From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

A coin is tossed 9 times. Hence $2^{9}$ different outcomes are possible. In how many cases 2 consecutive heads does not appear? [list=1] [*] 34 [*] 55 [*] 89 [*] None of these [/list]

Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

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]

Let $a, \ b \in \mathbb{Z_{+}}$. Denote $f(a, b)$ the number sequences $s_1, \ s_2, \ ..., \ s_a$, $s_i \in \mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \le b$. Show that $f(a, b)=f(b, a)$.

For $n \geq 3$ and $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$ given real numbers we have the following instructions: - place out the numbers in some order in a ring; - delete one of the numbers from the ring; - if just two numbers are remaining in the ring: let $S$ be the sum of these two numbers. Otherwise, if there are more the two numbers in the ring, replace Afterwards start again with the step (2). Show that the largest sum $S$ which can result in this way is given by the formula \[S_{max}= \sum^n_{k=2} \begin{pmatrix} n -2 \\ [\frac{k}{2}] - 1\end{pmatrix}a_{k}.\]

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 ?

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? $\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) \equal{} | S_1 \minus{} S_2 |$. a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd. b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$. c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ 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 ?

On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

On a table near the sea, there are $N$ glass boxes where $N<2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let $S$ be the sum of all values of $N$ for which Sowdha has a winning strategy and let $R$ be the sum of all values of $N$ for which Rafi has a winning strategy. What is the value of $\frac{R-S}{10}$?

Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?

Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\) \[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1p}\\ a_{21} & a_{22} & \ldots & a_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{array}\] such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\). Find \(N \pmod{p}\).

Mary typed a six-digit number, but the two $1$s she typed didn't show. What appeared was $2002$. How many different six-digit numbers could she have typed? $\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }15\qquad\textbf{(E) }20$

Given $m+n$ numbers: $a_i,$ $i = 1,2, \ldots, m,$ $b_j$, $j = 1,2, \ldots, n,$ determine the number of pairs $(a_i,b_j)$ for which $|i-j| \geq k,$ where $k$ is a non-negative integer.

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

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 $

Let $n$ be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular $2n$-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let $P(n)$ denote the number of different ways Luca can make $2n$ jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.) [b]a)[/b] Find the value of $P(n)$ if $n$ is odd. [b]b)[/b] Prove that if $n$ is even, then \[P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n}\left(\varphi\left(\frac{n}d\right)\binom{2d}{d}\right).\] [i]Proposed by Péter Csikvári and Kartal Nagy, Budapest[/i]

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

Consider the digits $1, 2, 3, 4, 5, 6, 7$. (a) Determine the number of seven-digit numbers with distinct digits that can be constructed using the digits above. (b) If we place all of these seven-digit numbers in increasing order, find the seven-digit number which appears in the $2022^{\text{th}}$ position.