Olympiad problems

2008 India Regional Mathematical Olympiad, 4

Determine all the natural numbers $n$ such that $21$ divides $2^{2^{n}}+2^n+1.$

2015 Indonesia MO Shortlist, A2

Tags: natural number , polynomial , algebra , functional equation

Suppose $a$ real number so that there is a non-constant polynomial $P (x)$ such that $\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}$ for each real number $x$, with $x+\pi \ne 0$ and $P(x+\pi)\ne 0$. Show that $a$ is a natural number.

2024 Nepal TST, P2

Tags: natural number , inequalities , function

Let $f: \mathbb{N} \to \mathbb{N}$ be an arbitrary function. Prove that there exist two positive integers $x$ and $y$ which satisfy $f(x+y) \le f(2x+f(y))$. [i](Proposed by David Anghel, Romania)[/i]

1997 Tuymaada Olympiad, 2

Tags: system of equations , natural number , diophantine equation , algebra

Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .

2010 IFYM, Sozopol, 4

Tags: natural number , algebra

Let $x,y\in \mathbb{N}$ and $k=\frac{x^2+y^2}{2xy+1}$. Determine all natural values of $k$.

1998 Czech and Slovak Match, 4

Tags: natural number , functional equation in n , algebra

Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$ .

2011 Ukraine Team Selection Test, 8

Tags: sequence , natural number , algebra

Is there an increasing sequence of integers $ 0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots $ for which the following two conditions are satisfied simultaneously: 1) any natural number can be given as $ {{a} _{i}} + {{a} _{j}} $ for some (possibly equal) $ i \ge 0 $, $ j \ge 0$ , 2) $ {{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16} $ for all natural $ n $?

2019 India PRMO, 12

Tags: natural number , function

A natural number $k > 1$ is called [i]good[/i] if there exist natural numbers $$a_1 < a_2 < \cdots < a_k$$ such that $$\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1$$. Let $f(n)$ be the sum of the first $n$ [i][good[/i] numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer.

2012 IFYM, Sozopol, 2

Tags: number theory , natural number

Find all natural numbers, which cannot be expressed in the form $\frac{a}{b}+\frac{a+1}{b+1}$ where $a,b\in \mathbb{N}$.

2015 Caucasus Mathematical Olympiad, 1

Tags: prime , natural number , product , prime number , number theory

Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.

2022 239 Open Mathematical Olympiad, 4

Tags: prime , natural number , number theory

Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair $(x+y,x)$ or a pair $(2x+y+1,x+y+1)$ from a pair $(x,y).$ At the beginning, the pair $(1,1)$ is presented on the calculator. Prove that for any natural $n$ there is exactly one pair $(n,k)$ that can be obtained using a calculator.