Olympiad problems

PEN A Problems, 112

Tags: algebra , Vieta , induction , number theory , relatively prime , Divisibility Theory , pen

Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.

PEN A Problems, 8

Tags: quadratics , search , modular arithmetic , algebra , Divisibility Theory , pen

The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.

PEN P Problems, 17

Tags: quadratics , Additive Number Theory , pen

Let $p$ be a prime number of the form $4k+1$. Suppose that $r$ is a quadratic residue of $p$ and that $s$ is a quadratic nonresidue of $p$. Show that $p=a^{2}+b^{2}$, where \[a=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-r)}{p}\right), b=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-s)}{p}\right).\] Here, $\left( \frac{k}{p}\right)$ denotes the Legendre Symbol.

PEN H Problems, 13

Tags: modular arithmetic , geometry , 3D geometry , Diophantine Equations , pen

Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]

PEN J Problems, 14

Tags: Divisor Functions , pen , imo shortlist 2001

Find all positive integers $n$ such that ${d(n)}^{3} =4n$.

PEN S Problems, 13

Tags: function , inequalities , induction , Miscellaneous Problems , pen

The sum of the digits of a natural number $n$ is denoted by $S(n)$. Prove that $S(8n) \ge \frac{1}{8} S(n)$ for each $n$.

PEN H Problems, 78

Tags: Diophantine Equations , pen

Let $x, y$, and $z$ be integers with $z>1$. Show that \[(x+1)^{2}+(x+2)^{2}+\cdots+(x+99)^{2}\neq y^{z}.\]

PEN K Problems, 5

Tags: function , arithmetic sequence , Functional Equations , pen

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]

PEN H Problems, 10

Tags: modular arithmetic , Diophantine Equations , pen

Prove that there are unique positive integers $a$ and $n$ such that \[a^{n+1}-(a+1)^{n}= 2001.\]

PEN H Problems, 14

Tags: Diophantine Equations , pen

Show that the equation $x^2 +y^5 =z^3$ has infinitely many solutions in integers $x, y, z$ for which $xyz \neq 0$.

PEN A Problems, 3

Tags: algebra , polynomial , Vieta , function , induction , Divisibility Theory , pen

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.

2018-IMOC, N4

Tags: algebra unsolved , algebra , number theory , pen

Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition \[0 < a_{n+1} - a_n \leq 2001\] for all $n \in \mathbb{N}^{*}$ Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.

PEN A Problems, 48

Tags: Divisibility Theory , pen

Let $n$ be a positive integer. Prove that \[\frac{1}{3}+\cdots+\frac{1}{2n+1}\] is not an integer.

PEN A Problems, 23

Tags: quadratics , Gauss , modular arithmetic , number theory , relatively prime , Divisibility Theory , pen

(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.

PEN E Problems, 7

Tags: modular arithmetic , Euler , pen

Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.

PEN E Problems, 31

Tags: pen , number theory

Suppose $n$ and $r$ are nonnegative integers such that no number of the form $n^2+r-k(k+1) \text{ }(k\in\mathbb{N})$ equals to $-1$ or a positive composite number. Show that $4n^2+4r+1$ is $1$, $9$, or prime.

PEN H Problems, 62

Tags: Diophantine Equations , pen

Solve the equation $7^x -3^y =4$ in positive integers.

PEN A Problems, 117

Tags: Divisibility Theory , pen

Find the smallest positive integer $n$ such that \[2^{1989}\; \vert \; m^{n}-1\] for all odd positive integers $m>1$.

PEN H Problems, 85

Tags: inequalities , Diophantine Equations , pen

Find all integer solutions to $2(x^5 +y^5 +1)=5xy(x^2 +y^2 +1)$.

PEN H Problems, 25

Tags: geometry , 3D geometry , modular arithmetic , Diophantine Equations , pen

What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \cdots, x_{t}$ with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]

2016 Moldova Team Selection Test, 2

Tags: floor function , quadratics , function , pen

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

PEN H Problems, 74

Tags: Diophantine Equations , pen

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{a^{a}}= b^{b}.\]

PEN C Problems, 6

Tags: algebra , polynomial , absolute value , Quadratic Residues , pen

Let $a, b, c$ be integers and let $p$ be an odd prime with \[p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.\] Show that \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]

PEN O Problems, 2

Tags: pen , number theory

Let $p$ be a prime. Find all positive integers $k$ such that the set $\{1,2, \cdots, k\}$ can be partitioned into $p$ subsets with equal sum of elements.

PEN H Problems, 32

Tags: Diophantine Equations , pen

Let $n$ be a natural number. Solve in whole numbers the equation \[x^{n}+y^{n}=(x-y)^{n+1}.\]