Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

Which of the following derived units is equivalent to units of velocity? $ \textbf {(A) } \dfrac {\text {W}}{\text {N}} \qquad \textbf {(B) } \dfrac {\text {N}}{\text {W}} \qquad \textbf {(C) } \dfrac {\text {W}}{\text {N}^2} \qquad \textbf {(D) } \dfrac {\text {W}^2}{\text {N}} \qquad \textbf {(E) } \dfrac {\text {N}^2}{\text {W}^2} $ [i]Problem proposed by Ahaan Rungta[/i]

Triangle $ABC$ is inscribed in circle $(O)$. $ P$ is a point on arc $BC$ that does not contain $ A$ such that $AP$ is the symmedian of triangle $ABC$. $E ,F$ are symmetric of $P$ wrt $CA, AB$ respectively . $K$ is symmetric of $A$ wrt $EF$. $L$ is the projection of $K$ on the line passing through $A$ and parallel to $BC$. Prove that $PA=PL$.

Square trinomial $f(x)$ is such that the polynomial (f(x))^5 - f(x) has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.

$M$ is a finite set of points in a plane. Point $O$ in the plane is called an “almost centre of symmetry” of set $M$ if it is possible to remove from $M$ one point in such a way that among the remaining members $O$ is the centre of symmetry in the usual sense. How many such “almost centres of symmetry” may a finite point set in a plane have? Indicate all such points. (V Prasolov, Moscow)

Real numbers $a$, $b$, $c$ satisfy the equation $$2a^3-b^3+2c^3-6a^2b+3ab^2-3ac^2-3bc^2+6abc=0$$. If $a<b$, find which of the numbers $b$, $c$ is larger.

How many moles of oxygen gas are produced by the decomposition of $245$ g of potassium chlorate? \[\ce{2KClO3(s)} \rightarrow \ce{2KCl(s)} + \ce{3O2(g)}\] Given: Molar Mass/ $\text{g} \cdot \text{mol}^{-1}$ $\ce{KClO3}$: $122.6$ $ \textbf{(A)}\hspace{.05in}1.50 \qquad\textbf{(B)}\hspace{.05in}2.00 \qquad\textbf{(C)}\hspace{.05in}2.50 \qquad\textbf{(D)}\hspace{.05in}3.00 \qquad $

Find all positive integers $n$ such that there exist two permutations $a_0,a_1,\ldots,a_{n-1}$ and $b_0,b_1,\ldots,b_{n-1}$ of the set $\lbrace0,1,\ldots,n-1\rbrace$, satisfying the condition $$ia_i\equiv b_i\pmod{n}$$ for all $0\le i\le n-1$. [i]Proposed by Fysty[/i]

Let $\Gamma$ and $\Gamma'$ be two circles with centres $O$ and $O'$, such that $O$ belongs to $\Gamma'$. Let $M$ be a point on $\Gamma'$, outside of $\Gamma$. The tangents to $\Gamma$ that go through $M$ touch $\Gamma$ in two points $A$ and $B$, and cross $\Gamma'$ again in two points $C$ and $D$. Finally, let $E$ be the crossing point of the lines $AB$ and $CD$. Prove that the circumcircles of the triangles $CEO'$ and $DEO'$ are tangent to $\Gamma'$.

Numbers $1, 2, \dots , n$ are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?

Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.

Let $a,b,c,d $-reals positive numbers. Prove inequality: $\frac{a^2+b^2+c^2}{ab+bc+cd}+\frac{b^2+c^2+d^2}{bc+cd+ad}+\frac{a^2+c^2+d^2}{ab+ad+cd}+\frac{a^2+b^2+d^2}{ab+ad+bc} \geq 4$

Does every positive polynomial in two real variables attain its lower bound in the plane?

A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square’s perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly $2021$ regions. Compute the smallest possible value of $n$.

In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?

A game consists in $9$ coins (blacks or whites) arrenged in the following position (see picture 1). If you choose $1$ coin on the border of the square, this coin and it's neighbours change their color. If you choose the coin at the centre, it doesn't change it's color, but the other $8$ coins do. Here is an example of $9$ white coins, and the changes of their colors, choosing the coin said: (see picture 2). Is it possible, starting with $9$ white coins, to have $9$ black coins?.

Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

Let $f$ be a continuous real-valued function on $\mathbb R^3$.  Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals zero.  Must $f(x,y,z)$ be identically zero?

The length of the hypotenuse of a right triangle is $h$ , and the radius of the inscribed circle is $r$. The ratio of the area of the circle to the area of the triangle is $\textbf{(A) }\frac{\pi r}{h+2r}\qquad\textbf{(B) }\frac{\pi r}{h+r}\qquad\textbf{(C) }\frac{\pi}{2h+r}\qquad\textbf{(D) }\frac{\pi r^2}{r^2+h^2}\qquad\textbf{(E) }\text{none of these}$

Anton wrote $4$ positive integers on the board. Oleksii calculated their product, while Fedir calculated the sum of their fourth powers. Is it possible that Oleksii's number and Fedir's number have the same number of digits and that these numbers are written as digit-reversals of each other? [i]Proposed by Fedir Yudin and Mykhailo Shtandenko[/i]

Let $p$ and $q$ be two different primes. Prove that \[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]

How many digits are in the product $4^5 \cdot 5^{10}$? $ \textbf{(A)} 8 \qquad\textbf{(B)} 9 \qquad\textbf{(C)} 10 \qquad\textbf{(D)} 11 \qquad\textbf{(E)} 12 $

In the acute-angled triangle $ABC$, let $M$ be the midpoint of the atlitude $h_b$ through $B$ and $N$ be the midpoint of the height $h_c$ through $C$. Further let $P$ be the intersection of $AM$ and $h_c$ and $Q$ be the intersection of $AN$ and $h_b$. Show that $M, N, P$ and $Q$ lie on a circle.

A quadrilateral \( ABCD \) with no parallel sides is inscribed in a circle \( \Omega \). Circles \( \omega_a, \omega_b, \omega_c, \omega_d \) are inscribed in triangles \( DAB, ABC, BCD, CDA \), respectively. Common external tangents are drawn between \( \omega_a \) and \( \omega_b \), \( \omega_b \) and \( \omega_c \), \( \omega_c \) and \( \omega_d \), and \( \omega_d \) and \( \omega_a \), not containing any sides of quadrilateral \( ABCD \). A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle \( \Gamma \). Prove that the lines joining the centers of \( \omega_a \) and \( \omega_c \), \( \omega_b \) and \( \omega_d \), and the centers of \( \Omega \) and \( \Gamma \) all intersect at one point. \\