# 2023-10 A pair of primes

Find all pairs of prime numbers $$(p, q)$$ such that $$pq$$ divides $$p^p + q^q + 1$$.

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# 2019-11 Smallest prime

Find the smallest prime number $$p \geq 5$$ such that there exist no integer coefficient polynomials $$f$$ and $$g$$ satisfying
$p | ( 2^{f(n)} + 3^{g(n)})$
for all positive integers $$n$$.

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# 2019-05 Convergence with primes

Let $$p_n$$ be the $$n$$-th prime number, $$p_1 = 2, p_2 = 3, p_3 = 5, \dots$$. Prove that the following series converges:
$\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.$

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# 2013-08 Minimum of a set involving polynomials with integer coefficients

Let $$p$$ be a prime number. Let $$S_p$$ be the set of all positive integers $$n$$ satisfying
$x^n – 1 = (x^p – x + 1) f(x) + p g(x)$
for some polynomials $$f$$ and $$g$$ with integer coefficients. Find all $$p$$ for which $$p^p -1$$ is the minimum of $$S_p$$.

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# 2011-9 Distinct prime factors

Prove that there is a constant c>1 such that if  $$n>c^k$$ for positive integers n and k, then the number of distinct prime factors of $$n \choose k$$ is at least k.

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# 2008-6 Many primes

Let f(x) be a polynomial with integer coefficients. Prove that if f(x) is not constant, then there are infinitely many primes p such that $$f(x)\equiv 0\pmod p$$ has a solution x.

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Let $$n$$ be a positive integer. Let $$a_1,a_2,\ldots,a_k$$ be distinct integers larger than $$n^{n-1}$$ such that $$|a_i-a_j|<n$$ for all $$i,j$$.
Prove that the number of primes dividing $$a_1a_2\cdots a_k$$ is at least $$k$$.
$$n$$은 양의 정수라 하자. $$n^{n-1}$$보다 큰 $$k$$개의 서로 다른 정수 $$a_1,a_2,\ldots,a_k$$가 모든 $$i,j$$에 대해서 $$|a_i-a_j|<n$$을 만족한다고 하자.
이때 $$a_1a_2\cdots a_k$$의 약수인 소수의 개수는 $$k$$개 이상임을 보여라.