Let $N_7$ be the answer to problem 7. Each side of a regular $N_7$-gon is colored with a single color from a set of two given colors. Two colorings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. Compute the number of possible different colorings.

$a_1,a_2,\dots,a_k$ and $b_1,b_2,\dots,a_n$ are integer sequences with $a_1=b_1=0$, $a_k=26,b_n=25$, and $k+n=23$. It is known that $a_{i+1}-a_i=2\enspace\text{or}\enspace3$ and $b_{j+1}-b_j=2\enspace\text{or}\enspace3$ for all applicable $i$ and $j$, and the numbers $a_2,\dots,a_k,b_2,\dots,b_n$ are pairwise different. Determine the total number of such pairs of sequences.

If there is only $1$ complex solution to the equation $8x^3 + 12x^2 + kx + 1 = 0$, what is $k$?

Consider the function $f : N \to N$ given by (i) $f(1) = 1$, (ii) $f(2n) = f(n)$ for any $n \in N$, (iii) $f(2n+1) = f(2n)+1$ for any $n \in N$. (a) Find the maximum value of $f(n)$ for $1 \le n \le 1995$; (b) Find all values of $f$ on this interval.

Four distinct points are arranged in a plane so that the segments connecting them has lengths $a,a,a,a,2a,$ and $b$. What is the ratio of $b$ to $a$? $ \textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi $

For a positive integer $n$, written in decimal base, we denote by $p(n)$ the product of its digits. a) Prove that $p(n) \leq n$; b) Find all positive integers $n$ such that \[ 10p(n) = n^2+ 4n - 2005. \] [i]Eugen Păltănea[/i]

Consider a right cylinder with height $5\sqrt3$. A plane intersects each of the bases of the cylinder at exactly one point, and the cylindric section (the intersection of the plane and the cylinder) forms an ellipse. Find the product of the sum and the dierence of the lengths of the major and minor axes of this ellipse. [i]Note:[/i] An ellipse is a regular oval shape resulting when a cone is cut by an oblique plane which does not intersect the base. The major axis is the longer diameter and the minor axis the shorter.

Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively. Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively. Prove, that $I, O, H$ lies on one line.

For each $m\in \mathbb N$ we define $rad\ (m)=\prod p_i$, where $m=\prod p_i^{\alpha_i}$. [b]abc Conjecture[/b] Suppose $\epsilon >0$ is an arbitrary number, then there exist $K$ depinding on $\epsilon$ that for each 3 numbers $a,b,c\in\mathbb Z$ that $gcd (a,b)=1$ and $a+b=c$ then: \[ max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon} \] Now prove each of the following statements by using the $abc$ conjecture : a) Fermat's last theorem for $n>N$ where $N$ is some natural number. b) We call $n=\prod p_i^{\alpha_i}$ strong if and only $\alpha_i\geq 2$. c) Prove that there are finitely many $n$ such that $n,\ n+1,\ n+2$ are strong. d) Prove that there are finitely many rational numbers $\frac pq$ such that: \[ \Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3} \]

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

The sum of the roots of the equation $ 4x^2\plus{}5\minus{}8x\equal{}0$ is equal to: $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ -5 \qquad \textbf{(C)}\ -\dfrac{5}{4} \qquad \textbf{(D)}\ -2 \qquad \textbf{(E)}\ \text{None of these}$

$ABC$ is a triangle with an area of $1$ square meter. Given the point $D$ on $BC$, point $E$ on $CA$, point $F$ on $AB$, such that quadrilateral $AFDE$ is cyclic. Prove that the area of $DEF \le \frac{EF^2}{4 AD^2}$. [i](holmes)[/i]

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M,$ where $r$ and $M$ are positive integers and $M$ is not divisible by $3.$ What is $r?$ $\textbf{(A) }5 \qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

The number $2022$ is written on the white board. Ivan and Peter play a game, Ivan starts and they alternate. On a move, Ivan erases the number $b$, written on the board, throws a dice which shows some number $a$, and writes the residue of $(a+b) ^2$ modulo $5$. Similarly, Peter throws a dice which shows some number $a$, and changes the previously written number $b$ to the residue of $a+b$ modulo $3$. The first player to write a $0$ wins. What is the probability of Ivan winning the game?

Suppose $ABC$ is an acute triangle. Points $A_1, B_1$ and $C_1$ are chosen on sides $BC, AC$ and $AB$ respectively so that the rays $A_1A, B_1B$ and $C_1C$ are bisectors of triangle $A_1B_1C_1$. Prove that $AA_1, BB_1$ and $CC_1$ are altitudes of triangle $ABC$. (6)

There are 36 contestants in the CMU Puyo-Puyo Tournament, each with distinct skill levels. The tournament works as follows: First, all $\binom{36}{2}$ pairings of players are written down on slips of paper and are placed in a hat. Next, a slip of paper is drawn from the hat, and those two players play a match. It is guaranteed that the player with a higher skill level will always win the match. We continue drawing slips (without replacement) and playing matches until the results of the match completely determine the order of skill levels of all 36 contestants (i.e. there is only one possible ordering of skill levels consistent with the match results), at which point the tournament immediately finishes. What is the expected value of the number of matches played before the stopping point is reached? [i]Proposed by Dilhan Salgado[/i]

Prove that for any natural number $n$ and real numbers $a$ and $x$ satisfying $a^{n+1} \le x \le 1$ and $0 < a < 1$ it holds that $$\prod_{k=1}^n \left|\frac{x-a^k}{x+a^k}\right| \le \prod_{k=1}^n \frac{1-a^k}{1+a^k}$$

Let $ABC$ be an acute triangle with $AB\neq AC$, $M$ be the median of $BC$, and $H$ be the orthocenter of $\triangle ABC$. The circumcircle of $B$, $H$, and $C$ intersects the median $AM$ at $N$. Show that $\angle ANH=90^\circ$.

Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic. (D.Prokopenko)

Let $a$, $b$, $c$ be real numbers. Prove that \begin{align*}&\quad\,\,\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\\&\ge \sqrt{3[(a+b)^2+(b+c)^2+(c+a)^2]}.\end{align*}

Let $ABC$ be an acute triangle with orthocenter $H$. The circumcircle of the triangle $BHC$ intersects $AC$ a second time in point $P$ and $AB$ a second time in point $Q$. Prove that $H$ is the circumcenter of the triangle $APQ$. [i](Karl Czakler)[/i]

Compute the number of ways of tiling the $2\times 10$ grid below with the three tiles shown. There is an infinite supply of each tile, and rotating or reflecting the tiles is not allowed. [img]https://cdn.artofproblemsolving.com/attachments/5/a/bb279c486fc85509aa1bcabcda66a8ea3faff8.png[/img]

Suppose that $7$ boys and $13$ girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $GBBGGGBGBGGGBGBGGBGG$ we have $S=12$. The average value of $S$ (if all possible orders of the 20 people are considered) is closest to $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13 $

What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? $ \text{(A)}\ 30\qquad\text{(B)}\ 45\qquad\text{(C)}\ 60\qquad\text{(D)}\ 75\qquad\text{(E)}\ 90 $

Find all pairs of positive integers $(x,y)$ , such that $x^2$ is divisible by $2xy^2 -y^3 +1$.