There are two types of items in Alik's collection: badges and bracelets and there are more badges than bracelets. Alik noticed that if he increases the number of bracelets some (not necessarily integer) number of times without changing the number of icons, then in its collection will be $100$ items. And if, on the contrary, he increases the initial number of badges by the same number of times, leaving the same number of bracelets, then he will have $101$ items. How many badges and how many bracelets could there be in Alik's collection?

$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.

Prove that for any convex quadrilateral it is always possible to cut out three smaller quadrilaterals similar to the original one with the scale factor equal to 1/2. (The angles of a smaller quadrilateral are equal to the corresponding original angles and the sides are twice smaller then the corresponding sides of the original quadrilateral.)

Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|$.

Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.

Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively. Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively. Prove, that $I, O, H$ lies on one line.

Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[xy+x+y = 71\qquad\text{and}\qquad x^2y+xy^2 = 880.\]

Gus is an inhabitant on an $11$ by $11$ grid of squares. He can walk from one square to an adjacent square (vertically or horizontally) in $1$ unit of time. There are also two vents on the grid, one at the top left and one at the bottom right. If Gus is at one vent, he can teleport to the other vent in $0.5$ units of time. Let an ordered pair of squares $(a,b)$ on the grid be [i]sus [/i] if the fastest path from $a$ to $b$ requires Gus to teleport between vents. Walking on top of a vent does not count as teleporting between vents. What is the total number of ordered pairs of squares that are [i]sus[/i]? Note that the pairs $(a_1,b_1)$ and $(a_2,b_2)$ are considered distinct if and only if $a_1 \ne a_2$ or $b_1 \ne b_2$.

Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac{1}{p}, $ where $p$ is a real parameter. Investigate for which values of $p$ solutions exist and how many solutions exist. (Of course, the last question ''how many solutions exist'' should be understood as ''how many solutions exists modulo $2\pi $''.)

There are $k$ heaps on the table, each containing a different positive number of stones. Juri and Mari make moves alternatingly, Juri starts. On each move, the player making the move has to pick a heap and remove one or more stones in it from the table; in addition, the player is allowed to distribute any number of remaining stones from that heap in any way between other non-empty heaps. The player to remove the last stone from the table wins. For which positive integers $k$ does Juri have a winning strategy for any initial state that satisfies the conditions?

On a circle there are some black and white points (there are at least $12$ points). Each point has $10$ neighbors ($5$ left and $5$ right neighboring points), $5$ being black and $5$ white. Prove that the number of points on the circle is divisible by $4$.

Decompose the expression into real factors: \[E = 1 - \sin^5(x) - \cos^5(x).\]

Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i,-2+i$ and $-1-2i$. The fourth number is A. $2+i$ B. $2-i$ C. $1-2i$ D. $-1+2i$ E. $-2-i$

Find the number of positive integral solutions to the equation $\sum \limits_{i=1}^{2019} 10^{a_i}=\sum \limits_{i=1}^{2019} 10^{b_i}$, such that $a_1<a_2<\cdots <a_{2019}$ , $b_1<b_2<\cdots <b_{2019}$ and $a_{2019} < b_{2019}$ [list=1] [*] 1 [*] 2 [*] 2019 [*] None of the above [/list]

Let $x_1=1$, $x_2$, $x_3$, $\ldots$ be a sequence of real numbers such that for all $n\geq 1$ we have \[ x_{n+1} = x_n + \frac 1{2x_n} . \] Prove that \[ \lfloor 25 x_{625} \rfloor = 625 . \]

In triangle $ABC$, with $AB<AC$, $I$ is the incenter, $E$ is the intersection of $A$-excircle and $BC$. Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$. Point $Q$ lies on $BC$ such that $\angle AIQ=90$. Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$, circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$. if $M$ is the Midpoint od the arc $BC$, which does not contain $A$, prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$. Proposed by Amirmahdi Mohseni

In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$- and $y$- axes, and call a set of points $nice$ if no two of them have the same $x$- or $y$- coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$. [i]Yannick Yao[/i]

Given is a square $ABCD$ with $E \in BC$, arbitrarily. On $CD$ lies the point $F$ is such that $\angle EAF = 45^o$. Prove that $EF$ is tangent to the circle with center $A$ and radius $AB$.

Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$.

Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

How many four-digit multiples of $8$ are greater than $2008$?

We have \( n \) chips that are initially placed on the number line at position 0. On each move, we select a position \( x \in \mathbb{Z} \) where there are at least two chips; we take two of these chips, then place one at \( x-1 \) and the other at \( x+1 \). a) Prove that after a finite number of moves, regardless of how the moves are chosen, we will reach a final position where no two chips occupy the same number on the number line. b) For every possible final position, let \( \Delta \) represent the difference between the numbers where the rightmost and the leftmost chips are located. Find all possible values of \( \Delta \) in terms of \( n \).

In triangle $ABC$ the circumcircle has radius $R$ and center $O$ and the incircle has radius $r$ and center $I\ne O$ . Let $G$ denote the centroid of triangle $ABC$. Prove that $IG \perp BC$ if and only if $AB = AC$ or $AB + AC = 3BC$.

Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function, and suppose that the sequence $f^{(n)}$ of derivatives converges pointwise. Prove that $f^{(n)}(z)\to Ce^z$ pointwise for a suitable complex number $C$.

How many sets of two positive prime numbers $\{p,q\}$ have the property that $p + q = 100$?