Found problems: 30
2009 Tournament Of Towns, 7
Let ${n \choose k}$ be the number of ways that $k$ objects can be chosen (regardless of order) from a set of $n$ objects. Prove that if positive integers k and l are greater than $1$ and less than $n$, then integers ${n \choose k}$ and ${n \choose l}$ have a common divisor greater than $1$.
2015 Saudi Arabia GMO TST, 4
For each positive integer $n$, define $s(n) =\sum_{k=0}^n r_k$, where $r_k$ is the remainder when $n \choose k$ is divided by $3$. Find all positive integers $n$ such that $s(n) \ge n$.
Malik Talbi
2018 Abels Math Contest (Norwegian MO) Final, 4
Find all polynomials $P$ such that
$P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$
$=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$
for all real numbers $x$.
2004 Moldova Team Selection Test, 5
Let $n\in\mathbb{N}$, the set $A=\{(x_1,x_2...,x_n)|x_i\in\mathbb{R}_{+}, i=1,2,...,n\}$ and the function $$f:A\rightarrow\mathbb{R}, f(x_1,...,x_n)=\frac{1}{x_1}+\frac{1}{2x_2}+\ldots+\frac{1}{(n-1)x_{n-1}}+\frac{1}{nx_n}.$$
Prove that $f(\textstyle\binom{n}{1},\binom{n}{2},...,\binom{n}{n-1},\binom{n}{n})=f(2^{n-1},2^{n-2},...,2,1).$
1997 Singapore MO Open, 2
Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.