Prove that for any positive integer $m$ there exists a positive integer $n$, such that any $n$ different points on a plane can be partitioned into $m$ non-empty sets, *convex hulls* of which would share a common point.
*Convex hull* of a finite set $X$ of points on a plane is a set of points that lie inside or on the border of at least one convex polygon with vertices in $X$, including degenerate ones, i.e. a segment and a point are considered to be convex polygons. No three vertices of a convex polygon are collinear. A polygon contains its border.
A convex polyhedron has at least 5 vertices, and exactly three edges meet at each vertex. Prove that rational numbers can be assigned to its vertices so that:
(i) at least one assigned number is 2020;
(ii) for every face, the product of the numbers at all its vertices is 1.
Let $ABC$ be a triangle. Points $D,E$ lie on line $AB$, in the order $D-A-B-E$, with $AD=AC$ and $BE=BC$. The circumcircles of $DBC,EAC$ meet at $X\neq C$, and the circumcircles of $DEC,ABC$ meet at $Y\neq C$. If $DY+EY=2XY$, determine $\angle ACB$.
In trapezoid $ABCD$, none of whose interior angles is right, diagonals $AC,BD$ meet at $E$. Let $P,Q$ be the feet of the perpendiculars from $A,B$ to lines $BC,AD$, respectively. The circumcircles of triangles $CEQ,DEP$ meet again at $F\neq E$. Prove that lines $AP,BQ,EF$ are concurrent or parallel.
For a positive integer $n$, let $v_2(n)$ be the greatest integer $k\geq0$ such that $2^k\mid n$. Suppose $f:\mathbb N\to\mathbb N$ satisfies:
(i) $f(x)\leq3x$ for all $x\in\mathbb N$;
(ii) $v_2(f(x)+f(y))=v_2(x+y)$ for all $x,y\in\mathbb N$.
Prove that for every positive integer $a$ there is exactly one positive integer $x$ such that $f(x)=3a$.
Let $k$ be a positive integer. Consider a game on an infinite one-dimensional board. Initially $n$ chips are placed in its cells, with multiple chips allowed in a cell. Each move is one of:
$(1^\circ)$ Choose two adjacent nonempty cells and move all chips from one to the other.
$(2^\circ)$ Choose a cell with at least two chips and move one chip $k$ cells left and one chip $k$ cells right.
a. If $n\leq k+1$, prove that only finitely many moves can occur.
b. For which values of $k$ can some $n$ and initial placement permit an infinite sequence of moves?
A real-coefficient polynomial $p(x)$ is called *Almerian* if it has the form
$$
p(x) = x^3 + ax^2 + bx + a
$$
and its three roots are positive real numbers in arithmetic progression. Find all Almerian polynomials satisfying $p(7/4)=0$.
Given an integer $n\gt 1$, perform the following operation: if $n$ is even, divide it by 2; if it is odd, add 5. If the resulting number is 1, stop; otherwise repeat the same operation indefinitely. Determine all starting values $n$ for which the process is finite, that is, eventually reaches 1.
Ana and Bernardo play with a bag containing $n\geq1$ stones. Starting with Ana, they alternate turns. If the number of stones is even, a player may take one stone or half the stones; if odd, the player must take one stone. The goal is to take the last stone. For which $n$ does Ana have a winning strategy?
Consider the polynomial
$$
p(x) = (x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a)
$$
Prove that $p(x) \geq 0$ for every $x \in \mathbb{R}$ if and only if $a = b = c$.
Consider the integer sequence $\{f(n)\}_{n=1}^{\infty}$ defined by:
* $f(1) = 1$.
* If $n$ is even, $f(n) = f(n/2)$.
* If $n\gt 1$ is odd and $f(n-1)$ is odd, then $f(n) = f(n-1) - 1$.
* If $n\gt 1$ is odd and $f(n-1)$ is even, then $f(n) = f(n-1) + 1$
a) Calculate $f(2^{2020} - 1)$.
b) Prove that $\{f(n)\}_{n=1}^{\infty}$ is not eventually periodic: there are no positive integers $t,n_0$ such that $f(n+t)=f(n)$ for every $n\geq n_0$.
Let $a_1,a_2,\ldots,a_{2020}$ be 2020 real numbers such that the sum of any 1009 of them is positive. Prove that the sum of all 2020 numbers is positive.
Determine all $n$ for which there exists a convex $n$-gon whose interior angles, measured in degrees, are integers in arithmetic progression and are not all equal.
Let $ABC$ be a triangle with $AB\lt AC$, and let $I$ be its incenter. The incircle touches $BC$ at $D$. Let $E$ be the unique point such that $D$ is the midpoint of $BE$. The line through $E$ perpendicular to $BC$ meets $CI$ at $P$. Prove that $BP\perp AD$.
Let $ABC$ be an acute triangle. Let $D$ be the foot of the altitude to $BC$, $M$ the midpoint of $BC$, and $F$ the intersection of the internal bisector of $\angle BAC$ with $BC$. Determine all triangles for which $F$ is the midpoint of $DM$.
Each point of $A=\{(x,y,z)\in\mathbb Z^3\}$, the integer-coordinate points in three-dimensional space, is assigned one of $p$ colors. Prove that there must be a rectangular parallelepiped, a six-faced polyhedron with every face a rectangle, whose vertices belong to $A$ and all have the same color.