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I swear this is the last Taiwan TST I'm gonna post

Source: 2020 Taiwan TST Round 1 Mock Exam P3

May 27, 2020

number theoryTaiwan

Problem Statement

Let

N>2^{5000}

be a positive integer. Prove that if

1\leq a_1<\cdots<a_k<100

are distinct positive integers then the number

\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)

has at least

k

distinct prime factors.
Note. Results with

2^{5000}

replaced by some other constant

N_0

will be awarded points depending on the value of

N_0

.
Proposed by Evan Chen

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