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This problem arised in a local forum, proposed by a user named zxt.
Let $f(n)$ be the number of nonnegative integer $k$ not greater than $n$ such that $n \mid \binom{n}{k}$. If for each positive integer $t<m$, we have $f(t)<f(m)$, then we say $m$ is a Weiming number. Prove that there are infinitely many composite...

Background
By the Curry–Howard correspondence, a proof of $A \to B$ is a term $p : A \to B$ in a suitable type theory. For a fixed pair of propositions $(A, B)$, there may be many distinct proof terms — corresponding to different proof strategies.
For example, the infinitude of primes has proofs via:...

This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid of the eigenvalues and turn it into a sequence of algebraic...

The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying
$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$
are absurdly high, namely $$(154476802108746166441951315019919837485664325669565431700026634898253202035277999,$$ $$36875131794129999827197811565225474825492979968971970996283137471637224634055579,$$ $$43736126779286972578612526023713901528165375581...