Olympiad problems

PEN A Problems, 11

Tags: pigeonhole principle , linear algebra , matrix , modular arithmetic , Divisibility Theory , pen

Let $a, b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$.

PEN H Problems, 36

Tags: Diophantine Equations , pen

Prove that the equation $a^2 +b^2 =c^2 +3$ has infinitely many integer solutions $(a, b, c)$.

PEN A Problems, 6

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

[list=a][*] Find infinitely many pairs of integers $a$ and $b$ with $1<a<b$, so that $ab$ exactly divides $a^{2}+b^{2}-1$. [*] With $a$ and $b$ as above, what are the possible values of \[\frac{a^{2}+b^{2}-1}{ab}?\] [/list]

PEN E Problems, 1

Tags: pen , composite numbers , number theory

Prove that the number $512^{3} +675^{3}+ 720^{3}$ is composite.

PEN B Problems, 3

Tags: modular arithmetic , Primitive Roots , pen

Show that for each odd prime $p$, there is an integer $g$ such that $1<g<p$ and $g$ is a primitive root modulo $p^n$ for every positive integer $n$.

PEN A Problems, 1

Tags: quadratics , Vieta , algebra , ratio , number theory , relatively prime , pen

Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.

2007 China Team Selection Test, 3

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 A Problems, 15

Tags: number theory , least common multiple , Divisibility Theory , pen

Suppose that $k \ge 2$ and $n_{1}, n_{2}, \cdots, n_{k}\ge 1$ be natural numbers having the property \[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\] Show that $n_{1}=n_{2}=\cdots=n_{k}=1$.

PEN L Problems, 7

Tags: Linear Recurrences , pen

Let $m$ be a positive integer. Define the sequence $\{a_{n}\}_{n \ge 0}$ by \[a_{0}=0, \; a_{1}=m, \; a_{n+1}=m^{2}a_{n}-a_{n-1}.\] Prove that an ordered pair $(a, b)$ of non-negative integers, with $a \le b$, gives a solution to the equation \[\frac{a^{2}+b^{2}}{ab+1}= m^{2}\] if and only if $(a, b)$ is of the form $(a_{n}, a_{n+1})$ for some $n \ge 0$.

PEN L Problems, 1

Tags: induction , algebra , binomial theorem , Linear Recurrences , pen

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{0}=0, \; a_{1}=1, \; a_{n+2}=2a_{n+1}+a_{n}\] Show that $2^{k}$ divides $a_{n}$ if and only if $2^{k}$ divides $n$.

PEN H Problems, 15

Tags: factorial , number theory , Diophantine Equations , pen

Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.

PEN H Problems, 61

Tags: Diophantine Equations , pen

Solve the equation $2^x -5 =11^{y}$ in positive integers.

PEN A Problems, 43

Tags: modular arithmetic , Divisibility Theory , pen

Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.

PEN C Problems, 3

Tags: quadratics , floor function , modular arithmetic , Quadratic Residues , pen

Let $p$ be an odd prime number. Show that the smallest positive quadratic nonresidue of $p$ is smaller than $\sqrt{p}+1$.

PEN H Problems, 40

Tags: Diophantine Equations , pen

Determine all pairs of rational numbers $(x, y)$ such that \[x^{3}+y^{3}= x^{2}+y^{2}.\]

PEN K Problems, 9

Tags: function , Functional Equations , pen

Find all functions $f: \mathbb{N}_{0}\rightarrow \mathbb{N}_{0}$ such that for all $n\in \mathbb{N}_{0}$: \[f(f(n))+f(n)=2n+6.\]

2007 China Team Selection Test, 3

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 I Problems, 11

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 K Problems, 1

Tags: function , Functional Equations , pen

Prove that there is a function $f$ from the set of all natural numbers into itself such that $f(f(n))=n^2$ for all $n \in \mathbb{N}$.

PEN K Problems, 6

Tags: function , algebra , polynomial , Functional Equations , pen

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]

PEN A Problems, 5

Tags: conics , hyperbola , symmetry , algebra , Vieta , number theory , pen

Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]

PEN P Problems, 3

Tags: function , algebra , floor function , Additive Number Theory , pen

Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.

PEN H Problems, 65

Tags: modular arithmetic , number theory , Diophantine Equations , pen

Determine all pairs $(x, y)$ of integers such that \[(19a+b)^{18}+(a+b)^{18}+(19b+a)^{18}\] is a nonzero perfect square.

PEN A Problems, 96

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

Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.

PEN A Problems, 39

Tags: Divisibility Theory , pen

Let $n$ be a positive integer. Prove that the following two statements are equivalent. [list][*] $n$ is not divisible by $4$ [*] There exist $a, b \in \mathbb{Z}$ such that $a^{2}+b^{2}+1$ is divisible by $n$. [/list]