From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$. (I Sharygin)

On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$

Finland is going to change the monetary system again and replace the Euro by the Finnish Mark. The Mark is divided into $100$ pennies. There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able to pay for any purchase less than one mark should be minimal. Determine the coin denominations.

In the $200\times 200$ table in some cells lays red or blue chip. Every chip "see" other chip, if they lay in same row or column. Every chip "see" exactly $5$ chips of other color. Find maximum number of chips in the table.

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

Let $T=\{a_1,a_2,\dots,a_{1000}\}$, where $a_1<a_2<\dots<a_{1000}$, be a uniformly randomly selected subset of $\{1,2,\dots,2018\}$ with cardinality $1000$. The expected value of $a_7$ can be written in reduced form as $\tfrac{m}{n}$. Find $m+n$.

In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$, what is $ CD/BD$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); pair A = (0,0); pair C = (2,0); pair B = dir(57.5)*2; pair E = waypoint(C--A,0.25); pair D = waypoint(C--B,0.25); pair T = intersectionpoint(D--A,E--B); label("$B$",B,NW);label("$A$",A,SW);label("$C$",C,SE);label("$D$",D,NE);label("$E$",E,S);label("$T$",T,2*W+N); draw(A--B--C--cycle); draw(A--D); draw(B--E);[/asy]$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {2}{9}\qquad \textbf{(C)}\ \frac {3}{10}\qquad \textbf{(D)}\ \frac {4}{11}\qquad \textbf{(E)}\ \frac {5}{12}$

There are 20 towns on the bay of a circular island. Each town has 20 teams for a mathematical duel. No two of these teams are of equal strength. When two teams meet in a duel, the stronger one wins. For a given number $n\in \mathbb{N}$ one town $A$ can be called [i]“n-stronger”[/i] than $B$, if there exist $n$ different duels between a team from $A$ and team from $B$, for which the team from $A$ wins. Find the maximum value of $n$, for which it is possible for each town to be [i]n-stronger[/i] by its neighboring one clockwise.

Four mathematicians, four physicists, and four programmers gather in a classroom. The $12$ people organize themselves into four teams, with each team having one mathematician, one physicist, and one programmer. How many possible arrangements of teams can exist?

Let $b$ and $c$ be any two positive integers. Define an integer sequence $a_n$, for $n\geq 1$, by $a_1=1$, $a_2=1$, $a_3=b$ and $a_{n+3}=ba_{n+2}a_{n+1}+ca_n$. Find all positive integers $r$ for which there exists a positive integer $n$ such that the number $a_n$ is divisible by $r$.

Let $x_1, x_2, \ldots ,x_n(n\ge 2)$ be real numbers greater than $1$. Suppose that $|x_i-x_{i+1}|<1$ for $i=1, 2,\ldots ,n-1$. Prove that \[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1\]

A certain value of $x$ satisfies $1 + x + 5 - 1 = 7$. What is this value of $x$? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{impossible to determine}$

Compute the second smallest positive whole number that has exactly $6$ positive whole number divisors (including itself).

Prove that the sum of digits of the number $1972^n$ is not bounded from above when $n$ tends to infinity.

Each vertex of a regular $1997$-gon is labeled with an integer, so that the sum of the integers is $1$. We write down the sums of the first $k$ integers read counterclockwise, starting from some vertex $(k=1,2,\ldots,1997)$. Can we always choose the starting vertex so that all these sums are positive? If yes, how many possible choices are there?

A regular octagon $ ABCDEFGH$ has sides of length two. Find the area of $ \triangle{ADG}$. $ \textbf{(A)}\ 4 \plus{} 2 \sqrt{2} \qquad \textbf{(B)}\ 6 \plus{} \sqrt{2} \qquad \textbf{(C)}\ 4 \plus{} 3 \sqrt{2} \qquad \textbf{(D)}\ 3 \plus{} 4 \sqrt{2} \qquad \textbf{(E)}\ 8 \plus{} \sqrt{2}$

We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are $2022$ blue monochromatic triangles. How many red monochromatic triangles are there?

In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$? [asy] defaultpen(linewidth(0.7)); size(120); pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC; draw(A--B--C--cycle); for(int i = 1; i < 4; ++i) { AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4); draw(AB[i-1] -- AC[i-1]); } filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7)); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S);[/asy]

For each $n\in\mathbb N$, determine the sign of $n^6+5n^5\sin n+1$. For which $n\in\mathbb N$ does it hold that $\frac{n^2+5n\cos n+1}{n^6+5n^5\sin n+1}\ge10^{-4}$.

Let $ a$, $ b$, $ c$, and $ d$ be real numbers with $ |a\minus{}b|\equal{}2$, $ |b\minus{}c|\equal{}3$, and $ |c\minus{}d|\equal{}4$. What is the sum of all possible values of $ |a\minus{}d|$? $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Kan Krao Park is a circular park that has $21$ entrances and a straight line walkway joining each pair of two entrances. No three walkways meet at a single point. Some walkways are paved with bricks, while others are paved with asphalt. At each intersection of two walkways, excluding the entrances, is planted lotus if the two walkways are paved with the same material, and is planted waterlily if the two walkways are paved with different materials. Each walkway is decorated with lights if and only if the same type of plant is placed at least $45$ different points along that walkway. Prove that there are at least $11$ walkways decorated with lights and paved with the same material.

Given a polynomial $P(x)$ with real coefficents satisfying:$$P(x).P(x+1)=P(x^2+x+1),\forall x\in \mathbb{R}.$$ Prove that: $\deg(P)$ is an even number and find $P(x).$

In a football tournament there are $8$ teams, each of which plays exacly one match against every other team. If a team $A$ defeats team $B$, then $A$ is awarded $3$ points and $B$ gets $0$ points. If they end up in a tie, they receive $1$ point each. It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.

Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.

No matter how two copies of a convex polygon are placed inside a square, they always have a common point. Prove that no matter how three copies of the same polygon are placed inside this square, they also have a common point.