Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that $2023$ red chameleons can agree on a sequence of bites between themselves, after which they will all turn blue. What is the smallest $k$ that can guarantee that $k$ red chameleons, biting only each other, can turn blue? (For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)

Andy flips a strange coin for which the probability of flipping heads is $\frac{1}{2^k+1}$, where $k$ is the number of heads that appeared previously. If Andy flips the coin repeatedly until he gets heads 10 times, what is the expected number of total flips he performs? [i]Proposed by Harry Kim[/i]

A prime number $p$ and a positive integer $n$ are given. Prove that one can colour every one of the numbers $1,2,\ldots,p-1$ using one of the $2n$ colours so that for any $i=2,3,\ldots,n$ the sum of any $i$ numbers of the same colour is not divisible by $p$.

Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$. [i]Ciprus[/i]

Prove that there exist $0 < \alpha< \beta<1$ numbers have the following properties. (i) for any sufficiently large n, n points can be specified in $\Bbb R^3$ , so that each point is equidistant from at least $n^\alpha$ other points. (ii) the above statement is no longer true with $n^\beta$ instead of $n^\alpha$

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

4. Given is an acute-angled triangle $A B C, H$ is its orthocenter. Let $P$ be an arbitrary point inside (and not on the sides) of the triangle $A B C$ that belongs to the circumcircle of the triangle $A B H$. Let $A^{\prime}, B^{\prime}$, $C^{\prime}$ be projections of point $P$ to the lines $B C, C A, A B$. Prove that the circumcircle of the triangle $A^{\prime} B^{\prime} C^{\prime}$ passes through the midpoint of segment $C P$. Alexey Zaslavsky

Consider the numbers from $1$ to $16$. The "solitar" game consists in the arbitrary grouping of the numbers in pairs and replacing each pair with the great prime divisor of the sum of the two numbers (i.e from $(1,2); (3,4); (5,6);...;(15,16)$ the numbers which result are $3,7,11,5,19,23,3,31$). The next step follows from the same procedure and the games continues untill we obtain only one number. Which is the maximum numbers with which the game ends.

Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$ a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$ b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$

Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$ $\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$

All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined? \[ \begin{tabular}{c c c c} {} & \textbf{Adams} & \textbf{Baker} & \textbf{Adams and Baker} \\ \textbf{Boys:} & 71 & 81 & 79 \\ \textbf{Girls:} & 76 & 90 & ? \\ \textbf{Boys and Girls:} & 74 & 84 & \\ \end{tabular} \] $ \textbf{(A)}\ 81 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 84 \quad\textbf{(E)}\ 85 $

Let $[a]$ represent the largest integer less than or equal to $a$, for any real number $a$. Let $\{a\} = a - [a]$. Are there positive integers $m,n$ and $n+1$ real numbers $x_0,x_1,\hdots,x_n$ such that $x_0=428$, $x_n=1928$, $\frac{x_{k+1}}{10} = \left[\frac{x_k}{10}\right] + m + \left\{\frac{x_k}{5}\right\}$ holds? Justify your answer.

In a scalene triangle $ABC$ with incenter $I$, the incircle is tangent to sides $CA$ and $AB$ at points $E$ and $F$. The tangents to the circumcircle of triangle $AEF$ at $E$ and $F$ meet at $S$. Lines $EF$ and $BC$ intersect at $T$. Prove that the circle with diameter $ST$ is orthogonal to the nine-point circle of triangle $BIC$. [i]Proposed by Evan Chen[/i]

How many paths are there from $A$ to $B$ that consist of $5$ horizontal segments and $5$ vertical segments of length $1$ each? (see figure) [img]https://cdn.artofproblemsolving.com/attachments/4/2/5b476ca2a232fc67fb2e2f6bb06111cab60692.png[/img]

Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that $\bullet$ $P$ passes through $A$ and $B$, $\bullet$ the focus $F$ of $P$ lies on $E$, $\bullet$ the orthocenter $H$ of $\vartriangle F AB$ lies on the directrix of $P$. If the major and minor axes of $E$ have lengths $50$ and $14$, respectively, compute $AH^2 + BH^2$.

Let $\Gamma$ be the circumcircle of a triangle $ABC$ and let $E$ and $F$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with $\Gamma$. If $EF$ is tangent to the incircle $\gamma$ of $\triangle ABC$, then find the value of $\angle BAC$.

Which of these describes the graph of $x^2(x+y+1)=y^2(x+y+1)$ ? $\textbf{(A)}\ \text{two parallel lines}$\\ $\textbf{(B)}\ \text{two intersecting lines}$\\$\textbf{(C)}\ \text{three lines that all pass through a common point}$\\ $\textbf{(D)}\ \text{three lines that do not all pass through a common point}$\\$\textbf{(E)}\ \text{a line and a parabola}$

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

If a square is drawn externally on each side of a parallelogram, prove that: (a) The quadrilateral formed by the centers of these squares is also a square. (b) The diagonals of the new square formed are concurrent with the diagonals of the original parallelogram.

Let us say that a deck of $52$ cards is arranged in a “regular” way if the ace of spades is on the very top of the deck and any two adjacent cards are either of the same value or of the same suit (top and bottom cards regarded adjacent as well). Prove that the number of ways to arrange a deck in regular way is a) divisible by $12!$ (3) b) divisible by $13!$ (5)

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

For each positive integer $ n$ find all positive integers $ m$ for which there exist positive integers $ x_1<x_2<...<x_n$ with: $ \frac{1}{x_1}\plus{}\frac{2}{x_2}\plus{}...\plus{}\frac{n}{x_n}\equal{}m.$

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

On an infinite white checkered sheet, a square $Q$ of size $12$ × $12$ is selected. Petya wants to paint some (not necessarily all!) cells of the square with seven colors of the rainbow (each cell is just one color) so that no two of the $288$ three-cell rectangles whose centers lie in $Q$ are the same color. Will he succeed in doing this? (Two three-celled rectangles are painted the same if one of them can be moved and possibly rotated so that each cell of it is overlaid on the cell of the second rectangle having the same color.) (Bogdanov. I)

There are $n^2$ empty boxes, each with a square base. The height and width of each box are integers between $1$ and $n$ inclusive, and no two boxes are identical. One box [i]fits inside[/i] another if its height and width are both smaller, and additionally, one of its dimensions is at least $2$ units smaller. In this way, we can form sequences of boxes (the first inside the second, the second inside the third, and so on). We place each of these sequences on a different shelf. How many shelves are needed to store all the boxes, with certainty?