Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle (i) whose vertices are all the same colour, and (ii) at least one side of the triangle does not contain a point of the opposite colour.

On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. [i]Proposed by Tonći Kokan, Croatia[/i]

On every side of a square $ABCD$, we consider three points different (to each other). a) Find the number of line segments defined with endpoints those points , that do not lie on sides of the square. b) If there are no three of the previous line segments passing through the same point, find how many of the intersection points of those segmens line in the interior of the square.

[b]p1.[/b] Solve the equation $3^x + 9^x = 27^x$. [b]p2.[/b] An equilateral triangle in inscribed in a circle of area $1$ m$^2$. Then the second circle is inscribed in the triangle. Find the radius of the second circle. [b]p3.[/b] Solve the inequality: $2\sqrt{x^2 - 5x + 4} + 3\sqrt{x^2 + 2x - 3} \le 5\sqrt{6 - x - x^2}$ [b]p4.[/b] Peter and John played a game. Peter wrote on a blackboard all integers from $1$ to $18$ and offered John to choose $8$ different integers from this list. To win the game John had to choose 8 integers such that among them the difference between any two is either less than $7$ or greater than $11$. Can John win the game? Justify your answer. [b]p5.[/b] Prove that given $100$ different positive integers such that none of them is a multiple of $100$, it is always possible to choose several of them such that the last two digits of their sum are zeros. [b]p6.[/b] Given $100$ different squares such that the sum of their areas equals $1/2$ m$^2$ , is it possible to place them on a square board with area $1$ m$^2$ without overlays? Justify your answer. PS. You should use hide for answers.

[u]Set 4[/u] [b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$. [b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a multiple of $99$. Find $A + B + C + D + E$. [b]G12.[/b] Let $A, B, C, D$ be four points satisfying $AB = 10$ and $AC = BC = AD = BD = CD = 6$. If $V$ is the volume of tetrahedron $ABCD$, then find $V^2$. [u]Set 5[/u] [b]G13.[/b] Nate the giant is running a $5000$ meter long race. His first step is $4$ meters, his next step is $6$ meters, and in general, each step is $2$ meters longer than the previous one. Given that his $n$th step will get him across the finish line, find $n$. [b]G14.[/b] In square $ABCD$ with side length $2$, there exists a point $E$ such that $DA = DE$. Let line $BE$ intersect side $AD$ at $F$ such that $BE = EF$. The area of $ABE$ can be expressed in the form $a -\sqrt{b}$ where $a$ is a positive integer and $b$ is a square-free integer. Find $a + b$. [b]G15.[/b] Patrick the Beetle is located at $1$ on the number line. He then makes an infinite sequence of moves where each move is either moving $1$, $2$, or $3$ units to the right. The probability that he does reach $6$ at some point in his sequence of moves is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 6[/u] [b]G16.[/b] Find the smallest positive integer $c$ greater than $1$ for which there do not exist integers $0 \le x, y \le9$ that satisfy $2x + 3y = c$. [b]G17.[/b] Jaeyong is on the point $(0, 0)$ on the coordinate plane. If Jaeyong is on point $(x, y)$, he can either walk to $(x + 2, y)$, $(x + 1, y + 1)$, or $(x, y + 2)$. Call a walk to $(x + 1, y + 1)$ an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point $(10, 10)$? [b]G18.[/b] Deja vu? Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is $B$. Find the integer closest to $100B$. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We are given $8$ different weights and a balance without a scale. (a) Find the smallest number of weighings necessary to find the heaviest weight. (b) How many weighting is further necessary to find the second heaviest weight?

Determine the number of ways to partition the $n^2$ squares of an $n\times n$ grid into $n$ connected pieces of sizes $1$, $3$, $5$, $\dots$, $2n-1$ so that each piece is symmetric across the diagonal connecting the bottom right to the top left corner of the grid. A connected piece is a set of squares that any two of them are connected by a sequence of adjacent squares in the set. Two squares are adjacent if and only if they share an edge.

Let ${n \choose k}$ be the number of ways that $k$ objects can be chosen (regardless of order) from a set of $n$ objects. Prove that if positive integers k and l are greater than $1$ and less than $n$, then integers ${n \choose k}$ and ${n \choose l}$ have a common divisor greater than $1$.

A teacher wishes to separate her $12$ students into groups. Yesterday, the teacher put the students into $4$ groups of $3$. Today, the teacher decides to put the students into $4$ groups of $3$ again. However, she doesn’t want any pair of students to be in the same group on both days. Find how many ways she could formthe groups today.

Let $ \left( a_i \right)_{1\le i\le n} $ and $ \left( b_i \right)_{1\le i\le n} $ be two sequences, the former being a decreasing sequence and the latter being an increasing sequence. All the terms of $ \left( a_i \right)_{1\le i\le n} $ and $ \left( b_i \right)_{1\le i\le n} $ form the set $ \{1,2,3,\ldots ,2n \} . $ Prove that: $$ \left| a_1-b_1 \right| +\left| a_2-b_2 \right| +\cdots +\left| a_n-b_n \right|=n^2 $$

In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.

A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.

In how many ways can the circles be coloured using three colours, so that no two circles connected by a line segment have the same colour? [img]https://cdn.artofproblemsolving.com/attachments/3/2/e2bd61786aa4269593233311e85204cff071ec.png[/img]

Consider a pyramid whose base is a $2013$-sided polygon. On each face of the pyramid the number $0$ is written. The following operation is carried out: a vertex is chosen from the pyramid and add or subtract $1$ from all the faces that contain that vertex. It's possible, after repeating a finite number of times the previous procedure, that all the faces of the pyramid have the number $1$ written?

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

There are $100$ people in the group. Is it possible that for each pair of people exist at least $50$ others, so every in that group knows exactly one person from the pair?

A tennis competition takes place over four days, the number of participants is $2n$ with $n \ge 5$. Each participant plays exactly once a day (a couple of participants may be more times). Prove that such competition can end with exactly one winner and exactly three players in second place and such that there are no players with four lost games,

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that [b]i.)[/b] no three points of $ S$ are collinear, and [b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

In any rectangular game board with black and white squares, call a row $X$ a mix of rows $Y$ and $Z$ whenever each cell in row $X$ has the same colour as either the cell of the same column in row $Y$ or the cell of the same column in row $Z$. Let a natural number $m \ge 3$ be given. In some rectangular board, black and white squares lie in such a way that all the following conditions hold. 1) Among every three rows of the board, one is a mix of two others. 2) For every two rows of the board, their corresponding cells in at least one column have different colours. 3) For every two rows of the board, their corresponding cells in at least one column have equal colours. 4) It is impossible to add a new row with each cell either black or white to the board in a way leaving both conditions 1) and 2) still in force Find all possibilities of what can be the number of rows of the board.

Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers? [i]Proposed by Morteza Saghafian [/i]

George has two coins, one of which is fair and the other of which always comes up heads. Jacob takes one of them at random and flips it twice. Given that it came up heads both times, what is the probability that it is the coin that always comes up heads?

Prove that for every positive integer $t$ there is a unique permutation $a_0, a_1, \ldots , a_{t-1}$ of $0, 1, \ldots , t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i \neq t+i$.

Can we put positive integers $1,2,3, \cdots 64$ into $8 \times 8$ grids such that the sum of the numbers in any $4$ grids that have the form like $T$ ( $3$ on top and $1$ under the middle one on the top, this can be rotate to any direction) can be divided by $5$?

A six-sided die is rolled four times. What is the probability that the minimum value of the four rolls is $4$?

A convex polygon is dissected into a finite number of triangles with disjoint interiors, whose sides have odd integer lengths. The triangles may have multiple vertices on the boundary of the polygon and their sides may overlap partially. [list=a] [*]Prove that the polygon's perimeter is an integer which has the same parity as the number of triangles in the dissection. [*]Determine whether part a) holds if the polygon is not convex. [/list] [i]Proposed by Marius Cavachi[/i] [i]Note: the junior version only included part a), with an arbitrary triangle instead of a polygon.[/i]