Found problems: 7
2010 China Team Selection Test, 3
Given positive integer $k$, prove that there exists a positive integer $N$ depending only on $k$ such that for any integer $n\geq N$, $\binom{n}{k}$ has at least $k$ different prime divisors.
1960 Putnam, B6
Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that
$$\sum_{n=1}^{\infty} b_n$$
converges.
2017 Vietnamese Southern Summer School contest, Problem 3
Prove that, for any integer $n\geq 2$, there exists an integer $x$ such that $3^n|x^3+2017$, but $3^{n+1}\not | x^3+2017$.
2010 China Team Selection Test, 3
Given positive integer $k$, prove that there exists a positive integer $N$ depending only on $k$ such that for any integer $n\geq N$, $\binom{n}{k}$ has at least $k$ different prime divisors.
2017-IMOC, N4
Find all integers $n$ such that $n^{n-1}-1$ is square-free.
2024 Bangladesh Mathematical Olympiad, P9
Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.
2014 USA TSTST, 6
Suppose we have distinct positive integers $a, b, c, d$, and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence \begin{align*}
ca &- db \\
ca^2 &- db^2 \\
ca^3 &- db^3 \\
ca^4 &- db^4 \\
&\vdots
\end{align*} and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.