You are given some equilateral triangles and squares, all with side length 1, and asked to form convex $n$ sided polygons using these pieces. If both types must be used, what are the possible values of $n$, assuming that there is sufficient supply of the pieces?

Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?

Find all pairs of functions $ f : \mathbb R \to \mathbb R$, $g : \mathbb R \to \mathbb R$ such that \[f \left( x + g(y) \right) = xf(y) - y f(x) + g(x) \quad\text{for all } x, y\in\mathbb{R}.\]

$ABCD$ is inscribed in unit circle $\Gamma$. Let $\Omega_1$, $\Omega_2$ be the circumcircles of $\vartriangle ABD$, $\vartriangle CBD$ respectively. Circles $\Omega_1$, $\Omega_2$ are tangent to segment $BD$ at $M$,$N$ respectively. Line A$M$ intersects $\Gamma$, $\Omega_1$ again at points $X_1$,$X_2$ respectively (different from $A$, $M$). Let $\omega_1$ be the circle passing through $X_1$, $X_2$ and tangent to $\Omega_1$. Line $CN$ intersects $\Gamma$, $\Omega_2$ again at points $Y_1$, $Y_2$ respectively (different from $C$, $N$). Let $\omega_2$ be the circle passing through $Y_1$, $Y_2$ and tangent to $\Omega_2$. Circles $\Omega_1$,$\Omega_2$, $\omega_1$, $\omega_2$ have radii $R_1$, $R_2$, $r_1$, $r_2$ respectively. Prove that $$r_1+r_2-R_1-R_2=1.$$ [img]https://cdn.artofproblemsolving.com/attachments/1/5/70471f2419fadc4b2183f5fe74f0c7a2e69ed4.png[/img] [url=https://www.geogebra.org/m/vxx8ghww]geogebra file[/url]

$2014$ triangles have non-overlapping interiors contained in a circle of radius $1$. What is the largest possible value of the sum of their areas?

Here is a problem we (me and my colleagues) suggested and was given at the competition this year. The problem statement is very natural and short. However, we have not seen such a problem before. A real $2024 \times 2024$ matrix $A$ is called nice if $(Av, v) = 1$ for every vector $v\in \mathbb{R}^{2024}$ with unit norm. a) Prove that the only nice matrix such that all of its eigenvalues are real is the identity matrix. b) Find an example of a nice non-identity matrix

Let $D_{1}, D_{2}, . . . , D_{n}$ be rectangular parallelepipeds in space, with edges parallel to the $x, y, z$ axes. For each $D_{i}$, let $x_{i} , y_{i} , z_{i}$ be the lengths of its projections onto the $x, y, z$ axes, respectively. Suppose that for all pairs $D_{i}$ , $D_{j}$, if at least one of $x_{i} < x_{j}$ , $y_{i} < y_{j}$, $z_{i} < z_{j}$ holds, then $x_{i} \leq x_{j}$ , $y_{i} \leq y_{j}$, and $z_{i} < z_{j}$ . If the volume of the region $\bigcup^{n}_{i=1}{D_{i}}$ equals 1997, prove that there is a subset $\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}$ of the set $\{D_{1}, . . . , D_{n}\}$ such that $(i)$ if $k \not= l $ then $D_{i_{k}} \cap D_{i_{l}} = \emptyset $, and $(ii)$ the volume of $\bigcup^{m}_{k=1}{D_{i_{k}}}$ is at least 73.

Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$

On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take? $\textbf{(A) }50\qquad\textbf{(B) }70\qquad\textbf{(C) }80\qquad\textbf{(D) }90\qquad \textbf{(E) }100$

Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively. Suppose that $\frac{ax + b}{x} \in Q$ for every $x \in N^*$: Prove that there exist integers $A,B,C$ such that $\frac{ax + b}{x}= \frac{Ax + B}{Cx}$ for all $x \in N^* $

Points $A, B, C$ and $D$ lie on a circle, in that order clockwise, such that there is a point $E$ on segment $CD$ with the property that $AD = DE$ and $BC = EC$. Prove that the intersection point of the bisectors of the angles $\angle DAB$ and $\angle ABC$ is on the line $CD$.

Rumburak kidnapped $31$ members of party $A$ , $28$ members of party $B$, $23$ members of party $C$, $19$ members of Party $D$ and each of them in a separate cell. After work out occasionally they could walk in the yard and talk. Once three people started to talk to each other members of three different parties, Rumburak re-registered them to the fourth party as a punishment.(They never talked to each other more than three kidnapped.) a) Could it be that after some time all were abducted by members of one party? Which? b) Determine all four positive integers of which the sum is $101$ and which as the numbers of kidnapped members of the four parties allow the Rumburaks all of them became members of one party over time.

A hilly island has $2023$ lookouts. It is known that each of them is in line of sight with at least $42$ of the other lookouts. For any two distinct lookouts $X$ and $Y$ there is a positive integer $n$ and lookouts $A_1,A_2,\dots,A_{n+1}$ such that $A_1=X$ and $A_{n+1}=Y$ and $A_1$ is in line of sight with $A_2$, $A_2$ with $A_3$, $\dots$ and $A_n$ with $A_{n+1}$. The smallest such number $n$ is called the [i]viewing distance[/i] of $X$ and $Y$. Determine the largest possible viewing distance that can exist between two lookouts under these conditions.

Baron Munchausen has a set of $50$ coins. The mass of each is a distinct positive integer not exceeding $100$, and the total mass is even. The Baron claims that it is not possible to divide the coins into two piles with equal total mass. Can the Baron be right?

Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$. [i]Proposed by Evan Chen[/i]

A unit square is removed from the corner of an $n \times n$ grid, where $n \geq 2$. Prove that the remainder can be covered by copies of the figures of $3$ or $5$ unit squares depicted in the drawing below. [asy] import geometry; draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0)); draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy] [b]Note:[/b] Every square must be covered once and figures must not go over the bounds of the grid.

Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$, and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$. The tangent to the incircle at $P$ meets $BC$ at point $X$, and the tangent to the excircle at $Q$ meets $BC$ at $Y$. Prove that $MX=MY$.

A natural number $k > 1$ is called [i]good[/i] if there exist natural numbers $$a_1 < a_2 < \cdots < a_k$$ such that $$\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1$$. Let $f(n)$ be the sum of the first $n$ [i][good[/i] numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer.

In triangle $ABC$ $\omega$ is incircle and $\omega_1$,$\omega_2$,$\omega_3$ is tangents to $\omega$ and two sides of $ABC$. $r, r_1, r_2, r_3$ is radius of $\omega, \omega_1, \omega_2, \omega_3$. Prove that $\sqrt{r_1 r_2}+\sqrt{r_2 r_3}+\sqrt{r_3 r_1}=r$

Let $P$ be a function defined by $P(t)=a^t+b^t$, where $a$ and $b$ are complex numbers. If $P(1)=7$ and $P(3)=28$, compute $P(2)$. [i] Proposed by Justin Stevens [/i]

A circle is drawn on the plane. How to use only a ruler to draw a perpendicular from a given point outside the circle to a given line passing through the center of this circle?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two positive integers are written on the blackboard. Initially, one of them is $2000$ and the other is smaller than $2000$. If the arithmetic mean $ m$ of the two numbers on the blackboard is an integer, the following operation is allowed: one of the two numbers is erased and replaced by $ m$. Prove that this operation cannot be performed more than ten times. Give an example where the operation is performed ten times.

Let $C_1$ and $C_2$ be two circles intersecting at $A $ and $B$. Let $S$ and $T $ be the centres of $C_1 $ and $C_2$, respectively. Let $P$ be a point on the segment $AB$ such that $ |AP|\ne |BP|$ and $P\ne A, P \ne B$. We draw a line perpendicular to $SP$ through $P$ and denote by $C$ and $D$ the points at which this line intersects $C_1$. We likewise draw a line perpendicular to $TP$ through $P$ and denote by $E$ and F the points at which this line intersects $C_2$. Show that $C, D, E,$ and $F$ are the vertices of a rectangle.

Solve the equation $$\left[\frac{x^2+1}{x}\right]-\left[\frac{x}{x^2+1}\right]=3.$$