Mike rides a bike for $30$ minutes, traveling $8$ miles. He started riding at $20$ miles per hour, but by the end of his journey he was only traveling at $10$ miles per hour. What was his average speed, in miles per hour? [i]Proposed by Nathan Ramesh

Let $f(x)=x^2+x+1$. Determine, with proof, all positive integers $n$ such that $f(k)$ divides $f(n)$ whenever $k$ is a positive divisor of $n$.

A rectangular sheet with sides $a$ and $b$ is fold along a diagonal. Compute the area of the overlapping triangle.

If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$

In the sequence $ 2001, 2002, 2003, \ldots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $ 2001 \plus{} 2002 \minus{} 2003 \equal{} 2000$. What is the $ 2004^\text{th}$ term in this sequence? $ \textbf{(A)} \minus{} \! 2004 \qquad \textbf{(B)} \minus{} \! 2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 4003 \qquad \textbf{(E)}\ 6007$

In the Game of Life, each square in an infinite grid of squares is either shaded or blank. Every day, if a square shares an edge with exactly zero or four shaded squares, it becomes blank the next day. If a square shares an edge with exactly two or three shaded squares, it becomes shaded the next day. Otherwise, it does not change. On day $1$, each square is randomly shaded or blank with equal probability. If the probability that a given square is shaded on day 2 is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, find $a + b.$

Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.

In kindergarten, nurse took $n>1$ identical cardboard rectangles and distributed them to $n$ children; every child got one rectangle. Every child cut his (her) rectangle into several identical squares (squares of different children could be different). Finally, the total number of squares was prime. Prove that initial rectangles was squares.

Four cars participate in a rally on a circular racecourse. They start simultaneously from the same point and go with a constant (but different) speeds. It is known that any three of them meet at some point. Prove that all four of them will meet again at some point.

Given is an acute-angled triangle $ABC$ in which holds $|BC|: |AC| = 3:$2. Let $D$ be the midpoint of the side $\overline{AC}$, and P the midpoint of the segment $\overline{BD}$. A point $X$ is given on the line $AC$ so that $|AX| = |BC|$, where $A$ is between $X$ and $C$. The line $XP$ intersects the side $\overline{BC}$ at point $E$. The line $DE$ intersects the line $AP$ at point $Y$. Prove that the points $A$, $X$, $Y$, $E$ lie on one circle if and only if $|AB| = |BC|$.

Let be given $a\in\{0,1,2, 3,..., 100\}.$ Find all $n \in\{1,2, 3,..., 2013\}$ such that $C_n^{2013} > C_a^{2013}$ , where $C_k^m=\frac{m!}{k!(m -k)!}$.

A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, then $\angle BPC=50^{\circ}$, $\angle CQD=45^{\circ}$, $\angle DRE=40^{\circ}$, $\angle ESF=35^{\circ}$. Let $T$ be an intersection of $BE$ and $CF$. Find $\angle BTC$.

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

Let $n$ be a positive integer. A sequence $a_1$, $a_2$,$ ...$ , $a_n$ is called [i]good [/i] if the following conditions hold: $\bullet$ For each $i \in \{1, 2, ..., n\}$, $1 \le a_i \le n$ $\bullet$ For all positive integers $i, j$ with $1 \le i \le j \le n$, the expression $a_i + a_{i+1} + ...+ a_j$ is not divisible by $ n + 1$. Find the number of good sequences (in terms of $n$).

A lock has $16$ keys arranged in a $4 \times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too (see diagram). Show that no matter what the starting positions are, it is always possible to open this lock. (Only one key at a time can be switched.)

For all natural numbers $n$, let $$A_n=\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}\quad\text{(n many radicals)}$$ [b](a)[/b] Show that for $n\geqslant 2$, $$A_n=2\sin\frac{\pi}{2^{n+1}}$$ [b](b)[/b] Hence or otherwise, evaluate the limit $$\lim_{n\to\infty} 2^nA_n$$

In triangle $ABC$, $\measuredangle CBA=72^\circ$, $E$ is the midpoint of side $AC$, and $D$ is a point on side $BC$ such that $2BD=DC$; $AD$ and $BE$ intersect at $F$. The ratio of the area of triangle $BDF$ to the area of quadrilateral $FDCE$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(15,3), D=(5,1), A=7*dir(72)*dir(B--C), E=midpoint(A--C), F=intersectionpoint(A--D, B--E); draw(E--B--A--C--B^^A--D); label("$A$", A, dir(D--A)); label("$B$", B, dir(E--B)); label("$C$", C, dir(0)); label("$D$", D, SE); label("$E$", E, N); label("$F$", F, dir(80));[/asy] $\text{(A)} \ \frac 15 \qquad \text{(B)} \ \frac 14 \qquad \text{(C)} \ \frac 13 \qquad \text{(D)} \ \frac 25 \qquad \text{(E)} \ \text{none of these}$

The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$. For example, the result of the process applied to $938$ is $102$, since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$, applied three times the result is $1$, and applying it four times the result is $0$. When the process is applied one or more times to an integer $n$, the result is eventually $0$. The number obtained before obtaining $0$ is called the [i]house[/i] of $n$. How many integers less than $26000$ share the same [i]house[/i] as $2012$?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has limits at any point and has no local extrema. Show that: $a)$ $f$ is continuous; $b)$ $f$ is strictly monotone.

Let $n$ be a positive integer. Aavid has a card deck consisting of $ 2n$ cards, each colored with one of $n$ colors such that every color is on exactly two of the cards. The $2n$ cards are randomly ordered in a stack. Every second, he removes the top card from the stack and places the card into an area called the pit. If the other card of that color also happens to be in the pit, Aavid collects both cards of that color and discards them from the pit. Of the $(2n)!$ possible original orderings of the deck, determine how many have the following property: at every point, the pit contains cards of at most two distinct colors.

For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number?

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions: $$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$ for all real numbers $ x,y,t $ with $ t\in [0,1] . $ Prove that: [b]a)[/b] $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $ [b]b)[/b] for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds. $$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$

Let $f:\mathbb{R} \to \mathbb{R}$ be a monotonic function and $a,b,c,d$ be real numbers with $a$ and $c$ nonzero. Prove that if the equalities [center]$\int\limits_x^{x+\sqrt{3}} f(t) \mathrm{d}t=ax+b$ and $\int\limits_x^{x+\sqrt{2}} f(t) \mathrm{d}t=cx+d$[/center] hold for every real number $x,$ then $f$ is a polynomial function of degree one.

A game is played on a $1 \times 1000$ board. There are n chips, all of which are initially in a box near the board. Two players move in turn. The first may choose $17$ chips or less, from either on or off the board. She then puts them into unoccupied cells on the board so that there is no more than one chip in each of the cells. The second player may take off the board any number of chips occupying consecutive cells and put them back in the box. The first player wins if she can put all n chips on the board so that they occupy consecutive cells. (a) Show that she can win if $n = 98$. (b) For what maximal value of $n$ can she win? (A Shapovalov)