For an integer $a$ and positive integers $n$ and $k$, let $f_k(a, n)$ be the remainder when $a^k$ is divided by $n$. Find the largest composite integer $n\leq 100$ that guarantees the infinite sequence \[ f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots \] to be periodic for all integers $a$ (i.e., for each choice of $a$, there is some positive integer $T$ such that $f_k(a,n) = f_{k+T}(a,n)$ for all $k$).

Given a positive integer $ n$, denote by $ \sigma (n)$ the sum of all positive divisors of $ n$. We say that $ n$ is $ abundant$ if $ \sigma (n)>2n.$ (For example, $ 12$ is abundant since $ \sigma (12)\equal{}28>2 \cdot 12$.) Let $ a,b$ be positive integers and suppose that $ a$ is abundant. Prove that $ ab$ is abundant.

Let $ABCD$ be a square with $2cm$ side length and with center $T$. A rhombus $ARTE$ is drawn where point $E$ lies on line $DC$. What is the area of $ARTE$?

Alien Tanvi has a favorite number, but somehow she’s managed to forget it. She remembers that it can be written as $x^2+\tfrac{1}{x^2},$ where $x$ is a real number satisfying $x^4+4x^2+\tfrac{4}{x^2}+\tfrac{1}{x^4}=523.$ What is Alien Tanvi's favorite number?

Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.

Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? [img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img]

Let $$A=\left( \begin{array}{cc} 4 & -\sqrt{5} \\ 2\sqrt{5} & -3 \end{array} \right) $$ Find all pairs of integers \(m,n\) with \(n \geq 1\) and \(|m| \leq n\) such as all entries of \(A^n-(m+n^2)A\) are integer.

Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let $M$ and $N$ be the expected number of times Josh and Ron tap their screens, respectively. What is $|M-N|$?

Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

[b]p1.[/b] Lisa is playing the piano at a tempo of $80$ beats per minute. If four beats make one measure of her rhythm, how many seconds are in one measure? [b]p2.[/b] Compute the smallest integer $n > 1$ whose base-$2$ and base-$3$ representations both do not contain the digit $0$. [b]p3.[/b] In a room of $24$ people, $5/6$ of the people are old, and $5/8$ of the people are male. At least how many people are both old and male? [b]p4.[/b] Juan chooses a random even integer from $1$ to $15$ inclusive, and Gina chooses a random odd integer from $1$ to $15$ inclusive. What is the probability that Juan’s number is larger than Gina’s number? (They choose all possible integers with equal probability.) [b]p5.[/b] Set $S$ consists of all positive integers less than or equal to $ 2016$. Let $A$ be the subset of $S$ consisting of all multiples of $6$. Let $B$ be the subset of $S$ consisting of all multiples of $7$. Compute the ratio of the number of positive integers in $A$ but not $B$ to the number of integers in $B$ but not $A$. [b]p6.[/b] Three peas form a unit equilateral triangle on a flat table. Sebastian moves one of the peas a distance $d$ along the table to form a right triangle. Determine the minimum possible value of $d$. [b]p7.[/b] Oumar is four times as old as Marta. In $m$ years, Oumar will be three times as old as Marta will be. In another $n$ years after that, Oumar will be twice as old as Marta will be. Compute the ratio $m/n$. [b]p8.[/b] Compute the area of the smallest square in which one can inscribe two non-overlapping equilateral triangles with side length $ 1$. [b]p9.[/b] Teemu, Marcus, and Sander are signing documents. If they all work together, they would finish in $6$ hours. If only Teemu and Sander work together, the work would be finished in 8 hours. If only Marcus and Sander work together, the work would be finished in $10$ hours. How many hours would Sander take to finish signing if he worked alone? [b]p10.[/b]Triangle $ABC$ has a right angle at $B$. A circle centered at $B$ with radius $BA$ intersects side $AC$ at a point $D$ different from $A$. Given that $AD = 20$ and $DC = 16$, find the length of $BA$. [b]p11.[/b] A regular hexagon $H$ with side length $20$ is divided completely into equilateral triangles with side length $ 1$. How many regular hexagons with sides parallel to the sides of $H$ are formed by lines in the grid? [b]p12[/b]. In convex pentagon $PEARL$, quadrilateral $PERL$ is a trapezoid with side $PL$ parallel to side $ER$. The areas of triangle $ERA$, triangle $LAP$, and trapezoid $PERL$ are all equal. Compute the ratio $\frac{PL}{ER}$. [b]p13.[/b] Let $m$ and $n$ be positive integers with $m < n$. The first two digits after the decimal point in the decimal representation of the fraction $m/n$ are $74$. What is the smallest possible value of $n$? [b]p14.[/b] Define functions $f(x, y) = \frac{x + y}{2} - \sqrt{xy}$ and $g(x, y) = \frac{x + y}{2} + \sqrt{xy}$. Compute $g (g (f (1, 3), f (5, 7)), g (f (3, 5), f (7, 9)))$. [b]p15.[/b] Natalia plants two gardens in a $5 \times 5$ grid of points. Each garden is the interior of a rectangle with vertices on grid points and sides parallel to the sides of the grid. How many unordered pairs of two non-overlapping rectangles can Nataliia choose as gardens? (The two rectangles may share an edge or part of an edge but should not share an interior point.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Call a $ 3$-digit number [i]geometric[/i] if it has $ 3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

Solve $2a^2+3a-44=3p^n$ in positive integers where $p$ is a prime.

For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences \[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4. [i]Proposed by Gerhard Wöginger, Austria[/i]

A [i]change [/i] in a natural number $n$ consists of adding a pair of zeros between two digits or at the end of the decimal representation of $n$. A [i]countryman [/i] of $n$ is a number that can be obtained from one or more changes in $n$. For example. $40041$, $4410000$ and $4004001$ are all countrymen from $441$. Determine all the natural numbers $n$ for which there is a natural number m with the property that $n$ divides $m$ and all the countrymen of $m$.