$N$ teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order $\mathcal{A} = (a_1 \ge a_2 \ge \cdots \ge a_N )$ is known, as well as which team obtained which score. Find the number of sequences $\mathcal{A}$ such that the outcome of all matches is uniquely determined by this information. [I]Proposed by Dominic Yeo, United Kingdom.[/i]

Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied: $(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity. $(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even . [i]Proposed by Poland.[/i]

A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is $\textbf{(A)}\ \dfrac{5}{32} \qquad \textbf{(B)}\ \dfrac{11}{64} \qquad \textbf{(C)}\ \dfrac{3}{16} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{1}{2}$

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? $\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

The spectators are seated in a row with no empty places. Each is in a seat which does not match the spectator's ticket. An usher can order two spectators in adjacent seats to trade places unless one of them is already seated correctly. Is it true that from any initial arrangement, the spectators can be brought to their correct seats?

Let $n$ and $t$ be positive integers. What is the number of ways to place $t$ dominoes $(1\times 2$ or $2\times 1$ rectangles) in a $2\times n$ table so that there is no $2\times 2$ square formed by $2$ dominoes and each $2\times 3$ rectangle either does not have a horizontal domino in the middle and last cell in the first row or does not have a horizontal domino in the first and middle cell in the second row (or both)?

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]

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 $f(a, b)=f(b, a)$.

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$

In the reality show [i]Big Sister Brasil[/i], it is said that there is a [i]treta[/i] if two people are friends with each other and enemies with a third one. For audience purposes, the broadcaster wants a lot of [i]tretas[/i]. If friendship and enmity are reciprocal relationships, given $n$ people, what is the maximum number of [i]tretas[/i]?

An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

Let $n \geq 2$ be a positive integer, and suppose buildings of height $1, 2, \ldots, n$ are built in a row on a street. Two distinct buildings are said to be $\emph{roof-friendly}$ if every building between the two is shorter than both buildings in the pair. For example, if the buildings are arranged $5, 3, 6, 2, 1, 4,$ there are $8$ roof-friendly pairs: $(5, 3), (5, 6), (3, 6), (6, 2), (6, 4), (2, 1),$ $(2, 4), (1, 4).$ Find, with proof, the minimum and maximum possible number of roof-friendly pairs of buildings, in terms of $n.$

A $10$ digit number is called interesting if its digits are distinct and is divisible by $11111$. Then find the number of interesting numbers.

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

Let $x_1,\ldots,x_n$ be a sequence with each term equal to $0$ or $1$. Form a triangle as follows: its first row is $x_1,\ldots,x_n$ and if a row if $a_1, a_2, \ldots, a_m$, then the next row is $a_1 + a_2, a_2 + a_3, \ldots, a_{m-1} + a_m$ where the addition is performed modulo $2$ (so $1+1=0$). For example, starting from $1$, $0$, $1$, $0$, the second row is $1$, $1$, $1$, the third one is $0$, $0$ and the fourth one is $0$. A sequence is called good it is the same as the sequence formed by taking the last element of each row, starting from the last row (so in the above example, the sequence is $1010$ and the corresponding sequence from last terms is $0010$ and they are not equal in this case). How many possibilities are there for the sequence formed by taking the first element of each row, starting from the last row, which arise from a good sequence?

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.

How many triangles appear in the diagram below? [asy] import graph; size(4.4cm); real labelscalefactor = 0.5; pen dotstyle = black; draw((-2,5)--(-2,1)); draw((-2,5)--(2,5)); draw((2,5)--(2,1)); draw((-2,1)--(2,1)); draw((0,5)--(0,1)); draw((-2,3)--(2,3)); draw((-1,5)--(-1,1)); draw((1,5)--(1,1)); draw((-2,2)--(2,2)); draw((-2,4)--(2,4)); draw((1,5)--(-2,2)); draw((-2,2)--(-1,1)); draw((-1,1)--(2,4)); draw((2,4)--(1,5)); draw((-1,5)--(-2,4)); draw((-2,4)--(1,1)); draw((1,1)--(2,2)); draw((2,2)--(-1,5)); [/asy]

In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t), N_2(t), N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each t during the test, \[|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .\]

Let $N$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3 + 1\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$

In a class among 80 students number of boys is 40 and number of girls is 40. 50 of the students use spectacles. Which of the following is correct? [list=1] [*] Only 10 boys use spectacles [*] Only 20 girls use spectacles [*] At most 25 boys do not use spectacles [*] At most 30 girls do not use spectacles [/list]

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?

In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$