] $\textbf{(A)}\ \dfrac{1}{6}\qquad \textbf{(B)}\ \dfrac{1}{4} \qquad \textbf{(C)}\ \dfrac{1}{3} \qquad \textbf{(D)}\ \dfrac{2}{5} \qquad \textbf{(E)}\ \dfrac{1}{2}$

Consider triangle $ABC$ such that $AB \le AC$. Point $D$ on the arc $BC$ of thecircumcirle of $ABC$ not containing point $A$ and point $E$ on side $BC$ are such that $\angle BAD = \angle CAE < \frac12 \angle BAC$ . Let $S$ be the midpoint of segment $AD$. If $\angle ADE = \angle ABC - \angle ACB$ prove that $\angle BSC = 2 \angle BAC$ .

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

There is a pile with 2022 rocks. Ana y Beto play by turns to the following game, starting with Ana: in each turn, if there are $n$ rocks in the pile, the player can remove $S(n)$ rocks or $n-S(n)$ rocks, where $S(n)$ is the sum of the the digits of $n$. The person who removes the last rock wins. Determine which of the two players has a winning strategy and describe it.

A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.

Prove that $\sigma(n)\phi(n) < n^2$, but that there is a positive constant $c$ such that $\sigma(n)\phi(n) \ge c n^2$ holds for all positive integers $n$.

A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$. Prove that the lines $BC$ and $B_1C_1$ are parallel. [i](A. Voidelevich)[/i]