There are three men and eleven women taking a dance class. In how many different ways can each man be paired with a woman partner and then have the eight remaining women be paired into four pairs of two?

A set $S=\{ s_1,s_2,...,s_k\}$ of positive real numbers is "polygonal" if $k\geq 3$ and there is a non-degenerate planar $k-$gon whose side lengths are exactly $s_1,s_2,...,s_k$; the set $S$ is multipolygonal if in every partition of $S$ into two subsets,each of which has at least three elements, exactly one of these two subsets in polygonal. Fix an integer $n\geq 7$. (a) Does there exist an $n-$element multipolygonal set, removal of whose maximal element leaves a multipolygonal set? (b) Is it possible that every $(n-1)-$element subset of an $n-$element set of positive real numbers be multipolygonal?

[b]p1.[/b] Let $U = \{-2, 0, 1\}$ and $N = \{1, 2, 3, 4, 5\}$. Let $f$ be a function that maps $U$ to $N$. For any $x \in U$, $x + f(x) + xf(x)$ is an odd number. How many $f$ satisfy the above statement? [b]p2.[/b] Around a circle are written all of the positive integers from $ 1$ to $n$, $n \ge 2$ in such a way that any two adjacent integers have at least one digit in common in their decimal expressions. Find the smallest $n$ for which this is possible. [b]p3.[/b] Michael loses things, especially his room key. If in a day of the week he has $n$ classes he loses his key with probability $n/5$. After he loses his key during the day he replaces it before he goes to sleep so the next day he will have a key. During the weekend(Saturday and Sunday) Michael studies all day and does not leave his room, therefore he does not lose his key. Given that on Monday he has 1 class, on Tuesday and Thursday he has $2$ classes and that on Wednesday and Friday he has $3$ classes, what is the probability that loses his key at least once during a week? [b]p4.[/b] Given two concentric circles one with radius $8$ and the other $5$. What is the probability that the distance between two randomly chosen points on the circles, one from each circle, is greater than $7$ ? [b]p5.[/b] We say that a positive integer $n$ is lucky if $n^2$ can be written as the sum of $n$ consecutive positive integers. Find the number of lucky numbers strictly less than $2015$. [b]p6.[/b] Let $A = \{3^x + 3^y + 3^z|x, y, z \ge 0, x, y, z \in Z, x < y < z\}$. Arrange the set $A$ in increasing order. Then what is the $50$th number? (Express the answer in the form $3^x + 3^y + 3^z$). [b]p7.[/b] Justin and Oscar found $2015$ sticks on the table. I know what you are thinking, that is very curious. They decided to play a game with them. The game is, each player in turn must remove from the table some sticks, provided that the player removes at least one stick and at most half of the sticks on the table. The player who leaves just one stick on the table loses the game. Justin goes first and he realizes he has a winning strategy. How many sticks does he have to take off to guarantee that he will win? [b]p8.[/b] Let $(x, y, z)$ with $x \ge y \ge z \ge 0$ be integers such that $\frac{x^3+y^3+z^3}{3} = xyz + 21$. Find $x$. [b]p9.[/b] Let $p < q < r < s$ be prime numbers such that $$1 - \frac{1}{p} -\frac{1}{q} -\frac{1}{r}- \frac{1}{s}= \frac{1}{pqrs}.$$ Find $p + q + r + s$. [b]p10.[/b] In ”island-land”, there are $10$ islands. Alex falls out of a plane onto one of the islands, with equal probability of landing on any island. That night, the Chocolate King visits Alex in his sleep and tells him that there is a mountain of chocolate on one of the islands, with equal probability of being on each island. However, Alex has become very fat from eating chocolate his whole life, so he can’t swim to any of the other islands. Luckily, there is a teleporter on each island. Each teleporter will teleport Alex to exactly one other teleporter (possibly itself) and each teleporter gets teleported to by exactly one teleporter. The configuration of the teleporters is chosen uniformly at random from all possible configurations of teleporters satisfying these criteria. What is the probability that Alex can get his chocolate? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

Define the sequence $\{a_i\}$ by $a_0=1$, $a_1=4$, and $a_{n+1}=5a_n-a_{n-1}$ for all $n\geq 1$. Show that all terms of the sequence are of the form $c^2+3d^2$ for some integers $c$ and $d$.

A quarter of an checkered plane is given, infinite to the right and up. All its rows and columns are numbered starting from $0$. All cells with coordinates $(2n, n)$, were cut out from this figure, starting from $n = 1$. In each of the remaining cells they wrote a number, the number of paths from the corner cell to this one (you can only walk up and to the right and you cannot pass through the removed cells). Prove that for each removed cell the numbers to the left and below it differ by exactly $2$.

Let $ABC$ be a triangle. Let $X$ be the point on side $AB$ such that $\angle{BXC} = 60^{\circ}$. Let $P$ be the point on segment $CX$ such that $BP\bot AC$. Given that $AB = 6, AC = 7,$ and $BP = 4,$ compute $CP$.

If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily $\text{(A)} \ pq \qquad \text{(B)} \ \frac{1}{pq} \qquad \text{(C)} \ \frac{p}{q^2} \qquad \text{(D)} \ \frac{q}{p^2} \qquad \text{(E)} \ \frac{p}{q}$

Eight bulbs are arranged on a circle. In every step we perform the following operation: We simultaneously switch off all those bulbs whose two neighboring bulbs are in different states, and switch on the other bulbs. Prove that after at most four steps all the bulbs will be switched on.

What is the remainder when $128^{2023}$ is divided by $126$?

On top of a thin square cake are triangular chocolate chips which are mutually disjoint. Is it possible to cut the cake into convex polygonal pieces each containing exactly one chip?

Let $p$ a positive prime number and $\mathbb{F}_{p}$ the set of integers $mod \ p$. For $x\in \mathbb{F}_{p}$, define $|x|$ as the cyclic distance of $x$ to $0$, that is, if we represent $x$ as an integer between $0$ and $p-1$, $|x|=x$ if $x<\frac{p}{2}$, and $|x|=p-x$ if $x>\frac{p}{2}$ . Let $f: \mathbb{F}_{p} \rightarrow \mathbb{F}_{p}$ a function such that for every $x,y \in \mathbb{F}_{p}$ \[ |f(x+y)-f(x)-f(y)|<100 \] Prove that exist $m \in \mathbb{F}_{p}$ such that for every $x \in \mathbb{F}_{p}$ \[ |f(x)-mx|<1000 \]

Let $ABC$ be an acute scalene triangle with $O$ as its circumcenter. Point $P$ lies inside triangle $ABC$ with $\angle PAB = \angle PBC$ and $\angle PAC = \angle PCB$. Point $Q$ lies on line $BC$ with $QA = QP$. Prove that $\angle AQP = 2\angle OQB$.

Let $P$ be a polynomial satisfying $P(x + 1) + P(x - 1) = x^3$ for all real numbers $x$. Find the value of $P(12)$.

Let $g_1$ and $g_2$ be some lines, which intersect in point $A$. A circle $k_1$ is tangent to $g_1$ at point $A$ and intersects $g_2$ for a second time in $C$. A circle $k_2$ is tangent to $g_2$ at point $A$ and intersects $g_1$ for a second time in $D$. The circles $k_1$ and $k_2$ intersect for a second time in point $B$. Prove that, if $\frac{AC}{AD}=\sqrt{2}$, then $\frac{BC}{BD}=2$.

A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil $ or $\lfloor x\rfloor $ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil $ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$.)

During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic, \[x^2-sx+p,\] with roots $r_1$ and $r_2$. He notices that \[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\cdots=r_1^{2007}+r_2^{2007}.\] He wonders how often this is the case, and begins exploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of \[\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}}.\] Help Michael by computing this maximum.

Find the largest value of $a^b$ such that the positive integers $a,b>1$ satisfy $$a^bb^a+a^b+b^a=5329$$

A triangle with vertices $(6,5)$, $(8,-3)$, and $(9,1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles? $\textbf{(A) }9\qquad \textbf{(B) }\dfrac{28}{3}\qquad \textbf{(C) }10\qquad \textbf{(D) }\dfrac{31}{3}\qquad \textbf{(E) }\dfrac{32}{3}\qquad$

Show that a set $A$ consisting of $16$ consecutive non-negative integers can be partitioned in two disjoint sets $X$ and $Y$ each containing $8$ elements so that \(\sum\limits_{x\in X}x^k=\sum\limits_{y\in Y} y^k,\) for $k=1,2,3.$

If $a$, $6$, and $b$, in that order, form an arithmetic sequence, compute $a+b$.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

A set $S$ consists of $1992$ positive integers among whose units digits all $10$ digits occur. Show that there is such a set $S$ having no nonempty subset $S_{1}$ whose sum of elements is divisible by $2000$.

Consider the family of nonisosceles triangles $ABC$ satisfying the property $AC^2 + BC^2 = 2 AB^2$. Points $M$ and $D$ lie on side $AB$ such that $AM = BM$ and $\angle ACD = \angle BCD$. Point $E$ is in the plane such that $D$ is the incenter of triangle $CEM$. Prove that exactly one of the ratios \[ \frac{CE}{EM}, \quad \frac{EM}{MC}, \quad \frac{MC}{CE} \] is constant.

Let $a=1+10^{-4}$. Consider some $2023\times 2023$ matrix with each entry a real in $[1,a]$. Let $x_i$ be the sum of the elements of the $i$-th row and $y_i$ be the sum of the elements of the $i$-th column for each integer $i\in [1,n]$. Find the maximum possible value of $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$ (the answer may be expressed in terms of $a$).