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.

The Mictlán is an $n\times n$ board and each border of each $1\times 1$ cell is painted either purple or orange. Initially, a catrina lies inside a $1\times 1$ cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple? [i]Proposed by CDMX[/i]

Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}

In an acute triangle $ABC$, let $ \overline {AK}, \overline {BL}, \overline {CM} $ be the altitudes of the triangle concurrent at the point $ H $ and let $ P $ the midpoint of $ \overline {AH} $. Let's define $ S = \overline {BH} \cap \overline {MK} $ and $ T = \overline {LP} \cap \overline {AB} $. Show that $ \overline {TS} \perp \overline {BC} $

For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.

Find all pairs of real numbers $k, l$ such that inequality $ka^2 + lb^2> c^2$ applies to the lengths of sides $a, b, c$ of any triangle.

Bertalan thought about a $4$-digit positive number. Then he draw a simple graph on $4$ vertices and wrote the digits of the number to the vertices of the graph in such a way that every vertex received exactly the degree of the vertex. In how many ways could he think about? In a simple graph every edge connects two different vertices, and between two vertices at most one edge can go.