Ana and Beto play the following game with a stick of length $15$. Ana starts, and on her first turn, she cuts the stick into two pieces with integer lengths. Then, on each player's turn, they must cut one of the pieces, of their choice, into two new pieces with integer lengths. The player who, on their turn, leaves at least one piece with length equal to $1$ loses. Determine which of the two players has a winning strategy.

Each of the positive integers from $1$ to $2023,$ inclusive, are randomly colored either blue or red. For each nonempty subset of $S=\{1,2,\cdots,2023\},$ we define the score of that subset to be the positive difference between the number of blue integers and the number of red integers in that subset. Let $X$ be the expected value of the sum of the scores of all the nonempty subsets of $S$. What is the maximum integer $k$ such that $2^k$ divides $2^{2023}\cdot X$? [i]Proposed by Kyle Lee[/i]

Cody has an unfair coin that flips heads with probability either $\tfrac13$ or $\tfrac23$, but he doesn't know which one it is. Using this coin, what is the fewest number of independent flips needed to simulate a coin that he knows will flip heads with probability $\tfrac13$?

Let $ABC$ be a triangle ($\angle A\neq 90^\circ$). $BE,CF$ are the altitudes of the triangle. The bisector of $\angle A$ intersects $EF,BC$ at $M,N$. Let $P$ be a point such that $MP\perp EF$ and $NP\perp BC$. Prove that $AP$ passes through the midpoint of $BC$. [i]Proposed by Iman Maghsoudi, Hooman Fattahi[/i]

Find the maximum possible value of the product of distinct positive integers whose sum is $2004$.

Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression: (b-a)(b^3+3a^3)

The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ . . . a_n \times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k$, and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$ ${{ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ 4} $

$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$. Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$ (V. Prasolov )

The Fibonacci sequence $F_0, F_1, \ldots$ satisfies $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all $n \ge 0$. Compute the number of triples $(a, b, c)$ with $0 \le a < b < c \le 100$ for which $F_a, F_b, F_c$ is an arithmetic progression.

$3n-1$ points are given in the plane, no three are collinear. Prove that one can select $2n$ of them whose convex hull is not a triangle.

Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that: (a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$; (b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$. [hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]

Find the number of all $\text{permutations}$ of $\left \{ 1,2,\cdots ,n \right \}$ like $p$ such that there exists a unique $i \in \left \{ 1,2,\cdots ,n \right \}$ that : $$p(p(i)) \geq i$$

Let $ABCD$ be a convex quadrilateral. Let $E$ be the intersection of its diagonals. Suppose that $CD = BC = BE$. Prove that $AD + DC\ge AB$. [i]Proposed by Dominik Burek - Poland[/i]

Determine all positive integers $a,b,c$ such that $ab + ac + bc$ is a prime number and $$\frac{a+b}{a+c}=\frac{b+c}{b+a}.$$

Find the volume of the region of points $(x,y,z)$ such that \[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]

An equation $P(x)=Q(y)$ is called [b]Interesting[/b] if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in $\mathbb{N}$. An interesting equation $P(x)=Q(y)$ [b]yields in[/b] interesting equation $F(x)=G(y)$ if there exists polynomial $R(x) \in \mathbb{Q} [x]$ such that $F(x) \equiv R(P(x))$ and $G(x) \equiv R(Q(x))$. (a) Suppose that $S$ is an infinite subset of $\mathbb{N} \times \mathbb{N}$.$S$ [i]is an answer[/i] of interesting equation $P(x)=Q(y)$ if each element of $S$ is an answer of this equation. Prove that for each $S$ there's an interesting equation $P_0(x)=Q_0(y)$ such that if there exists any interesting equation that $S$ is an answer of it, $P_0(x)=Q_0(y)$ yields in that equation. (b) Define the degree of an interesting equation $P(x)=Q(y)$ by $max\{deg(P),deg(Q)\}$. An interesting equation is called [b]primary[/b] if there's no other interesting equation with lower degree that yields in it. Prove that if $P(x)=Q(y)$ is a primary interesting equation and $P$ and $Q$ are monic then $(deg(P),deg(Q))=1$. Time allowed for this question was 2 hours.

The quiz scores of a class with $k>12$ students have a mean of $8.$ The mean of a collection of $12$ of these quiz scores is $14.$ What is the mean of the remaining quiz scores in terms of $k$? $\textbf{(A) } \frac{14-8}{k-12} \qquad \textbf{(B) } \frac{8k-168}{k-12} \qquad \textbf{(C) } \frac{14}{12} - \frac{k}{8} \qquad \textbf{(D) } \frac{14(k-12)}{k^2} \qquad \textbf{(E) } \frac{14(k-12)}{8k}$

Let $x_1,x_2,\ldots x_n$ be positive integers such that no one of them is an initial fragment of any other (for example, $12$ is an initial fragment of $\underline{12},\underline{12}5$ and $\underline{12}405$). Prove that \[\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}<3. \]

Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.

In a regular hexagon, segments with lengths from $1$ to $6$ were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon. [img]https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png[/img]

The plane is divided by three infinite sets of parallel lines into equilateral triangles of equal area. Let $M$ be the set of their vertices, and $A$ and $B$ be two vertices of such an equilateral triangle. One may rotate the plane through $120^o$ around any vertex of the set $M$. Is it possible to move the point $A$ to the point $B$ by a number of such rotations (N Vasiliev, Moscow)

Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?

For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\] [i]Proposed by A. Antropov[/i]

Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$. $\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$

A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\] are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),\\T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of $|S-T|$.