Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?

A non self-intersecting polygon is given in a Cartesian coordinate system such that its perimeter contains no lattice points, and its vertices have no integer coordinates. A point is called semi-integer if exactly one of its coordinates is an integer. Let $P_1, P_2,\ldots, P_k$ denote the semi-integer points on the perimeter of the polygon. Let ni denote the floor of the non-integer coordinate of $P_i$. Prove that integers $n_1,n_2,\ldots ,n_k$ can be divided into two groups with the same sum. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Let $C_1 , C_2$ be two circles in the plane intersecting at two distinct points. Let $P$ be the midpoint of a variable chord $AB$ of $C_2$ with the property that the circle on $AB$ as diameter meets $C_1$ at a point $T$ such that $P T$ is tangent to $C_1$ . Find the locus of $P$ .

Let $\Delta ABC$ be an acute scalene triangle with $AC<BC$, an orthocenter $H$ and altitudes $AE$, $BF$. The points $E'$ and $F'$ are symmetrical to $E$ and $F$ with respect to $A$ and $B$ respectively. Point $O$ is the center of the circumscribed circle of $ABC$ and $M$ is the midpoint of $AB$. Let $N$ be the midpoint of $OM$. Prove that the tangent through $H$ to the circumscribed circle of $\Delta E'HF'$ is perpendicular to line $CN$.

Each sidelength of a convex quadrilateral $ABCD$ is not less than $1$ and not greater than $2$. The diagonals of this quadrilateral meet at point $O$. Prove that $S_{AOB}+ S_{COD} \le 2(S_{AOD}+ S_{BOC})$.

$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$

Suppose $a, b, c,$ and $d$ are pairwise distinct positive perfect squares such that $a^b = c^d.$ Compute the smallest possible value of $a + b + c + d.$

Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$. $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $

Aaron is trying to write a program to compute the terms of the sequence defined recursively by $a_0=0$, $a_1=1$, and \[a_n=\begin{cases}a_{n-1}-a_{n-2}&n\equiv0\pmod2\\2a_{n-1}-a_{n-2}&\text{else}\end{cases}\] However, Aaron makes a typo, accidentally computing the recurrence by \[a_n=\begin{cases}a_{n-1}-a_{n-2}&n\equiv0\pmod3\\2a_{n-1}-a_{n-2}&\text{else}\end{cases}\] For how many $0\le k\le2016$ did Aaron coincidentally compute the correct value of $a_k$?

circle is inscribed in an equilateral triangle of side length $1$. Tangents to the circle are drawn that cut off equilateral triangles at each corner. Circles are inscribed in each of these equilateral triangles. If this process is repeated infinitely many times, what is the sum of the areas of all the circles? [img]https://cdn.artofproblemsolving.com/attachments/c/e/ef4000989155708db8cfa674dd00857afb9919.png[/img]

Triangle $ABC$ is a right triangle with $AC=7,$ $BC=24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD=BD=15.$ Given that the area of triangle $CDM$ may be expressed as $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$

Given a positive integer \( n \), we denote by \( P(n) \) the result of multiplying all the digits of \( n \). Find a number \( m \) with ten digits, none of them zero, with the following property: $$P\left(m+P(m)\right)= P (m)$$

Let $x$ be a real number such that $x^2 -x+1 = 7$ and $x^2 +x+1 = 13$. Compute the value of $x^4$.

Let $ABC$ be an acute-angled triangle and $F$ a point in its interior such that $$ \angle AFB = \angle BFC = \angle CFA = 120^{\circ}.$$ Prove that the Euler lines of the triangles $AFB, BFC$ and $CFA$ are concurrent.

Points $A$, $B$, $C$, and $D$ lie on a circle $\Gamma$, in that order, with $AB=5$ and $AD=3$. The angle bisector of $\angle ABC$ intersects $\Gamma$ at point $E$ on the opposite side of $\overleftrightarrow{CD}$ as $A$ and $B$. Assume that $\overline{BE}$ is a diameter of $\Gamma$ and $AC=AE$. Compute $DE$.

Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.

There are $ n$ voters and $ m$ candidates. Every voter makes a certain arrangement list of all candidates (there is one person in every place $ 1,2,...m$) and votes for the first $ k$ people in his/her list. The candidates with most votes are selected and say them winners. A poll profile is all of this $ n$ lists. If $ a$ is a candidate, $ R$ and $ R'$ are two poll profiles. $ R'$ is $ a\minus{}good$ for $ R$ if and only if for every voter; the people which in a worse position than $ a$ in $ R$ is also in a worse position than $ a$ in $ R'$. We say positive integer $ k$ is monotone if and only if for every $ R$ poll profile and every winner $ a$ for $ R$ poll profile is also a winner for all $ a\minus{}good$ $ R'$ poll profiles. Prove that $ k$ is monotone if and only if $ k>\frac{m(n\minus{}1)}{n}$.

Let $n$ be a natural number. An integer $a>2$ is called $n$-decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$, where $d\neq n$ is a natural divisor of $n$. Find all composite $n\in \mathbb{N}$, for which there's an $n$-decomposable number.

[b]N[/b]ew this year at HMNT: the exciting game of RNG baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number n uniformly at random from $\{0, 1, 2, 3, 4\}$. Then, the following occurs: • If $n>0$, then the player and everyone else currently on the field moves (counterclockwise) around the square by n bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point. • If $n=0$ (a strikeout), then the game ends immediately; the team does not score any more points. What is the expected number of points that a given team will score in this game?

Consider triangles whose each side length squared is a rational number. Is it true that (a) the square of the circumradius of every such triangle is rational; (b) the square of the inradius of every such triangle is rational?

In a triangle $ABC$, the incircle centered at $I$ is tangent to the sides $BC, AC$ and $AB$ at $D, E$ and $F$, respectively. Let $X, Y$ and $Z$ be the feet of the perpendiculars drawn from $A, B$ and $C$ to a line $\ell$ passing through $I$. Prove that $DX, EY$ and $FZ$ are concurrent.

Find all prime numbers $p$ such that $\frac{3^{p-1} - 1}{p}$ is a perfect square.

Let $f(x) = x^2 + ax + b cos x$. Find all values of parameter$ a$ and $b$, for which the equations $f(x) = 0$ and $f(f(x)) = 0 $have the same non-empty sets of real roots.

Let $ x,y,z$ be positive real numbers satisfying $ x\plus{}y\plus{}z\equal{}xyz$. Prove that: \[\frac1{\sqrt{1+x^2}}+\frac1{\sqrt{1+y^2}}+\frac1{\sqrt{1+z^2}}\leq\frac{3}{2}\]

On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.