Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$.

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form $$ \frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)} $$ where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determine $a+b+c+p+q$.

Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

Odysseus on his ten year return voyage to Ithaca sailed between two monsters. On one side, the creature Charybdis periodically sucked the oceans such that a whirlpool formed. On the opposing side the creature Scylla would lunge down from above and devour one sailor in each of her many mouths. Odysseus opted to sail near Scylla skirting Charybdis by 500 m. At this distance, the maximum drop in water level of the ocean was 0.2 m from between when Charybdis was draining the oceans and when he was not. At Charybdis' mouth the funnel of the whirlpool is 25 m wide. Assume that the oceans are perfectly calm and that there are no intermolecular attractions between water molecules. (a) How deep is Charybdis under water? (b) The boat with a crew of 40 men weighs 5,000 kg. Each crew member displaces 3 kg of water at a velocity of 5 m/s every stroke every second. If at some point Odysseus is traveling radially away from Charybdis, what is the closest his ship can be without being sucked in? Assume that Odysseus' vessel has an extremely shallow draft (low friction). [i]Problem proposed by Brian Yue[/i]

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

Solve the following Diophantines. [b]a)[/b] $ x^2+y^2=6z^2 $ [b]b)[/b] $ x^2+y^2-2x+4y-1=0 $ [i]Dan Negulescu[/i]

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.

Let $p \geq 5$ be a prime and let \begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*} where $q_i$ are primes. Prove, \begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}

An $8$ by $8$ board (with $64$ $1$ by $1$ squares) is painted white. We are allowed to choose any rectangle consisting of $3$ of the $64$ squares and paint each of the $3$ squares in the opposite colour (the white ones black, the black ones white). Is it possible to paint the entire board black by means of such operations? (IS Rubanov, Kirov)

If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?

In triangle $ ABC$, $ AB\equal{}AC$, $ \angle A\equal{}40^\circ$. Point $ O$ is within the triangle with $ \angle OBC \cong \angle OCA$. The number of degrees in angle $ BOC$ is: $ \textbf{(A)}\ 110 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 70$

Given a convex polyhedron with 2022 faces. In 3 arbitary faces, there are already number $26; 4$ and $2022$ (each face contains 1 number). They want to fill in each other face a real number that is an arithmetic mean of every numbers in faces that have a common edge with that face. Prove that there is only one way to fill all the numbers in that polyhedron.

Determine all $n \in \mathbb{N}$ for which [list][*] $n$ is not the square of any integer, [*] $\lfloor \sqrt{n}\rfloor ^3$ divides $n^2$. [/list]

Positive integers are put into the following table. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{tabular} Find the number of the line and column where the number $2015$ stays.