In the plane, consider a point $ X$ and a polygon $ \mathcal{F}$ (which is not necessarily convex). Let $ p$ denote the perimeter of $ \mathcal{F}$, let $ d$ be the sum of the distances from the point $ X$ to the vertices of $ \mathcal{F}$, and let $ h$ be the sum of the distances from the point $ X$ to the sidelines of $ \mathcal{F}$. Prove that $ d^2 \minus{} h^2\geq\frac {p^2}{4}.$

Let $y(x)$ be a continuously differentiable real-valued function of a real variable $x$. Show that if $y'(x)^2 +y(x)^3 \to 0$ as $x\to \infty,$ then $y(x)$ and $y'(x) \to 0$ as $x \to \infty.$

Let $ M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace$. Find the smallest $ n \in \mathbb{N}$ such that any $ n$-element subset of $ M$ contains 3 pairwise disjoint 4-element subsets $ S, T, U$ such that [b]I.[/b] For any 2 elements in $ S$, the larger number is a multiple of the smaller number. The same applies for $ T$ and $ U$. [b]II.[/b] For any $ s \in S$ and $ t \in T$, $ (s,t) \equal{} 1$. [b]III.[/b] For any $ s \in S$ and $ u \in U$, $ (s,u) > 1$.

The numbers $1$ to $12$ are arranged in a sequence. The number of ways this can be done equals $12 \times11 \times 10\times ...\times 1$. We impose the condition that in the sequence there should be exactly one number that is smaller than the number directly preceding it. How many of the $12 \times11 \times 10\times ...\times 1$ sequences satisfy this condition?

Let $f$ be a polynomial such that, for all real number $x$, $f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019$. Compute $f(2018)$.

We are given a polyhedron with at least $5$ vertices, such that exactly $3$ edges meet in each of the vertices. Prove that we can assign a rational number to every vertex of the given polyhedron such that the following conditions are met: $(i)$ At least one of the numbers assigned to the vertices is equal to $2020$. $(ii)$ For every polygonal face, the product of the numbers assigned to the vertices of that face is equal to $1$.

Let $x, y$ be positive integers such that $\frac{x^2}{y}+\frac{y^2}{x}$ is an integer. Prove that $y|x^2$.

Every point of the plane is painted with one of three colors. Can we always find two points a distance $1$ cm apart which are of the same color?

A space station consists of two living modules attached to a central hub on opposite sides of the hub by long corridors of equal length. Each living module contains $N$ astronauts of equal mass. The mass of the space station is negligible compared to the mass of the astronauts, and the size of the central hub and living modules is negligible compared to the length of the corridors. At the beginning of the day, the space station is rotating so that the astronauts feel as if they are in a gravitational field of strength $g$. Two astronauts, one from each module, climb into the central hub, and the remaining astronauts now feel a gravitational field of strength $g'$ . What is the ratio $g'/g$ in terms of $N$?[asy] import roundedpath; size(300); path a = roundedpath((0,-0.3)--(4,-0.3)--(4,-1)--(5,-1)--(5,0),0.1); draw(scale(+1,-1)*a); draw(scale(+1,+1)*a); draw(scale(-1,-1)*a); draw(scale(-1,+1)*a); filldraw(circle((0,0),1),white,black); filldraw(box((-2,-0.27),(2,0.27)),white,white); draw(arc((0,0),1.5,+35,+150),dashed,Arrow); draw(arc((0,0),1.5,-150,-35),dashed,Arrow);[/asy] $ \textbf{(A)}\ 2N/(N-1) $ $ \textbf{(B)}\ N/(N-1) $ $ \textbf{(C)}\ \sqrt{(N-1)/N} $ $ \textbf{(D)}\ \sqrt{N/(N-1)} $ $ \textbf{(E)}\ \text{none of the above} $

Find the sum of the positive integer solutions to the equation $\left\lfloor\sqrt[3]{x}\right\rfloor+\left\lfloor\sqrt[4]{x}\right\rfloor=4.$

In trapezoid $ABCD$, $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$. Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$. [i]Proposed by Lewis Chen[/i]

In an infinite sequence of positive integers every element (starting with the second) is the harmonic mean of its neighbors. Show that all the numbers must be equal.

Show that $$\sqrt{n + \left[ \sqrt{n} +\frac12\right]}$$ is an irrational number, for every positive integer $n$.

Let $A_{1}A_{2} \cdots A_{14}$ be a regular $14-$gon. Prove that $A_{1}A_{3}\cap A_{5}A_{11}\cap A_{6}A_{9}\ne \emptyset$.

Consider the additive group $\mathbb{Z}^{2}$. Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$. Let $H_{xy}$ be the smallest subgroup containing $(0,x)$ and $(1,y)$. Find some pair $(x,y)$ with $x>0$ such that $H=H_{xy}$.

In a triangle \( ABC \), \( D \) is the foot of the altitude from \( C \). Let \( P \in \overline{CD} \). \( Q \) is the intersection of \( \overline{AP} \) and \( \overline{CB} \), and \( R \) is the intersection of \( \overline{BP} \) and \( \overline{CA} \). Prove that \( \angle RDC = \angle QDC \).

Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

Let $a_1,a_2,...$ and $b_1,b_2,..$. be two arbitrary infinite sequences of natural numbers. Prove that there exist different indices $r$ and $s$ such that $a_r \ge a_s$ and $b_r \ge b_s$.

Let $f:\mathbb{N}\to\mathbb{N}_0$ be a function that satisfies \[ f(mn) = mf(n) + nf(m)\] for all positive integers $m,n$ and $f(2024)=10120$. Prove that there are two integers $m,n$ with $m\ne n$ such that $f(m)=f(n)$.

In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.

We say that three different integers are [i]friendly[/i] if one of them divides the product of the other two. Let $n$ be a positive integer. a) Show that, between $n^2$ and $n^2+n$, exclusive, does not exist any triplet of friendly numbers. b) Determine if for each $n$ exists a triplet of friendly numbers between $n^2$ and $n^2+n+3\sqrt{n}$ , exclusive.

Let $B,C,D,E$ be four pairwise distinct collinear points and let $A$ be a point not on line $BC$. Now let the circumcircle of $ABC$ meet $AD$ and $AE$ respectively again at $F$ and $G$ Show that $DEFG$ is cyclic if and only if $AB=AC$

Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

Prove that the area of the circle inscribed in a regular hexagon is greater than $90\%$ of the area of the hexagon.