The incircle of triangle $ABC$ with $AB\neq AC$ has center $I$ and is tangent to $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. The circumcircle of triangle $ADI$ intersects $AB$ and $AC$ again at $X$ and $Y.$ Prove that $EF$ bisects $XY.$

Given a positive integer number $k$, define the function $f$ on the set of all positive integer numbers to itself by \[f(n)=\begin{cases}1, &\text{if }n\le k+1\\ f(f(n-1))+f(n-f(n-1)), &\text{if }n>k+1\end{cases}\] Show that the preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integers.

Let $M=\{x^2+x \mid x\in \mathbb N^{\star} \}$. Prove that for every integer $k\geq 2$ there exist elements $a_{1}, a_{2}, \ldots, a_{k},b_{k}$ from $M$, such that $a_{1}+a_{2}+\cdots+a_{k}=b_{k}$.

Faces of a cube $9\times 9\times 9$ are partitioned onto unit squares. The surface of a cube is pasted over by $243$ strips $2\times 1$ without overlapping. Prove that the number of bent strips is odd. [i]A. Poliansky[/i]

A group of aliens from Gliese $667$ Cc come to Earth to test the hypothesis that mathematics is indeed a universal language. To do this, they give you the following information about their mathematical system: $\bullet$ For the purposes of this experiment, the Gliesians have decided to write their equations in the same syntactic format as in Western math. For example, in Western math, the expression “$5+4$” is interpreted as running the “$+$” operation on numbers $5$ and $4$. Similarly, in Gliesian math, the expression $\alpha \gamma \beta$ is interpreted as running the “$\gamma $” operation on numbers $\alpha$ and $ \beta$. $\bullet$ You know that $\gamma $ and $\eta$ are the symbols for addition and multiplication (which works the same in Gliesian math as in Western math), but you don’t know which is which. By some bizarre coincidence, the symbol for equality is the same in Gliesian math as it is in Western math; equality is denoted with an “$=$” symbol between the two equal values. $\bullet$ Two symbols that look exactly the same have the same meaning. Two symbols that are different have different meanings and, therefore, are not equal. They then provide you with the following equations, written in Gliesian, which are known to be true: [img]https://cdn.artofproblemsolving.com/attachments/b/e/e2e44c257830ce8eee7c05535046c17ae3b7e6.png[/img]

The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours. [i]Proposed by S. Berlov, S. Ivanov[/i]

Let $ g(n,k)$ denote the number of strongly connected, $ \textit{simple}$ directed graphs with $ n$ vertices and $ k$ edges. ($ \textit{Simple}$ means no loops or multiple edges.) Show that \[ \sum_{k=n}^{n^2-n}(-1)^kg(n,k)=(n-1)!.\] [i]A. A. Schrijver[/i]

Let $ABCD$ and $A_1B_1C_1D_1$ be convex quadrangles in a plane, such that $AB=A_1B_1$, $BC=B_1C_1$, $CD=C_1D_1$ and $DA=D_1A_1$. Given that diagonals $AC$ and $BD$ are perpendicular to each other, prove that the same holds for diagonals $A_1C_1$ and $B_1D_1$.

In a dance class there are $10$ boys and $10$ girls. It is known that for each $1\le k\le 10$ and for each group of $k$ boys, the number of girls who are friends with at least one boy in the group is not less than $k$. Prove that it is possible to pair up the boys and the girls for a dance so that each pair consists of a boy and a girl who are friends.

Each vertex of a regular $11$-gon is colored black or gold. All possible triangles are formed using these vertices. Prove that there are either two congruent triangles with three black vertices or two congruent triangles with three gold vertices.

Meena owns a bottle cap collection. While on a vacation, she finds a large number of bottle caps, increasing her collection size by $40\%$. Later on her same vacation, she decides that she does not like some of the bottle caps, so she gives away $20\%$ of her current collection. Suppose that Meena owns $21$ more bottle caps after her vacation than before her vacation. How many bottle caps did Meena have before her vacation?

Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

There are two values of $ a$ for which the equation $ 4x^2 \plus{} ax \plus{} 8x \plus{} 9 \equal{} 0$ has only one solution for $ x$. What is the sum of these values of $ a$? $ \textbf{(A)}\ \minus{}16\qquad \textbf{(B)}\ \minus{}8\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 20$

A tournament has $5$ players and is in round-robin format (each player plays each other exactly once). Each game has a $\frac13$ chance of player $A$ winning, a $\frac13$ chance of player $B$ winning, and a$ \frac13$ chance of ending in a draw. The probability that at least one player draws all of their games can be written in simplest form as $\frac{m}{3^n}$ where $m, n$ are positive integers. Find $m + n$.

Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.

Let $AB$ be the diameter of circle $\Gamma$ centred at $O$. Point $C$ lies on ray $\overrightarrow{AB}$. The line through $C$ cuts circle $\Gamma$ at $D$ and $E$, with point $D$ being closer to $C$ than $E$ is. $OF$ is the diameter of the circumcircle of triangle $BOD$. Next, construct $CF$, cutting the circumcircle of triangle $BOD$ at $G$. Prove that $O,A,E,G$ are concyclic. (Possibly proposed by Pak Wono)

Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality? [i](Walther Janous)[/i]

A positive integer $M$ has been represented as a product of primes. Each of these primes is increased by 1 . The product $N$ of the new multipliers is divisible by $M$ . Prove that if we represent $N$ as a product of primes and increase each of them by 1 then the product of the new multipliers will be divisible by $N$ . Alexandr Gribalko

Consider acute $\triangle ABC$ with altitudes $AA_1, BB_1$ and $CC_1$ ($A_1 \in BC,B_1 \in AC,C_1 \in AB$). A point $C' $ on the extension of $B_1A_1$ beyond $A_1$ is such that $A_1C' = B_1C_1$. Analogously, a point $B'$ on the extension of A$_1C_1$ beyond $C_1$ is such that $C_1B' = A_1B_1$ and a point $A' $ on the extension of $C_1B_1$ beyond $B_1$ is such that $B_1A' = C_1A_1$. Denote by $A'', B'', C''$ the symmetric points of $A' , B' , C'$ with respect to $BC, CA$ and $AB$ respectively. Prove that if $R, R'$ and R'' are circumradiii of $\triangle ABC, \triangle A'B'C'$ and $\triangle A''B''C''$, then $R, R'$ and $R'' $ are sidelengths of a triangle with area equals one half of the area of $\triangle ABC$.

Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$, $p$ and $q$, there exist not necessarily distinct integers $a \neq 0$, $b$, $c$ belonging to $S$, such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$. Determine the maximum number of elements that $S$ can have.

Chelsea goes to La Verde's at MIT and buys $100$ coconuts, each weighing $4$ pounds, and $100$ honeydews, each weighing $5$ pounds. She wants to distribute them among $n$ bags, so that each bag contains at most $13$ pounds of fruit. What is the minimum $n$ for which this is possible?

Emily has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any previous boxes. How many balls in total are in the boxes as a result of Emily's $2017$th step? $\text{(A) }9\qquad\text{(B) }11\qquad\text{(C) }13\qquad\text{(D) }15\qquad\text{(E) }17$

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$. let $X$ is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$. prove that $\widehat{BAD}=\widehat{CAX}$

Suppose $ \vartriangle ABC$ is similar to $\vartriangle DEF$, with $ A$, $ B$, and $C$ corresponding to $D, E$, and $F$ respectively. If $\overline{AB} = \overline{EF}$, $\overline{BC} = \overline{FD}$, and $\overline{CA} = \overline{DE} = 2$, determine the area of $ \vartriangle ABC$.

How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$) $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$