The Quadrumland map is a 6 × 6 square where each square cell is either a kingdom or a disputed territory. There are 27 kingdoms and 9 disputed territories. Each disputed territory is claimed by those and only those kingdoms that are neighbouring with it (adjacent by an edge or a vertex). Is it possible that for each disputed territory the numbers of claims are different? You can discuss your solutions here

If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}} $$

There is a row of $n$ chairs, numbered in order from left to right from $1$ to $n$. Additionally, the $n$ numbers from $1$ to $n$ are distributed on the backs of the chairs, one number per chair, such that the number on the back of a chair never matches the number of the chair itself. There is a child sitting on each chair. Every time the teacher claps, each child checks the number on the back of the chair they are sitting on and moves to the chair corresponding to that number. Prove that for any $m$ that is not a power of a prime, with $1 < m \leqslant n$, it is possible to distribute the numbers on the backrests such that, after the teacher claps $m$ times, for the first time, all the children are sitting in the chairs where they initially started. (During the process, it may happen that some children return to their original chairs, but they do not all do so simultaneously until the $m^{\text{th}}$ clap.)

Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.

$1600$ bananas are distributed among $100$ monkeys (it is possible that some monkeys do not receive bananas). Prvove that at least four monkeys receive the same amount of bananas.

Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa. [url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]

Let $F$ be the set of all $n$-tuples $(A_1, \ldots, A_n)$ such that each $A_{i}$ is a subset of $\{1, 2, \ldots, 1998\}$. Let $|A|$ denote the number of elements of the set $A$. Find \[ \sum_{(A_1, \ldots, A_n)\in F} |A_1\cup A_2\cup \cdots \cup A_n| \]

A game consists in pushing a flat stone along a sequence of squares $S_0, S_1, S_2, . . .$ that are arranged in linear order. The stone is initially placed on square $S_0$. When the stone stops on a square $S_k$ it is pushed again in the same direction and so on until it reaches $S_{1987}$ or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly $n$ squares is $\frac{1}{2^n}$. Determine the probability that the stone will stop exactly on square $S_{1987}.$

Let $H$ be the orthocenter of a triangle $ABC$, and $M$, $N$ be the midpoints of segments $BC$, $AH$ respectively. The perpendicular from $N$ to $MH$ meets $BC$ at point $A^{\prime}$. Points $B^{\prime}$ and $C^{\prime}$ are defined similarly. Prove that $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ are collinear. Proposed by: F.Ivlev

Let $n \in N, n\ge 2$, and $a_1, a_2, ..., a_n, b_1, b_2, ..., b_n$ be real positive numbers such that $$\frac{a_1}{b_1} \le \frac{a_2}{b_2} \le ... \le\frac{a_n}{b_n}.$$ Find the largest real $c$ so that $$(a_1-b_1c)x_1+(a_2-b_2c)x_2+...+(a_n-b_nc)x_n \ge 0,$$ for every $x_1, x_2,..., x_n > 0$, with $x_1\le x_2\le ...\le x_n$.

Let $ABCD$ be a parallelogram with $AB>BC$ and $\angle DAB$ less than $\angle ABC$. The perpendicular bisectors of sides $AB$ and $BC$ intersect at the point $M$ lying on the extension of $AD$. If $\angle MCD=15^{\circ}$, find the measure of $\angle ABC$

Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and $$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$ Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$

Compute $$\left(20+\frac{1}{23}\right)\cdot\left(23+\frac{1}{20}\right)-\left(20-\frac{1}{23}\right)\cdot\left(23-\frac{1}{20}\right)$$ [i]Proposed by Andy Xu[/i]

Three vertices of parallelogram $PQRS$ are $P(-3,-2)$, $Q(1,-5)$, $R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is: $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 9$

Knights, who always tell truth, and liars, who always lie, live on an island. They have been distributed into two teams $A$ and $B$ for a game of tennis, and team $A$ had more members than team $B$. Two players from different teams started the game, whenever a player loses the game, he leaves it forever and he is replaces by a member of his team (that has never played before). The team, all of whose members left the game, loses. After the tournament, every member of team $A$ was asked: "Is it true that you have lost to a liar in some game?", and every member of team $B$ was asked: "Is it true that you have won at least two games, in which your opponent was a knight?". It turns out that every single answer was positive. Which team won?

Let $S$ be a fixed point on a sphere $\Sigma$ with center $\Omega$. Consider all tetrahedra $SABC$ inscribed in $\Sigma$ such that $SA,SB,SC$ are pairwise orthogonal. (a) Prove that all the planes $ABC$ pass through a single point. (b) In one such tetrahedron, $H$ and $O$ are the orthogonal projections of $S$ and $\Omega$ onto the plane $ABC$, respectively. Let $R$ denote the circumradius of $\triangle ABC$. Prove that $R^2=OH^2+2SH^2$.

Vova has a square grid $72\times 72$. Unfortunately, $n$ cells are stained with coffee. Determine if Vova always can cut out a clean square $3\times 3$ without its central cell, if a) $n=699$; b) $n=750$.

Each vertex of a right $n$-gon $(n\geq 3)$ is colored in yellow, blue or red. On each turn are chosen two adjacent vertices in different color and then are recolored in the third. For which $n$ can we get from an arbitrary coloring of the $n$-gon a monochromatic one (in one color)?

We say that a number $n \ge 2$ has the property $(P)$ if, in its prime factorization, at least one of the factors has an exponent $3$. a) Determine the smallest number $N$ with the property that, no matter how we choose $N$ consecutive natural numbers, at least one of them has the property $(P).$ b) Determine the smallest $15$ consecutive numbers $a_1, a_2, \ldots, a_{15}$ that do not have the property $(P),$ such that the sum of the numbers $5 a_1, 5 a_2, \ldots, 5 a_{15}$ is a number with the property $(P).$

For a positive integer $n$, let $A_n$ and $B_n$ be the families of $n$-element subsets of $S_n=\{1,2,\ldots ,2n\}$ with respectively even and odd sums of elements. Compute $|A_n|-|B_n|$.

Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that: \[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]

Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.

Joe and Kathryn work part-time jobs at the local mall to make some money for college. Joe works at GameStop, while Kathryn works at Bath and Body Works. However, neither of them usually ever leaves on pay day without spending a healthy portion of their check at their own store, especially angering Joe’s parents, who think video games are for Neanderthals or children under $8$. Joe makes $\$8$ an hour, while Kathryn makes $\$10$ an hour. Both work $20$ hours a week. Every week, Joe has a $20\%$ probability of purchasing a used $\$25$ video game, and Kathryn has a $25\%$ probability of purchasing a $\$30$ skin moisturizer. Find the expected value, in dollars, of their combined weekly “take-home pay.” (Take-home pay is total pay minus in-store spending.)

Let $x=\frac pq$ for $p$, $q$ coprime. Find $p+q$. [asy] import olympiad; size(200); pen qq=font("phvb"); defaultpen(linewidth(0.6)+fontsize(10pt)); pair A=(-2.25,7),B=(-5,0),C=(5,0),D=waypoint(A--B,3/7), E=waypoint(A--C,1/2),F=intersectionpoint(C--D, B--E); draw(A--B--C--cycle^^B--E^^C--D); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,NE); label("$F$",F,N); label(scale(2.5)*"X",centroid(A,D,E),qq); label(scale(2.5)*"3",centroid(B,D,F),0.5*N,qq); label(scale(2.5)*"6",centroid(B,F,C),0.25*dir(180),qq); label(scale(2.5)*"2",centroid(C,E,F),dir(140),qq); [/asy]

Does there exist a natural number $ n$ such that $ n>2$ and the sum of squares of some $ n$ consecutive integers is a perfect square?