A diagonal of a 100-gon is called good if it divides the 100-gon into two polygons each with an odd number of sides. A 100-gon was split into triangles with non-intersecting diagonals, exactly 49 of which are good. The triangles are colored into two colors such that no two triangles that border each other are colored with the same color. Prove that there is the same number of triangles colored with one color as with the other. Fresh translation; slightly reworded.

A well-known puzzle asks for a partition of a cross into four parts which are to be reassembled into a square. One solution is exhibited on the picture. [img]https://cdn.artofproblemsolving.com/attachments/9/1/3b8990baf5e37270c640e280c479b788d989ba.png[/img] Show that there are infinitely many solutions. (Some solutions split the cross into four equal parts!)

Every bacterium has a horizontal body with natural length and some nonnegative number of vertical feet, each with nonnegative (!) natural length, that lie below its body. In how many ways can these bacteria fill an $m\times n$ table such that no two of them overlap? [i]proposed by Mahyar Sefidgaran[/i]

Pedro and Tiago are playing a game with a deck of n cards, numbered from $1$ to $n$. Starting with Pedro, they choose cards alternately, and receive the number of points indicated by the cards. However, whenever the player chooses the card with the highest number among those remaining in the deck, he is forced to pass his next turn, not choosing any card. When the deck runs out, the player with the most points wins. Knowing that Tiago can at least draw, regardless of Pedro's moves, how many cards are in the deck? Indicates all possibilities,

A partition of a positive integer $n$ is a summing $n_1+\ldots+n_k=n$, where $n_1\ge n_2\ge\ldots\ge n_k$. Call a partition [i]perfect[/i] if every $m\le n$ can be represented uniquely as a sum of some subset of the $n_i$'s. How many perfect partitions are there of $n=307$?

In the Intergalactic Lottery, $7$ numbers are drawn out of $55$. R2-D2 and C-3PO decide that they want to win this lottery, so they fill out lottery tickets separately such that for each possible draw one of them does have a winning ticket for that draw. Prove that one of them has $7$ tickets with all different numbers.

[b]p1.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$. [b]p2.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If f(x) is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$? [b]p3.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers. [b]p4.[/b] In triangle $ABC$, points $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle $BEC$ has area $45$ and triangle $ADC$ has area $72$, and lines $CH$ and $AB$ meet at $F$. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with $c$ squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$. [b]p5.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer $x$ such that $x^2 + 18x + y$ is a perfect fourth power. [b]p6.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a, b$ with $b$ squarefree, then find $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that: (i) For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$. (ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors. Prove that $k\leq 2$.

Let $x_1, x_2, \ldots$ be a sequence of $0,1$, such that it satisfies the following three conditions: 1) $x_2=x_{100}=1$, $x_i=0$ for $1 \leq i \leq 100$ and $i \neq 2,100$; 2) $x_{2n-1}=x_{n-50}+1, x_{2n}=x_{n-50}$ for $51 \leq n \leq 100$; 3) $x_{2n}=x_{n-50}, x_{2n-1}=x_{n-50}+x_{n-100}$ for $n>100$. Show that the sequence is periodic.

$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.

A rook is randomly placed on an otherwise empty $8 \times 8$ chessboard. Owen makes moves with the rook by randomly choosing $1$ of the $14$ possible moves. Find the expected value of the number of moves it takes Owen to move the rook to the top left square. Note that a rook can move any number of squares either in the horizontal or vertical direction each move.

A unit square is cut by $n$ straight lines . Prove that in at least one of these parts one can completely fit a square with side $\frac{1}{n+1}$ [hide=original wording]Одиничний квадрат розрізано $n$ прямими на частини. Доведіть, що хоча б в одній з цих частин можна повністю розмістити квадрат зі стороною $\frac{1}{n+1}$[/hide] [hide=notes] The selection panel jury made a mistake because the solution known to it turned out to be incorrect. As it turned out, the assertion of the problem is still correct, although it cannot be proved by simple methods, see. article: Keith Ball. Тhe plank problem for symmetric bodies // Іпѵепііопез МаіЬешаІіеае. — 1991. — Ѵоі. 104, по. 1. — Р. 535-543. [url]https://arxiv.org/abs/math/9201218[/url][/hide]

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

Each of the subsets $ A_1$, $ A_2$, $ \dots,$ $ A_n$ of a 2009-element set $ X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $ X$ there is a 24-element subset $ B$ containing neither of the sets $ A_1$, $ A_2$, $ \dots,$ $ A_n$.

A cube has one of its vertices and all edges connected to that vertex deleted. How many ways can the letters from the word "$AMONGUS$" be placed on the remaining vertices of the cube so that one can walk along the edges to spell out "$AMONGUS$"? Note that each vertex will have at most $1$ letter, and one vertex is deleted and not included in the walk

Given $n$ sets $A_i$, with $| A_i | = n$, prove they may be indexed $A_i = \{a_{i,j} \mid j=1,2,\ldots,n \}$, in such way that the sets $B_j = \{a_{i,j} \mid i=1,2,\ldots,n \}$, $1\leq j\leq n$, also have $| B_j | = n$. (Anonymous)

Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$. Find $T(n+1)-T(n)$.

Some bishops and knights are placed on an infinite chessboard, where each square has side length $1$ unit. Suppose that the following conditions hold: [list] [*] For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight). [*] For each knight, there exists a bishop that is exactly $\sqrt{5}$ units away from it. [*] If any piece is removed from the board, then at least one of the above conditions is no longer satisfied. [/list] If $n$ is the total number of pieces on the board, find all possible values of $n$. [i]Sheldon Kieren Tan[/i]

[b]p19.[/b] $A_1A_2...A_{12}$ is a regular dodecagon with side length $1$ and center at point $O$. What is the area of the region covered by circles $(A_1A_2O)$, $(A_3A_4O)$, $(A_5A_6O)$, $(A_7A_8O)$, $(A_9A_{10}O)$, and $(A_{11}A_{12}O)$? $(ABC)$ denotes the circle passing through points $A,B$, and $C$. [b]p20.[/b] Let $N = 2000... 0x0 ... 00023$ be a $2023$-digit number where the $x$ is the $23$rd digit from the right. If$ N$ is divisible by $13$, compute $x$. [b]p21.[/b] Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between $12$ PM and $1$ PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between $0$ and $30$ minutes. What is the probability that they will meet? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a committee there are $n$ members. Each pair of members are either friends or enemies. Each committee member has exactly three enemies. It is also known that for each committee member, an enemy of his friend is automatically his own enemy. Find all possible value(s) of $n$

Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$. [i](H. Lesov)[/i]

Three numbers are selected at random from the interval $[0,1]$. What is the probability that they form the lengths of the sides of a triangle?

We call a coloring of an $8\times 8$ board in three colors good if in any corner of five cells contains cells of all three colors. (A five-square corner is a shape made from a $3 \times 3$ square by cutting square $ 2\times 2$.) Prove that the number of good colorings is not less than than $68$.

A $k \times k$ array contains each of the numbers $1, 2, \dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \in \mathbb{N}^+$)? [i]Attila Sztranyák and Péter Erben[/i]

If there are several heaps of stones on the table, it is said that there are $\textit{many}$ stones on the table, if we can find $50$ piles and number them with the numbers from $1$ to $50$ so that the first pile contains at least one stone, the second - at least two stones,..., the $50$-th has at least $50$ stones. Let the table be initially contain $100$ piles of $100$ stones each. Find the largest $n \leq 10 000$ such that after removing any $n$ stones, there will still be $\textit{many}$ stones left on the table.