Let $k \le n$ be positive integers and $x$ be a real number with $0 \le x < 1/n$. Prove that $${n \choose 0} - {n \choose 1} x +{n \choose 2} x^2 - ... + (-1)^k {n \choose k} x^k > 0$$

Prove that for any prime $p$ in the interval $\left]n, \frac{4n}{3}\right]$, $p$ divides \[\sum^{n}_{j=0}{{n}\choose{j}}^{4}.\]

Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$, (i)$f(a,b) \leq a+b$; (ii)$f(a,f(b,c))=f(f(a,b),c)$ (iii)Both $\binom{f(a,b)}{a}$ and $\binom{f(a,b)}{b}$ are odd numbers.(where $\binom{n}{k}$ denotes the binomial coefficients)

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]

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]

The sum $$\sum_{k=84}^{8000}{k \choose 84}{{8084 - k} \choose 84}$$ can be written as a binomial coefficient $a \choose b$ for integers $a, b$. Find a possible pair $(a, b)$

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]

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

Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

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.

How does one show $$\text{lcm}\left(\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\right)=\frac{\text{lcm}(1,2,\ldots,n+1)}{n+1}$$

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.

Find the coefficient of the fourth term of the expansion of $(x+y)^{15}$.

Prove the equality $${n \choose 0}^2+ {n \choose 1}^2+ {n \choose 2}^2+...+{n \choose n}^2={2n \choose n}$$

Evaluate $$ \sum_{k=0}^{n} \binom{n}{k}k^{2}.$$

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

There is a rectangular plot of size $1 \times n$. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$, the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$. Let $t_n$ denote the number of ways this can be done. For example, clearly $t_1 = 2$ because we can have either a red or a blue tile. Also $t_2 = 5$ since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile. [list=a] [*]Prove that $t_{2n+1} = t_n(t_{n-1} + t_{n+1})$ for all $n > 1$. [*]Prove that $t_n = \sum_{d \ge 0} \binom{n-d}{d}2^{n-2d}$ for all $n >0$. [/list] Here, \[ \binom{m}{r} = \begin{cases} \dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\ 0, &\text{ otherwise} \end{cases}\] for integers $m,r$.

Show that for each natural number $n > 1$ $1 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}$

Find all positive integers $m,n$ with $m \leq 2n$ that solve the equation \[ m \cdot \binom{2n}{n} = \binom{m^2}{2}. \] [i](German MO 2016 - Problem 4)[/i]

Given an integer $k \geq 2$, determine the largest number of divisors the binomial coefficient $\binom{n}{k}$ may have in the range $n-k+1, \ldots, n$ , as $n$ runs through the integers greater than or equal to $k$.

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]

Given $$b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1,$$ prove that $$b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2.$$ Hence, as a corollary, show $$ \lim_{n \to \infty} b_n =2.$$

Define a sequence of integers by $a_0=1$ , and $a_n=\sum_{k=0}^{n-1} \binom{n}{k}a_k$ , $n \geq 1$ . Let $m$ be a positive integer , let $p$ be a prime , and let $q$ and $r$ be non-negative integers . Prove that : $$a_{p^mq+r} \equiv a_{p^{m-1}q+r} \pmod{p^m}$$

Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.