The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

In Happy City there are $2014$ citizens called $A_1, A_2, \dots , A_{2014}$. Each of them is either [i]happy[/i] or [i]unhappy[/i] at any moment in time. The mood of any citizen $A$ changes (from being unhappy to being happy or vice versa) if and only if some other happy citizen smiles at $A$. On Monday morning there were $N$ happy citizens in the city. The following happened on Monday during the day: the citizen $A_1$ smiled at citizen $A_2$, then $A_2$ smiled at $A_3$, etc., and, finally, $A_{2013}$ smiled at $A_{2014}$. Nobody smiled at anyone else apart from this. Exactly the same repeated on Tuesday, Wednesday and Thursday. There were exactly $2000$ happy citizens on Thursday evening. Determine the largest possible value of $N$.

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Let $ n$ be a positive integer. Show that the numbers \[ \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1}\] are congruent modulo $ 2^n$ to $ 1$, $ 3$, $ 5$, $ \ldots$, $ 2^n \minus{} 1$ in some order. [i]Proposed by Duskan Dukic, Serbia[/i]

$(BEL 6)$ Evaluate $\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}$ in two different ways and prove that $\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4$

Call a subset of $\{1,2,\ldots, n\}$ [i]unfriendly[/i] if no two of its elements are consecutive. Show that the number of unfriendly subsets with $k$ elements is $\binom{n-k+1}{k}.$

Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For example, one can choose 250 red balls, 187 white balls and 463 balls, or one can choose 360 red balls and 540 blue balls.

Show that the equation ${n \choose k}=m^{l}$ has no integral solution with $l \ge 2$ and $4 \le k \le n-4$.

For an odd prime $ p, $ show that $ \sum_{k=1}^{p-1} \frac{k^p-k}{p}\equiv \frac{1+p}{2}\pmod p . $

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number \[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} \] but $ 2^{3k \plus{} 1}$ does not. [i]Author: Waldemar Pompe, Poland[/i]

$a)$ Prove that for all positive integers $n \geq 3$ holds: $$\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2$$ where $\binom{n}{k}$ , with integer $k$ such that $n \geq k \geq 0$, is binomial coefficent $b)$ Let $n \geq 3$ be an odd positive integer. Prove that set $A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\}$ has odd number of odd numbers

For what positive integers $n, k$ (with $k < n$) are the binomial coefficients $${n \choose k- 1} \,\,\, , \,\,\, {n \choose k} \,\,\, , \,\,\, {n \choose k + 1}$$ three successive terms of an arithmetic progression?

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

Consider the binomial coefficients $\binom{n}{k}=\frac{n!}{k!(n-k)!}\ (k=1,2,\ldots n-1)$. Determine all positive integers $n$ for which $\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}$ are all even numbers.

Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles. [i]UK - Sahl Khan[/i]

Compute $\lim\limits_{n \to \infty} \frac{1}{\log \log n} \sum\limits_{k=1}^n (-1)^k \binom{n}{k} \log k.$

What is the remainder of the number $1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}$ when divided by $2008$?

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

Let $q=p^r$ for a prime number $p$ and positive integer $r$. Let $\zeta = e^{\frac{2\pi i}{q}}$. Find the least positive integer $n$ such that \[\sum_{\substack{1\leq k\leq q\\ \gcd(k,p)=1}} \frac{1}{(1-\zeta^k)^n}\] is not an integer. (The sum is over all $1\leq k\leq q$ with $p$ not dividing $k$.) [i]Victor Wang[/i]

Let $S_n$ be number of ordered sets of natural numbers $(a_1;a_2;....;a_n)$ for which $\frac{1}{a_1}+\frac{1}{a_2}+....+\frac{1}{a_n}=1$. Determine 1)$S_{10} mod(2)$. 2)$S_7 mod(2)$. (1) is first problem in 10 grade, (2)- third in 9 grade.

Prove that for any positive integer $n$, the number \[ S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n \] is the sum of two consecutive perfect squares. [i]Dorin Andrica[/i]