This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 4

2024 Tuymaada Olympiad, 1
Prove that a positive integer of the form $n^4 +1$ can have more than $1000$ divisors of the form $a^4 +1$ with integral $a$.
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2023 Indonesia MO, 4
Determine whether or not there exists a natural number $N$ which satisfies the following three criteria: 1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$, 2. $N$ only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of $N$ are odd.
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2024 Indonesia MO, 2
The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a [i]fatal[/i] triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$. As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$. The positive integer $N$ is called [i]fatal[/i] if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$. (a) Prove that 16 is not [i]fatal[/i]. (b) Prove that all integers bigger than 16 which are [b]not[/b] an integer multiple of 6 are fatal.
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2024 Bangladesh Mathematical Olympiad, P8
Let $k$ be a positive integer. Show that there exist infinitely many positive integers $n$ such that $\frac{n^n-1}{n-1}$ has at least $k$ distinct prime divisors. [i]Proposed by Adnan Sadik[/i]
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