Olympiad problems

1953 Moscow Mathematical Olympiad, 241

Prove that the polynomial $x^{200} y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.

2016 BAMO, 3

Tags: prime factorization , prime factor , factor , number theory

The ${\textit{distinct prime factors}}$ of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$. Let $A=2^k - 2$ and $B= 2^k \cdot A$, where $k$ is an integer ($k \ge 2$). Show that, for every integer $k$ greater than or equal to $2$, [list=i] [*] $A$ and $B$ have the same set of distinct prime factors. [*] $A+1$ and $B+1$ have the same set of distinct prime factors. [/list]

1950 Polish MO Finals, 1

Tags: factor , algebra , polynomial

Decompose the polynomial $$x^8 + x^4 +1$$ to factors of at most second degree.

2018 Singapore Junior Math Olympiad, 4

Tags: factor , sum , divisor , 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: number of divisors , divisor , factor , prime factorization , inequalities , prime , number theory

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)

2014 India PRMO, 1

Tags: number theory , prime , factor , maximum

A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?

1939 Moscow Mathematical Olympiad, 048

Tags: algebra , factoring , factor

Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients

2017 Grand Duchy of Lithuania, 4

Tags: number theory , odd , multiple , factor

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.)

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , factor , positive

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

2012 Swedish Mathematical Competition, 2

Tags: number theory , factor

The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$. Find $m$.

2014 Contests, 1

Tags: number theory , prime , factor , maximum

A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?

1994 Tuymaada Olympiad, 2

Tags: number theory , factor , product , factorization

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.