In a warehouse there are many empty cans of $4$ colors: red, green, Blue and yellow. Some boys play to build towers in which no two cans of the same color, with a can in each floor and at any height. How many different towers can be built?

Let $T$ a set with 2007 points on the plane, without any 3 collinear points. Let $P$ any point which belongs to $T$. Prove that the number of triangles that contains the point $P$ inside and its vertices are from $T$, is even.

There are $n$ points on a circle ($n > 1$). Denote them with $P_1,P_2, P_3, ..., P_n$ such that the polyline $P_1P_2P_3... P_n$ does not intersect itself. In how many ways is this possible?

Let $U$ be a set of $m$ triangles. Prove that there exists a subset $W$ of $U$ which satisfies the following. (i). The number of triangles in $W$ is at least $0.45m^{\frac{4}{5}}$ (ii) There are no points $A, B, C, D, E, F$ such that triangles $ABC$, $BCD$, $CDE$, $DEF$, $EFA$, $FAB$ are all in $W$.

[b]p1.[/b] Suppose $A$, $B$ and $C$ are the angles of a triangle. Prove that $$1 - 8 \cos A\cos B \cos C = sin^2(B - C) + (cos(B - C) - 2 cosA)^2.$$ [b]p2.[/b] Let $x_1, x_2,..., x_{100}$ be integers whose values are either $0$ or $1$. (a) Show that $$x_1 + x_2 + ... + x_{100} - (x_1x_2 + x_2x_3 + ... + x_{99}x_{100} + x_{100}x_1)\le 50.$$ (b) Give specific values for $x_1, x_2,..., x_{100}$ that give equality. [b]p3.[/b] Let $ABCD$ be a trapezoid whose area is $32$ square meters. Suppose the lengths of the parallel segments $AB$ and $DC$ are $2$ meters and $6$ meters, respectively, and $P$ is the intersection of the diagonals $AC$ and $BD$. If a line through $P$ intersects $AD$ and $BC$ at $E$ and $F$, respectively, determine, with a proof, the minimum possible area for quadrilateral $ABFE$. [b]p4.[/b] Let $n$ be a positive integer and $x$ be a real number. Show that $$\lfloor nx \rfloor = \lfloor x \rfloor +\left\lfloor x + \frac{1}{n} \right\rfloor + \left\lfloor x + \frac{2}{n} \right\rfloor + ... + \left\lfloor x + \frac{n - 1}{n} \right\rfloor$$ where $\lfloor a \rfloor$ is the greatest integer less than or equal to $a$. (For example, $\lfloor 4.5\rfloor = 4$ and $\lfloor - 4.5 \rfloor = -5$.) [b]p5.[/b] A $3n$-digit positive integer (in base $10$) containing no zero is said to be [i]quad-perfect[/i] if the number is a perfect square and each of the three numbers obtained by viewing the first $n$ digits, the middle $n$ digits and the last $n$ digits as three $n$-digit numbers is in itself a perfect square. (For example, when $n = 1$, the only quad-perfect numbers are $144$ and $441$.) Find all $9$-digit quad-perfect numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer, and let $[n]=\{1,2,\dots,n\}$. Find the maximum posible cardinality of a subset $S$ of $[n]$ with the property that there aren't any distinct $a,b,c\in S$ such that $a+b=c$.

The number of maps $f$ from $1,2,3$ into the set $1,2,3,4,5$ such that $f(i)\le f(j)$ whenever $i\le j$ is $\textbf{(A)}~60$ $\textbf{(B)}~50$ $\textbf{(C)}~35$ $\textbf{(D)}~30$

We have $n$ countries. Each country have $m$ persons who live in that country ($n>m>1$). We divide $m \cdot n$ persons into $n$ groups each with $m$ members such that there don't exist two persons in any groups who come from one country. Prove that one can choose $n$ people into one class such that they come from different groups and different countries.

Let $a$ be a permutation on $\{0,1,\ldots ,2015\}$ and $b,c$ are also permutations on $\{1,2,\ldots ,2015\}$. For all $x\in \{1,2,\ldots ,2015\}$, the following conditions are satisfied: (i) $a(x)-a(x-1)\neq 1$,\\ (ii) if $b(x)\neq x$, then $c(x)=x$,\\ Prove that the number of $a$'s is equal to the number of ordered pairs of $(b,c)$.

Beto has rectangular chessboard where the number of rows and columns are consecutive numbers (for example, $30$ and $31$). Ana has tiles of two colors and different sizes: the red tiles are $5 \times 7$ rectangles and the blue tiles are $3 \times 5$ rectangles. Ana realized that she can cover all the squares of Beto’s board using only red tiles, which can be rotated, but must not overlap or extend beyond the board. Then, she realized she can also do the same using only blue tiles. What is the minimum number of squares that Beto’s board can have?

There are $2022$ students in winter school. Two arbitrary students are friend or enemy each other. Each turn, we choose a student $S$, make friends of $S$ enemies, and make enemies of $S$ friends. This continues until it satisfies the final condition. [b]Final Condition[/b] : For any partition of students into two non-empty groups $A$, $B$, there exist two students $a$, $b$ such that $a\in A$, $b\in B$, and $a$, $b$ are friend each other. Determine the minimum value of $n$ such that regardless of the initial condition, we can satisfy the final condition with no more than $n$ turns.

An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order. [asy] size(3cm); pair A=(0,0),D=(1,0),B,C,E,F,G,H,I; G=rotate(60,A)*D; B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A; draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]

Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.) For which $n$ is it possible for Mark to achieve this? [i]Proposed by Michael Ren[/i]

A square is placed in the plane and a point $P$ is marked in this plane with invisible ink. A certain person can see this point through special glasses. One can draw a straight line and this person will say on which side of the line the point $P$ lies. If $P$ lies on the line, the person says so. What is the minimal number of questions one needs to find out if $P$ lies inside the square or not? (Folklore)

A baseball league has $6$ teams. To decide the schedule for the league, for each pair of teams, a coin is flipped. If it lands head, they will play a game this season, in which one team wins and one team loses. If it lands tails, they don't play a game that season. Define the [i]imbalance[/i] of this schedule to be the minimum number of teams that will end up undefeated, i.e. lose $0$ games. Find the expected value of the imbalance in this league.

[b]p1.[/b] Due to the crisis, the salaries of the company's employees decreased by $1/5$. By what percentage should it be increased in order for it to reach its previous value? [b]p2.[/b] Can the sum of six different positive numbers equal their product? [b]p3.[/b] Points$ A, B, C$ and $B$ are marked on the straight line. It is known that $AC = a$ and $BP = b$. What is the distance between the midpoints of segments $AB$ and $CB$? List all possibilities. [b]p4.[/b] Find the last three digits of $625^{19} + 376^{99}$. [b]p5.[/b] Citizens of five different countries sit at the round table (there may be several representatives from one country). It is known that for any two countries (out of the given five) there will be citizens of these countries sitting next to each other. What is the smallest number of people that can sit at the table? [b]p6.[/b] Can any rectangle be cut into $1999$ pieces, from which you can form a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

In the country of Óxisz the minister of finance observed at the end of the tax census that the sum of properties of any two neighboring city counted in dinar is divisible by $1000$, and she also observed that the sum of properties of all cities is also divisible by $1000$. What is the least sum of properties of all cities if the map of the cities looks as follows? [img]https://cdn.artofproblemsolving.com/attachments/0/5/274730ebfdd52d0c3642dfbd0596fe587eb211.png[/img] [i]Remark: The cities may have non-integer properties, but it is also positive. On the map the points are the cities, and two cities are neighboring if there is a direct connection between them.[/i]

Whether there are integers $a_1$, $a_2$, $\cdots$, that are different from each other, satisfying: (1) For $\forall k\in\mathbb N_+$, $a_{k^2}>0$ and $a_{k^2+k}<0$; (2) For $\forall n\in\mathbb N_+$, $\left| a_{n+1}-a_n\right|\leqslant 2023\sqrt n$?

There are $11$ points equally spaced on a circle. Some of the segments having endpoints among these vertices are drawn and colored in two colors, so that each segment meets at an internal for it point at most one other segment from the same color. What is the greatest number of segments that could be drawn?

In the simple and connected graph $G$ let $x_i$ be the number of vertices with degree $i$. Let $d>3$ be the biggest degree in the graph $G$. Prove that if : $$x_d \ge x_{d-1} + 2x_{d-2}+... +(d-1)x_1$$ Then there exists a vertex with degree $d$ such that after removing that vertex the graph $G$ is still connected. Proposed by [i]Ali Mirzaie[/i]

Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction). Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.