Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors. Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.

Suppose we have a sequence $a_1, a2_, ...$ of positive real numbers so that for each positive integer $n$, we have that $\sum_{k=1}^{n} a_ka_{\lfloor \sqrt{k} \rfloor} = n^2$. Determine the first value of $k$ so $a_k > 100$.

On the Cartesian plane, all points with both integer coordinates are painted blue. Two blue points are said to be [i]mutually visible[/i] if the line segment that connects them has no other blue points . Prove that there is a set of $ 2019$ blue points that are mutually visible two by two. [hide=official wording]No plano cartesiano, todos os pontos com ambas coordenadas inteiras são pintados de azul. Dois pontos azuis são ditos mutuamente visíveis se o segmento de reta que os conecta não possui outros pontos azuis. Prove que existe um conjunto de 2019 pontos azuis que são mutuamente visíveis dois a dois.[/hide] PS. There is a comment about problem being wrong / incorrect [url=https://artofproblemsolving.com/community/c6h1957974p14780265]here[/url]

If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$?

Ninety-one students in a school are distributed in three classes. Each student took part in a competition. It is known that among any six students of the same sex some two got the same number of points. Show that here are four students of the same sex who are in the same class and who got the same number of points.

From the $9 \times 9$ chessboard, all $16$ unit squares whose row numbers and column numbers are both even have been removed. Disect the punctured board into rectangular pieces, with as few of them being unit squares as possible.

If $a$ and $b$ are the roots of $x^2 - 2x + 5$, what is $|a^8 + b^8|$?

Let $(x_n)$ be an infinite sequence of real numbers from interval $(0, 1)$. An infinite sequence $(a_n)$ of positive integers is defined as follows: $a_1 = 1$, and for $i \ge 1$, $a_{i+1}$ is equal to the smallest positive integer $m$, for which $[x_1 + x_2 + \ldots + x_m] = a_i$. Show that for any indexes $i, j$ holds $a_{i+j} \ge a_i + a_j$. [i]Proposed by Nazar Serdyuk[/i]

Let $p$ and $q$ be two different prime numbers. Prove that there are two positive integers, $a$ and $b$, such that the arithmetic mean of the divisors of $n=p^aq^b$ is an integer.

Several points are denoted on a white piece of paper. The distance between each two of the points is greater than $24$. A drop of ink was sprinkled over the paper covering an area smaller than $\pi$. Prove that there exists a vector $\overrightarrow v$ with $\overrightarrow v<1$, such that after translating all of the points by $v$ none of them is covered in ink.

How many $4-$digit numbers $\overline{abcd}$ are there such that $a<b<c<d$ and $b-a<c-b<d-c$ ?

Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.

$a,b,c,d$ are positive reals with sum $1$. Show that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}$ with equality iff $a=b=c=d=\frac{1}{4}$.

Find all pairs of real numbers $(x,y)$ satisfying the system of equations: \begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}

Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle (i) whose vertices are all the same colour, and (ii) at least one side of the triangle does not contain a point of the opposite colour.

On a table lie 2022 matches and a regular dice that has the number $a$ on top. Now Max and Moritz play the following game: Alternately, they take away matches according to the following rule, where Max begins: The player to make a move rolls the dice over one of its edges and then takes a way as many matches as the top number shows. The player that cannot make legal move after some number of moves loses. For which $a$ can Moritz force Max to lose?

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that \[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.

Given a triangle $ABC$ let \begin{eqnarray*} x &=& \tan\left(\dfrac{B-C}{2}\right) \tan \left(\dfrac{A}{2}\right) \\ y &=& \tan\left(\dfrac{C-A}{2}\right) \tan \left(\dfrac{B}{2}\right) \\ z &=& \tan\left(\dfrac{A-B}{2}\right) \tan \left(\dfrac{C}{2}\right). \end{eqnarray*} Prove that $x+ y + z + xyz = 0$.

Positive reals $a,b,c \leq 1$ satisfy $\frac{a+b+c-abc}{1-ab-bc-ca} = 1$. Find the minimum value of $$\bigg(\frac{a+b}{1-ab} + \frac{b+c}{1-bc} + \frac{c+a}{1-ca}\bigg)^{2}$$ Proposed by Harry Chen (Extile)

[b]carpeting[/b] suppose that $S$ is a figure in the plane such that it's border doesn't contain any lattice points. suppose that $x,y$ are two lattice points with the distance $1$ (we call a point lattice point if it's coordinates are integers). suppose that we can cover the plane with copies of $S$ such that $x,y$ always go on lattice points ( you can rotate or reverse copies of $S$). prove that the area of $S$ is equal to lattice points inside it. time allowed for this question was 1 hour.

If $a$ and $b$ are real numbers, the equation $3x-5+a=bx+1$ has a unique solution $x$ [The symbol $a \neq 0$ means that $a$ is different from zero]: $ \textbf{(A) }\text{for all a and b} \qquad\textbf{(B) }\text{if a }\neq\text{2b}\qquad\textbf{(C) }\text{if a }\neq 6\qquad$ $\textbf{(D) }\text{if b }\neq 0\qquad\textbf{(E) }\text{if b }\neq 3 $

The order of $\log_{\sin1}\cos1,\log_{\sin1}\tan1,\log_{\cos1}\sin1,\log_{\cos1}\tan1$ is (form small to large) $\text{(A)}\log_{\sin1}\cos1<\log_{\cos1}\sin1<\log_{\sin1}\tan1<\log_{\cos1}\tan1$ $\text{(B)}\log_{\cos1}\sin1<\log_{\cos1}\tan1<\log_{\sin1}\cos1<\log_{\sin1}\tan1$ $\text{(C)}\log_{\sin1}\tan1<\log_{\cos1}\tan1<\log_{\cos1}\sin1<\log_{\sin1}\cos1$ $\text{(D)}\log_{\cos1}\tan1<\log_{\sin1}\tan1<\log_{\sin1}\cos1<\log_{\cos1}\sin1$

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.