Let $\omega_1$ and $\omega_2$ with center $O_1$ and $O_2$ respectively, meet at points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. Lines $XA$ and $Y A$ meet $\omega_2$ at $Z$ and $W$, respectively, such that $A$ lies between $X$ and $Z$ and between $Y$ and $W$. Let $M$ be the midpoint of $O_1O_2$, $S$ be the midpoint of $XA$ and $T$ be the midpoint of $W A$. Prove that $MS = MT$ if and only if $X,~ Y ,~ Z$ and $W$ are concyclic.

Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\tfrac12$ the highest note that is lower than the note he just played. What is the probability that he plays the highest note on the piano before playing the lowest note?

Prove that for any positive $ x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ the inequality \[ \sum_{i\equal{}1}^n\frac1{x_iy_i}\ge\frac{4n^2}{\sum_{i\equal{}1}^n(x_i\plus{}y_i)^2} \] holds.

What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}$? $\textbf{(A) }1020\qquad\textbf{(B) }1120\qquad\textbf{(C) }1220\qquad\textbf{(D) }2240\qquad\textbf{(E) }3360$

Consider 2 non-constant polynomials $P(x),Q(x)$, with nonnegative coefficients. The coefficients of $P(x)$ is not larger than $2021$ and $Q(x)$ has at least one coefficient larger than $2021$. Assume that $P(2022)=Q(2022)$ and $P(x),Q(x)$ has a root $\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)$. Prove that $|p|+n|q|\le Q(n)-P(n)$ for all $n=1,2,...,2021$

Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)

Let $H$ be a regular hexagon with side length $1$. The sum of the areas of all triangles whose vertices are all vertices of $H$ can be expressed as $A\sqrt{B}$ for positive integers $A$ and $B$ such that $B$ is square-free. What is $1000A +B$?

Suppose $S$ is the set of all positive integers. For $a,b \in S$, define \[a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}\] For example $8*12=6$. Show that [b]exactly two[/b] of the following three properties are satisfied: (i) If $a,b \in S$, then $a*b \in S$. (ii) $(a*b)*c=a*(b*c)$ for all $a,b,c \in S$. (iii) There exists an element $i \in S$ such that $a *i =a$ for all $a \in S$.

Initially, on the blackboard are written all natural numbers from $1$ to $20$. A move consists of selecting $2$ numbers $a<b$ written on the blackboard such that their difference is at least $2$, erasing these numbers and writting $a+1$ and $b-1$ instead. What is the maximum numbers of moves one can perform?

Let $O$ be the center and $R$ be the radius of circumcircle $\omega$ of triangle $ABC$. Circle $\omega_1$ with center $O_1$ and radius $R$ pass through points $A, O$ and intersects the side $AC$ at point $K$. Let $AF$ be the diameter of circle $\omega$ and points $F, K, O_1$ are collinear. Determine $\angle ABC$:

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

Let $ABC$ be a triangle, and let $D$ be the point where the incircle meets side $BC$. Let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the angle bisector of $\angle BAC$.

Find all triples of natural $ (x,y,n)$ satisfying the condition: \[ \frac {x! \plus{} y!}{n!} \equal{} 3^n \] Define $ 0! \equal{} 1$

Let the [i]square decomposition[/i] of a number be defined as the sequence of numbers given by the following algorithm. Given a positive integer $n$, add the largest possible perfect square that is less than or equal to $n$ to a sequence, and then subtract that number from $n$. Repeat as many times as necessary until your current $n$ is $0$. So for example, the square decomposition of $60$ would be $49, 9, 1, 1$. Define the size of a square decomposition to be the number of numbers in the sequence. Say that the maximal size of a square decomposition of a number in the range $[1, 2020]$ is $m$. Find the largest number in the range $[1, 2020]$ that has a square decomposition of size $m$. [i]Proposed by Timothy Qian[/i]

Compute $\cos(\tfrac{\pi}{9})-\cos(\tfrac{2\pi}{9})+\cos(\tfrac{3\pi}{9})-\cos(\tfrac{4\pi}{9})$.

There exists a unique line tangent to the graph of $y=x^4-20x^3+24x^2-20x+25$ at two distinct points. Compute the product of the $x$-coordinates of the two tangency points.

Let $\vartriangle BMT$ be a triangle with $BT = 1$ and height $1$. Let $O_0$ be the centroid of $\vartriangle BMT$, and let $\overline{BO_0}$ and $\overline{TO_0}$ intersect $\overline{MT}$ and $\overline{BM}$ at $B_1$ and $T_1$, respectively. Similarly, let $O_1$ be the centroid of $\vartriangle B_1MT_1$, and in the same way, denote the centroid of $\vartriangle B_nMT_n$ by $O_n$, the intersection of $\overline{BO_n}$ with $\overline{MT}$ by $B_{n+1}$, and the intersection of $\overline{TO_n}$ with $\overline{BM}$ by $T_{n+1}$. Compute the area of quadrilateral $MBO_{2021}T$.

In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.

We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points. Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

There are $1000$ students in a school. Every student has exactly $4$ friends. A group of three students $ \left \{A,B,C \right \}$ is said to be a [i]friendly triplet[/i] if any two students in the group are friends. Determine the maximal possible number of friendly triplets. [i]Proposed by Nikola Velov[/i]

Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$

In a triangle $ABC$, let $D$ and $E$ trisect $BC$, so $BD = DE = EC$. Let $F$ be the point on $AB$ such that $\frac{AF}{F B}= 2$, and $G$ on $AC$ such that $\frac{AG}{GC} =\frac12$ . Let $P$ be the intersection of $DG$ and $EF$, and extend $AP$ to intersect $BC$ at a point $X$. Find $\frac{BX}{XC}$

A variable tetrahedron $ABCD$ has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 70 $