4. A triangle $A B C$ with angle $A$ equal to $60^{\circ}$ is given. Its incircle is tangent to side $A B$ at point $D$, while its excircle tangent to side $A C$, is tangent to the extension of side $A B$ at point $E$. Prove that the perpendicular to side $A C$, passing through point $D$, meets the incircle again at a point equidistant from points $E$ and $C$. (The excircle is the circle tangent to one side of the triangle and to the extensions of two other sides.) Azamat Mardanov

In five years, Tom will be twice as old as Cindy. Thirteen years ago, Tom was three times as old as Cindy. How many years ago was Tom four times as old as Cindy?

Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.

Show that the polynomial over variables $x,y,z$ \[ x^4(x-y)(x-z) + y^4(y-z)(y-x) + z^4(z-x)(z-y) \] can't be written as a finite sum of squares of real polynomials over $x,y,z$.

In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$? [asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label("$A$",A,SW); label("$B$", B, NW); label("$C$",C,NE); label("$D$",D,SE); label("$P$",P,S); [/asy] $\textbf{(A) }3\sqrt 5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }6\sqrt 5 \qquad \textbf{(D) }20\qquad \textbf{(E) }25$

Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.

Each of $100$ stones has a sticker showing its true weight. No two stones weight the same. Mischievous Greg wants to rearrange stickers so that the sum of the numbers on the stickers for any group containing from $1$ to $99$ stones is different from the true weight of this group. Is it always possible?

The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\circ}$ and $ SB \equal{} SC \equal{} 1$. Find the volume of the tetrahedron.

The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$

Let $A\in M_n(\mathbb{C}) $ and $a\in \mathbb{C} $ such that $A-A^*=2aI_n $, where $A^*=(\overline{A})^T $ and $I_n$ is identity matrix. (i) Show that $|\det A|\ge |a|^n $. (ii) Show that if $|\det A|=|a|^n $ then $A=aI_n$.

Find the least positive integer with the property that if its digits are reversed and then $450$ is added to this reversal, the sum is the original number. For example, $621$ is not the answer because it is not true that $621 = 126 + 450$.

Big Chungus has been thinking of a new symbol for BMT, and the drawing below is what he came up with. If each of the $16$ small squares in the grid are unit squares, what is the area of the shaded region?

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is [i]not[/i] marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane. (i) Can Evan construct* the reflection of $A$ over $\ell$? (ii) Can Evan construct the foot of the altitude from $A$ to $\ell$? *To construct a point, Evan must have an algorithm which marks the point in finitely many steps. [i]Proposed by Zack Chroman[/i]

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Denote $\Delta$ the tangent at $A$ to the circle $\Gamma$. $\Gamma_1$ is a circle tangent to the lines $\Delta$, $(AB)$ and $(BC)$, and $E$ its touchpoint with the line $(AB)$. Let $\Gamma_2$ be a circle tangent to the lines $\Delta$, $(AC)$ and $(BC)$, and $F$ its touchpoint with the line $(AC)$. We suppose that $E$ and $F$ belong respectively to the segments $[AB]$ and $[AC]$, and that the two circles $\Gamma_1$ and $\Gamma_2$ lie outside triangle $ABC$. Show that the lines $(BC)$ and $(EF)$ are parallel.

Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f (999)$ ?

In the reality show [i]Big Sister Brasil[/i], it is said that there is a [i]treta[/i] if two people are friends with each other and enemies with a third one. For audience purposes, the broadcaster wants a lot of [i]tretas[/i]. If friendship and enmity are reciprocal relationships, given $n$ people, what is the maximum number of [i]tretas[/i]?

Find all ordered triples $(a,b,c)$ of real numbers such that \[\begin{cases} a+b=c,\\ a^2+b^2=c^2-c-6,\\ a^3+b^3 = c^3-2c^2-5c. \\ \end{cases}\] [i]Proposed by Evan Fang

Let $x{}$ and $y{}$ be natural numbers greater than 1. It turns out that $x^2+y^2-1$ is divisible by $x+y-1$. Prove that $x+y-1$ is composite. [i]From the folklore[/i]

There are $n\ge 3$ particles on a circle situated at the vertices of a regular $n$-gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before. Let $s$ be the smallest number of collisions after which all particles return to their original positions. Find $s$. [i]Proposed by N.V. Tejaswi[/i]

What is the number of ways in which one can color the squares of a $4\times 4$ chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares?

Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i\geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?

Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$. How many times does the digit $1$ occur in its decimal representation?

Given five points inside an equilateral triangle of side length $2$, show that there are two points whose distance from each other is at most $ 1$.