Olympiad problems

1994 Tuymaada Olympiad, 2

The set of numbers $M=\{4k-3 | k\in N\}$ is considered. A number of of this set is called “simple” if it is impossible to put in the form of a product of numbers from $M$ other than $1$. Show that in this set, the decomposition of numbers in the product of "simple" factors is ambiguous.

2017 Grand Duchy of Lithuania, 4

Tags: number theory , odd , factors , multiple

Show that there are infinitely many positive integers $n$ such that the number of distinct odd prime factors of $n(n + 3)$ is a multiple of $3$. (For instance, $180 = 2^2 \cdot 3^2 \cdot 5$ has two distinct odd prime factors and $840 = 2^3 \cdot 3 \cdot 5 \cdot 7$ has three.)

2018 Singapore Junior Math Olympiad, 4

Tags: Sum , factors , Divisors , number theory

Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.

2019 Tournament Of Towns, 1

Tags: inequalities , number theory , prime , number of divisors , prime factorization , factors , Divisors

Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity a) not greater than the complexity of $n$. b) less than the complexity of $n$. (Boris Frenkin)

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: factors , positive , number theory

For every positive integer $n$ let $\tau (n)$ denote the number of its positive factors. Determine all $n \in N$ that satisfy the equality $\tau (n) = \frac{n}{3}$