Olympiad problems

2017 China Team Selection Test, 4

Given integer $d>1,m$,prove that there exists integer $k>l>0$, such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$

2017 China Team Selection Test, 4

Given integer $d>1,m$,prove that there exists integer $k>l>0$, such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$

number theory kobayashi Euler's theorem factorization inequalities modular arithmetic p-adic valuation prime factorization

2019 Iran MO (3rd Round), 3

Let $S$ be an infinite set of positive integers and define: $T=\{ x+y|x,y \in S , x \neq y \} $ Suppose that there are only finite primes $p$ so that: 1.$p \equiv 1 \pmod 4$ 2.There exists a positive integer $s$ so that $p|s,s \in T$. Prove that there are infinity many primes that divide at least one term of $S$.

number theory kobayashi mathematical induction modular arithmetic parity pigeonhole principle

STEMS 2021 Math Cat B, Q2

Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$

polynomial power of 2 kobayashi divisibility prime number modular arithmetic p-adic valuation polynomials prime numbers