Five people are sitting at a round table. Let $f \ge 0$ be the number of people sitting next to at least one female and $m \ge 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

For a natural number $n{},$ determine the number of ordered pairs $(S,T)$ of subsets of $\{1,2,\ldots,n\}$ for which $s>|T|$ for any element $s\in S$ and $t>|S|$ for any element $t\in T.$

Thee tiles are marked $ X$ and two other tiles are marked $ O$. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $ XOXOX$? $ \textbf{(A)}\ \frac{1}{12}\qquad \textbf{(B)}\ \frac{1}{10}\qquad \textbf{(C)}\ \frac{1}{6}\qquad \textbf{(D)}\ \frac{1}{4}\qquad \textbf{(E)}\ \frac{1}{3}$

If a $5 \times n$ rectangle can be tiled using $n$ pieces like those shown in the diagram, prove that $n$ is even. Show that there are more than $2 \cdot 3^{k-1}$ ways to file a fixed $5 \times 2k$ rectangle $(k \geq 3)$ with $2k$ pieces. (symmetric constructions are supposed to be different.)

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$? $ \textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad $

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn$?$ $\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$

$ABC$ is an equilateral triangle with side length $11$ units. Consider the points $P_1,P_2, \dots, P_10$ dividing segment $BC$ into $11$ parts of unit length. Similarly, define $Q_1, Q_2, \dots, Q_10$ for the side $CA$ and $R_1,R_2,\dots, R_10$ for the side $AB$. Find the number of triples $(i,j,k)$ with $i,j,k \in \{1,2,\dots,10\}$ such that the centroids of triangles $ABC$ and $P_iQ_jR_k$ coincide.

Let $p, k$ and $x$ be positive integers such that $p \geq k$ and $x < \left[ \frac{p(p-k+1)}{2(k-1)} \right]$, where $[q]$ is the largest integer no larger than $q$. Prove that when $x$ balls are put into $p$ boxes arbitrarily, there exist $k$ boxes with the same number of balls.

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.

Sophie has $20$ indistinguishable pairs of socks in a laundry bag. She pulls them out one at a time. After pulling out $30$ socks, the expected number of unmatched socks among the socks that she has pulled out can be expressed in simplest form as $\tfrac{m}{n}$. Find $m+n$.

Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$. [i] (Vlad Robu) [/i]

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

How many [b]distinguishable[/b] rearrangements of the letters in CONTEST have both the vowels first? (For instance, OETCNST is a one such arrangements but OTETSNC is not.) $ \textbf{(A)}\ 60\qquad \textbf{(B)}\ 120\qquad \textbf{(C)}\ 240\qquad \textbf{(D)}\ 720\qquad \textbf{(E)}\ 2520$

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur? $\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$

Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi And Gamatatsu land on?

A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice [b]nice[/b] if at the end nobody has her own ball and it is called [b]tiresome[/b] if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.

We say that a positive integer is guarani if the sum of the number with its reverse is a number that only has odd digits. For example, $249$ and $30$ are guarani, since $249 + 942 = 1191$ and $30 + 03 = 33$. a) How many $2021$-digit numbers are guarani? b) How many $2023$-digit numbers are guarani?

Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl?

For any natural number $ n\ge 2, $ define $ m(n) $ to be the minimum number of elements of a set $ S $ that simultaneously satisfy: $ \text{(i)}\quad \{ 1,n\} \subset S\subset \{ 1,2,\ldots ,n\} $ $ \text{(ii)}\quad $ any element of $ S, $ distinct from $ 1, $ is equal to the sum of two (not necessarily distinct) elements from $ S. $ [b]a)[/b] Prove that $ m(n)\ge 1+\left\lfloor \log_2 n \right\rfloor ,\quad\forall n\in\mathbb{N}_{\ge 2} . $ [b]b)[/b] Prove that there are infinitely many natural numbers $ n\ge 2 $ such that $ m(n)=m(n+1). $ $ \lfloor\rfloor $ denotes the usual integer part.

In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied? [i]Proposed by Gerhard Wöginger, Austria[/i]

A very well known family of mathematicians has three children called [i]Antonia, Bernhard[/i] and [i]Christian[/i]. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days. Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day. Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.