Let a point $M$ not lying on coordinates axes be given. Points $Q$ and $P$ move along $Y$ - and $X$-axis respectively so that angle $P M Q$ is always right. Find the locus of points symmetric to $M$ wrt $P Q$.

Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $

Let $X$ be a set of $n + 1$ elements, $n\geq 2$. Ordered $n$-tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ formed from distinct elements of $X$ are called[i] disjoint [/i]if there exist distinct indices $1\leq i \neq j\leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples.

Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, their sum is $ 0$.) $ \textbf{(A)}\ \frac{3}{8} \qquad \textbf{(B)}\ \frac{1}{2} \qquad \textbf{(C)}\ \frac{43}{72} \qquad \textbf{(D)}\ \frac{5}{8} \qquad \textbf{(E)}\ \frac{2}{3}$

Solve the equation for positive integers $m, n$: \[\left \lfloor \frac{m^2}n \right \rfloor + \left \lfloor \frac{n^2}m \right \rfloor = \left \lfloor \frac mn + \frac nm \right \rfloor +mn\]

For each $i=1,2,..N$, let $a_i,b_i,c_i$ be integers such that at least one of them is odd. Show that one can find integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $\frac{4}{7}N$ different values of $i$.

Jerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis. Dennis then announces, "Aaron's number is at least three times Jerry's number." Aaron says, "I still don't know Jerry's number." Jerry then replies, "Oh, now I know Aaron's number." What is the sum of their numbers? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

A rhombus $\mathcal R$ has short diagonal of length $1$ and long diagonal of length $2023$. Let $\mathcal R'$ be the rotation of $\mathcal R$ by $90^\circ$ about its center. If $\mathcal U$ is the set of all points contained in either $\mathcal R$ or $\mathcal R'$ (or both; this is known as the [i]union[/i] of $\mathcal R$ and $\mathcal R'$) and $\mathcal I$ is the set of all points contained in both $\mathcal R$ and $\mathcal R'$ (this is known as the [i]intersection[/i] of $\mathcal R$ and $\mathcal R'$, compute the ratio of the area of $\mathcal I$ to the area of $\mathcal U$. [i]Proposed by Connor Gordon[/i]

In triangle $ABC$ bisectors $AA_1, BB_1$ and $CC_1$ are drawn. Given $\angle A : \angle B : \angle C = 4 : 2 : 1$, prove that $A_1B_1 = A_1C_1$. [i](7 points)[/i]

Find all values ​​of the parameter $a$ for which the equation $$(a-1)^2x^4 + (a^2-a) x^3 + 3x - 1 = 0$$ has a unique solution and for these $a$ solve the equation.

The circle $\omega_1$ intersects the circle $\omega_2$ in the points $A$ and $B$, a tangent line to this circles intersects $\omega_1$ and $\omega_2$ in the points $E$ and $F$ respectively. Suppose that $A$ is inside of the triangle $BEF$, let $H$ be the orthocenter of $BEF$ and $M$ is the midpoint of $BH$. Prove that the centers of the circles $\omega_1$ and $\omega_2$ and the point $M$ are collinears.

Let $x$ be a natural number. We call $\{x_0,x_1,\dots ,x_l\}$ a [i]factor link [/i]of $x$ if the sequence $\{x_0,x_1,\dots ,x_l\}$ satisfies the following conditions: (1) $x_0=1, x_l=x$; (2) $x_{i-1}<x_i, x_{i-1}|x_i, i=1,2,\dots,l$ . Meanwhile, we define $l$ as the length of the [i]factor link [/i] $\{x_0,x_1,\dots ,x_l\}$. Denote by $L(x)$ and $R(x)$ the length and the number of the longest [i]factor link[/i] of $x$ respectively. For $x=5^k\times 31^m\times 1990^n$, where $k,m,n$ are natural numbers, find the value of $L(x)$ and $R(x)$.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Gamma$, and let $E$ be the midpoint of the diagonal $BD$. Let $I_1, I_2, I_3, I_4$ be the centers of the circles inscribed into triangles $\vartriangle ABE$, $\vartriangle ADE$, $\vartriangle BCE$, $\vartriangle CDE$, in that order. Prove that the circles $AI_1I_2$ and $CI_3I_4$ intersect $\Gamma$ at diametrically opposite points. Remark: For a circle $C$ and points $X, Y \in C$, we say that $X$ and $Y$ are diametrically opposite if $XY$ is a diameter of $C$.

Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ be a point on the base $BC$ such that $BD:DC = 2: 1$. Note on the segment $AD$ a point $P$ such that $\angle BAC= \angle BPD $. Prove that $\angle BPD = 2 \angle CPD$.

Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.

Are there any integers $a,b$ and $c$, not all of them $0$, such that $$a^2=2021b^2+2022c^2~~?$$ [i] (Cosmin Gavrilă)[/i]

Determine all real number $(x,y)$ pairs that satisfy the equation. $$2x^2+y^2+7=2(x+1)(y+1)$$

A random rectangle (not necessarily a square) with positive integer dimensions is selected from the $2\times4$ grid below. The probability that the selected rectangle contains only white squares can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [asy] fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,blue); fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,blue); draw((0,0)--(4,0),black); draw((0,0)--(0,2),black); draw((4,0)--(4,2),black); draw((4,2)--(0,2),black); draw((0,1)--(4,1),black); draw((1,0)--(1,2),black); draw((2,0)--(2,2),black); draw((3,0)--(3,2),black); [/asy]

Let $f(x)$ be an even twice differentiable function such that $f''(0)\ne0$. Prove that $f(x)$ has a local extremum at $x=0$.

$$\int_0^\infty \frac{\sin(20x)\sin(21x)}{x^2}\,dx$$ [i]Proposed by Connor Gordon and Vlad Oleksenko[/i]

Let $ABCD$ be a cyclic quadrilateral so that $\overline{AC} \perp \overline{BD}$. Let $E$ be the intersection of $\overline{AC}$ and $\overline{BD}$, and let $F$ be the foot of the altitude from $E$ to $\overline{AB}$. Let $\overline{EF}$ intersect $\overline{CD}$ at $G$, and let the foot of the perpendiculars from $G$ to $\overline{AC}$ and $\overline{BD}$ be $H, I$ respectively. If $\overline{AB} = \sqrt{5}, \overline{BC} = \sqrt{10}, \overline{CD} = 3\sqrt{5}, \overline{DA} = 2\sqrt{10}$, find the length of $\overline{HI}$. [i]Proposed by Timothy Qian[/i]

Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is [asy] pair a = (0,0),b= (6,0),c=(0,1),d=(6,1); transform t = rotate(-45,(3,.5)); pair e = t*a,f=t*b,g=t*c,h=t*d; pair i = intersectionpoint(a--b,e--f),j=intersectionpoint(a--b,g--h),k=intersectionpoint(c--d,e--f),l=intersectionpoint(c--d,g--h); draw(a--b^^c--d^^e--f^^g--h); filldraw(i--j--l--k--cycle,blue); label("$\alpha$",i+(-.5,.2)); //commented out labeling because it doesn't look right. //path lbl1 = (a+(.5,.2))--(c+(.5,-.2)); //draw(lbl1); //label("$1$",lbl1);[/asy] $\text{(A)} \ \sin \alpha \qquad \text{(B)} \ \frac{1}{\sin \alpha} \qquad \text{(C)} \ \frac{1}{1 - \cos \alpha} \qquad \text{(D)} \ \frac{1}{\sin^2 \alpha} \qquad \text{(E)} \ \frac{1}{(1 - \cos \alpha)^2}$

Let $n>100$ be a positive integer and originally the number $1$ is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after $n$ moves, the denominator of the fraction left on the board is at most $2^n+50$, no matter how Vasya acts.