Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

Let $ABCDE$ be a regular pentagon with center $M$. A point $P$ (different from $M$) is chosen on the line segment $MD$. The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and the line through $P$ perpendicular to $CD$ in $P$ and $R$. Prove that $AR$ and $QR$ have same length. [i]proposed by Stephan Wagner[/i]

Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\] [i]G. Halasz[/i]

Prove that there are positive integers $a_1, a_2,\dots, a_{2020}$ such that $$\dfrac{1}{a_1}+\dfrac{1}{2a_2}+\dfrac{1}{3a_3}+\dots+\dfrac{1}{2020a_{2020}}=1.$$

21) A particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$. If the collision is perfectly $elastic$, what is the maximum possible fractional momentum transfer, $f_{max}$? A) $0 < f_{max} < \frac{1}{2}$ B) $f_{max} = \frac{1}{2}$ C) $\frac{1}{2} < f_{max} < \frac{3}{2}$ D) $f_{max} = 2$ E) $f_{max} \ge 3$

$ (a)$ Prove that $ \mathbb{N}$ can be partitioned into three (mutually disjoint) sets such that, if $ m,n \in \mathbb{N}$ and $ |m\minus{}n|$ is $ 2$ or $ 5$, then $ m$ and $ n$ are in different sets. $ (b)$ Prove that $ \mathbb{N}$ can be partitioned into four sets such that, if $ m,n \in \mathbb{N}$ and $ |m\minus{}n|$ is $ 2,3,$ or $ 5$, then $ m$ and $ n$ are in different sets. Show, however, that $ \mathbb{N}$ cannot be partitioned into three sets with this property.

Find all integers $x,y$ such that $x^3(y+1)+y^3(x+1)=19$. [i]Proposed by Bulgaria[/i]

Let $a$ and $b$ be positive integers such that $10< \gcd(a,b) < 20$ and $220 < \text{lcm}(a,b) < 230$. Find the difference between the smallest and largest possible values of $ab$.

Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle. [i]Proposed by Karthik Vedula[/i]

Inside the circle $\omega$ through points $A, B$ point $C$ is chosen. An arbitrary point $X$ is selected on the segment $BC$. The ray $AX$ cuts the circle in $Y$. Prove that all circles $CXY$ pass through a two fixed points that is they intersect and are coaxial, independent of the position of $X$.

Find the minimum value of $ \int_0^{\frac{\pi}{2}} |x\sin t\minus{}\cos t|\ dt\ (x>0).$

Determine all polynomials $P(x)$ with degree $n\geq 1$ and integer coefficients so that for every real number $x$ the following condition is satisfied $$P(x)=(x-P(0))(x-P(1))(x-P(2))\cdots (x-P(n-1))$$

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

Let $ ABC $ be a triangle, $ I_a $ be center of the $ A\text{-excircle}. $ This excircle intersects the lines $ AB, AC, $ at $ P, $ respectively, $ Q. $ The line $ PQ $ intersects the lines $ I_aB,I_aC $ at $ D, $ respectively, $ E. $ Let $ A_1 $ be the intersection of $ DC $ with $ BE, $ and define, analogously, $ B_1,C_1. $ Show that $ AA_1,BB_1,CC_1 $ are concurrent.

The $n$ participants of a tournament are numbered with $0$ through $n - 1$. At the end of the tournament it turned out that for every team, numbered with $s$ and having $t$ points, there are exactly $t$ teams having $s$ points each. Determine all possibilities for the final score list.

Find all regular $n$-gons with the following properties: (a) a diagonal is equal to the sum of two other diagonals (b) a diagonal is equal to the sum of a side and another diagonal

The incircle of a non-isosceles triangle $ABC$ touches $AB$ at point $C'$. The circle with diameter $BC'$ meets the incircle and the bisector of angle $B$ again at points $A_1$ and $A_2$ respectively. The circle with diameter $AC'$ meets the incircle and the bisector of angle $A$ again at points $B_1$ and $B_2$ respectively. Prove that lines $AB, A_1B_1, A_2B_2$ concur. (E. H. Garsia)

You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?

Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .

$x_1,...,x_n$ are non-negative reals and $n \geq 3$. Prove that at least one of the following inequalities is true: \[ \sum_{i=1} ^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}, \] \[ \sum_{i=1} ^n \frac{x_i}{x_{i-1}+x_{i-2}} \geq \frac{n}{2} . \]

If $ABCD$ is a rectangle and $P$ is a point inside it such that $AP=33, BP=16, DP=63$. Find $CP$.

A point $P$ lies on the internal angle bisector of $\angle BAC$ of a triangle $\triangle ABC$. Point $D$ is the midpoint of $BC$ and $PD$ meets the external angle bisector of $\angle BAC$ at point $E$. If $F$ is the point such that $PAEF$ is a rectangle then prove that $PF$ bisects $\angle BFC$ internally or externally.

Imagine a regular a $2015$-gon with edge length $2$. At each vertex, draw a unit circle centered at that vertex and color the circle’s circumference orange. Now, another unit circle $S$ is placed inside the polygon such that it is externally tangent to two adjacent circles centered at the vertices. This circle $S$ is allowed to roll freely in the interior of the polygon as long as it remains externally tangent to the vertex circles. As it rolls, $S$ turns the color of any point it touches into black. After it rolls completely around the interior of the polygon, the total length of the black lengths can be expressed in the form $\tfrac{p\pi}{q}$ for positive integers $p, q$ satisfying $\gcd(p, q) = 1$. What is $p + q$?

Let $a,b,c$ be real number greater than $1$. Let \[S=\log_a {bc}+\log_b {ca}+\log_c {ab}\] Find the minimum possible value of $S$.

In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.