[color=#000][b](i)[/b][/color] Let $a_1,a_2,...,a_n$ be n real numbers. Show that there exists some real number $\alpha$ such that $a_1+\alpha,a_2+\alpha,...,a_n+\alpha$ are all irrational. \\ [color=#000][b](ii)[/b][/color] Prove that such a satetement is not valid if all these are rquired to be rational. \\ [i]Hint (given in question) : Use Pigeon Hole Principle[/i]

A group of $67$ students pass their examination consisting of $6$ questions, labeled with the numbers $1$ to $6$. A correct answer to question $n$ is quoted $n$ points and for an incorrect answer to the same question a student loses $n$ point. a) Find the least possible positive difference between any $2$ final scores b) Show that at least $4$ participants have the same final score c) Show that at least $2$ students gave identical answer to all six questions.

Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $23$, prove that $M$ contains a subset of $4$ elements whose product is the $4$th power of an integer.

Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$

Let $S$ be a set of $9$ distinct integers all of whose prime factors are at most $3.$ Prove that $S$ contains $3$ distinct integers such that their product is a perfect cube.

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital $\textit{every}$ month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs. Hannah comes up with a plan that the committee likes. Beginning in October, the comittee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that $\textit{must}$ go by (including October when the first set of five teachers is selected, but not September) such that some five-member comittee $\textit{must have}$ taught together twice (all five members are the same during two different months)?

In an $N \times N$ table consisting of small unit squares, some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows, the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $N$?

Consider an $N\times N$ matrix, where $N$ is an odd positive integer, such that all its entries are $-1,0$ or $1$. Consider the sum of the numbers in every line and every column. Prove that at least two of the $2N$ sums are equal.

Find all positive integers $n$ with the property that the set \[\{n,n+1,n+2,n+3,n+4,n+5\}\] can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.

Let $G$ be a connected graph, with degree of all vertices not less then $m \geq 3$, such that there is no path through all vertices of $G$ being in every vertex exactly once. Find the least possible number of vertices of $G.$

Find the smallest integer $n$ satisfying the following condition: regardless of how one colour the vertices of a regular $n$-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.

A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples. What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }24\qquad \textbf{(E) }25$

Do there exist 12 rectangular parallelepipeds $P_1,\,P_2,\ldots,P_{12}$ with edges parallel to coordinate axes $OX,\,OY,\,OZ$ such that $P_i$ and $P_j$ have a common point iff $i\ne j\pm 1$ modulo 12?

Let $A_1, A_2, ..., A_k$ be $5$-element subsets of set $\{1, 2, ..., 23\}$ such that, for all $1 \le i < j \le k$ set $A_i \cap A_j$ has at most three elements. Show that $k \le 2018$.

Eight students solved $8$ problems. a) It turned out that each problem was solved by $5$ students. Prove that there are two students such that each problem is solved by at least one of them. b) If it turned out that each problem was solved by $4$ students, it can happen that there is no pair of students such that each problem is solved by at least one of them. (Give an example.) Proposed by S. Tokarev

A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles. [i]Calvin Deng.[/i]

There is graph $ G_0$ on vertices $ A_1, A_2, \ldots, A_n$. Graph $ G_{n \plus{} 1}$ on vertices $ A_1, A_2, \ldots, A_n$ is constructed by the rule: $ A_i$ and $ A_j$ are joined only if in graph $ G_n$ there is a vertices $ A_k\neq A_i, A_j$ such that $ A_k$ is joined with both $ A_i$ and $ A_j$. Prove that the sequence $ \{G_n\}_{n\in\mathbb{N}}$ is periodic after some term with period $ T \le 2^n$.

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

The sequence $\{x_{n}\}$ is defined by \[x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.\] Prove that the sequence is periodic if and only if $x_{0}$ is irrational.

Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

Exactly 1600 Coconuts are distributed on exactly 100 monkeys, where some monkeys also can have 0 coconuts. Prove that, no matter how you distribute the coconuts, at least 4 monkeys will always have the same amount of coconuts. (The original problem is written in German. So, I apologize when I've changed the original problem or something has become unclear while translating.)