When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.

In a sequence of positive integers, a inversion is a pair of positions, where the number in left is greater than the number in right. For example in the sequence $2, 5, 3, 1, 3$ has $5$ inversions{(5,1),(3,1),(5,3),(2,1),(5,3)}. Find the greatest number of inversions in a sequence where the sum of elements is $n$ a) where $n=7$ b) where $n=2019$

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$. [i]Proposed by Adrian Beker[/i]

Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.

A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? $ \displaystyle \textbf{(A)} \ \frac {3}{10} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {1}{2} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {7}{10}$

Define the sequence $A_1, A_2, A_3, \dots$ by $A_1 = 1$ and for $n=1,2,3,\dots$ $$A_{n+1}=\frac{A_n+2}{A_n +1}.$$ Define the sequences $B_1, B_2, B_3,\dots$ by $B_1=1$ and for $n=1,2,3,\dots$ $$B_{n+1}=\frac{B_n^2 +2}{2B_n}.$$ Prove that $B_{n+1}=A_{2^n}$ for all non-negative integers $n$.

Let $M$ be the set of positive integers which do not have a prime divisor greater than 3. For any infinite family of subsets of $M$, say $A_1,A_2,\ldots $, prove that there exist $i\ne j$ such that for each $x\in A_i$ there exists some $y\in A_j $ such that $y\mid x$.

Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

Let $ ABCD$ be a parallelogram with no angle equal to $ 60^{\textrm{o}}$. Find all pairs of points $ E, F$, in the plane of $ ABCD$, such that triangles $ AEB$ and $ BFC$ are isosceles, of basis $ AB$, respectively $ BC$, and triangle $ DEF$ is equilateral. [i]Valentin Vornicu[/i]

Let $ABC$ be an acute-angled triangle. The tangents to the circumcircle of triangle $ABC$ at $B$ and $C$ respectively meet at $D$. The circumcircles of triangles $ABD$ and $ACD$ meet line $BC$ at additional points $E$ and $F$ respectively. Lines $DB$ and $DC$ meet the circumcircle of triangle $DEF$ at additional points $X$ and $Y$ respectively. Let $O$ be the circumcentre of triangle $DEF$. Prove that the circumcircles of triangles $ABC$ and $OXY$ are tangent to each other.

Let $R$ be an associative non-commutative ring and let $n>2$ be a fixed natural number. Assume that $x^n=x$ for all $x\in R$. Prove that $xy^{n-1}=y^{n-1}x$ holds for all $x,y\in R$.

[b]p1.[/b] If $n$ is a positive integer such that $2n+1 = 144169^2$, find two consecutive numbers whose squares add up to $n + 1$. [b]p2.[/b] Katniss has an $n$-sided fair die which she rolls. If $n > 2$, she can either choose to let the value rolled be her score, or she can choose to roll a $n - 1$ sided fair die, continuing the process. What is the expected value of her score assuming Katniss starts with a $6$ sided die and plays to maximize this expected value? [b]p3.[/b] Suppose that $f(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, and that $f(1) = f(2) = f(3) = f(4) = f(5) = f(6) = 7$. What is $a$? [b]p4.[/b] $a$ and $b$ are positive integers so that $20a+12b$ and $20b-12a$ are both powers of $2$, but $a+b$ is not. Find the minimum possible value of $a + b$. [b]p5.[/b] Square $ABCD$ and rhombus $CDEF$ share a side. If $m\angle DCF = 36^o$, find the measure of $\angle AEC$. [b]p6.[/b] Tom challenges Harry to a game. Tom first blindfolds Harry and begins to set up the game. Tom places $4$ quarters on an index card, one on each corner of the card. It is Harry’s job to flip all the coins either face-up or face-down using the following rules: (a) Harry is allowed to flip as many coins as he wants during his turn. (b) A turn consists of Harry flipping as many coins as he wants (while blindfolded). When he is happy with what he has flipped, Harry will ask Tom whether or not he was successful in flipping all the coins face-up or face-down. If yes, then Harry wins. If no, then Tom will take the index card back, rotate the card however he wants, and return it back to Harry, thus starting Harry’s next turn. Note that Tom cannot touch the coins after he initially places them before starting the game. Assuming that Tom’s initial configuration of the coins weren’t all face-up or face-down, and assuming that Harry uses the most efficient algorithm, how many moves maximum will Harry need in order to win? Or will he never win? PS. You had better use hide for answers.

During the first week of the 2008-2009 school year at Jupiter Falls High School, the school holds a fire drill. The $2008$ students in attendance all leave the school and head for the football field. Wendy and several of her friends sit down in a circle on the ground and begin to chat. Wendy and her friend Lilly sit side-by-side, and after a little while decide to swap spots in order to make it easier to talk with different friends. This leads Lilly's boyfriend Nori to offer up a problem, "Suppose we all stood up and took the space that one of our neighboors had been sitting in. In how many ways could we do that?" "I think just four, " offers Wendy, oblivious that Nori is subtly voicing a complaint over Lilly's absence at his side. "We all either move one spot clockwise, or one spot counterclockwise. Unless we can sit on each other." Nori replies, "Oh, right. That's not really what I meant. What I meant was that we can also stay in our own spot, like Beth, Regan, Tom, Burt, and I just did. So, in how many ways can that happen? Assume no two people wind up in the same spot." Wendy pulls out a calculator and writes a program that cycles through all the possibilities. After a couple of minutes she announces, "There are $31$. $\textit{That's}$ a weird number." "Can you solve it generally?" asks Lilly. "Honestly, I'm not sure. I'd need to work on it a bit to know if I could," admits Wendy. Nori adds more complexity to the problem, "How about this: Let $k$ be the number of students in a circle. Then let $m$ be the number of ways we can rearrange ourselves so that each of us is in the same spot or within one spot of where we started, and no two people are ever in the same spot. If $m$ leaves a remainder of $1$ when divided by $5$, how many possible values are there of $k$, where $k$ is at least $3$ and at most $2008$?" Find the answer to Nori's problem.

Find all functions $f:\mathbb{R}\to \mathbb{R}$ so that for any reals $x,y$ the following holds: \[f(x\cdot f(x+y))+f(f(y)\cdot f(x+y))=(x+y)^2\]

Let $a_0, a_1, \ldots, a_9$ and $b_1 , b_2, \ldots,b_9$ be positive integers such that $a_9<b_9$ and $a_k \neq b_k, 1 \leq k \leq 8.$ In a cash dispenser/automated teller machine/ATM there are $n\geq a_9$ levs (Bulgarian national currency) and for each $1 \leq i \leq 9$ we can take $a_i$ levs from the ATM (if in the bank there are at least $a_i$ levs). Immediately after that action the bank puts $b_i$ levs in the ATM or we take $a_0$ levs. If we take $a_0$ levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of $n$ such that after finite number of takings money from the ATM there will be no money in it.

Given $1991$ distinct positive real numbers, the product of any ten of these numbers is always greater than $1$. Prove that the product of all $1991$ numbers is also greater than $1$.

Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.

Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i]

Let $A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})$. Prove that there exist $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}$ such that: $$\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2) $$

Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$

A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?

A table with $m$ rows and $n$ columns is given. At any move one chooses some empty cells such that any two of them lie in different rows and columns, puts a white piece in any of those cells and then puts a black piece in the cells whose rows and columns contain white pieces. The game is over if it is not possible to make a move. Find the maximum possible number of white pieces that can be put on the table.

Solve over non-negative integers the system $$ \begin{cases} x+y+z^2=xyz, \\ z\leq min(x,y). \end{cases} $$

Prove that, for all $n \in \mathbb{N}$ \begin{align*} \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{2n+1} \not\in \mathbb{Z} \end{align*}

Consider all possible functions defined for $x = 1, 2, ..., M$ and taking values $​​y = 1, 2, ..., n$. We denote the set of such functions by $T.$ By $T_0$ we denote the subset of $T$ consisting of functions whose value changes exactly by $ 1$ (in one direction or another) when the argument changes by $1$. Prove that if $M\ge 2n-4$, then among the functions from of the set $T$, there is a function that coincides at least at one point with any function from $T_0$. Specify at least one such function. Prove that if $M <2n-4$, then there is no such function.