Olympiad problems

1964 AMC 12/AHSME, 37

Given two positive number $a$, $b$ such that $a<b$. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than: $\textbf{(A) }\dfrac{(b+a)^2}{ab}\qquad\textbf{(B) }\dfrac{(b+a)^2}{8b}\qquad\textbf{(C) }\dfrac{(b-a)^2}{ab}$ $\textbf{(D) }\dfrac{(b-a)^2}{8a}\qquad \textbf{(E) }\dfrac{(b-a)^2}{8b}$

PEN A Problems, 18

Tags: quadratic , algebra , difference of squares , special factorizations , divisibility theory

Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.

2020 AMC 12/AHSME, 2

Tags: algebra , difference of squares , special factorizations

What is the value of the following expression? $$\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}$$ $\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{9950} \qquad \textbf{(C) } \frac{4780}{4779} \qquad \textbf{(D) } \frac{108}{107} \qquad \textbf{(E) } \frac{81}{80} $

2007 AMC 10, 23

Tags: algebra , difference of squares , special factorizations

How many ordered pairs $ (m,n)$ of positive integers, with $ m > n$, have the property that their squares differ by $ 96$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$

2014 Putnam, 3

Tags: limit , induction , inequalities , algebra , difference of squares

Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

2017 Canadian Mathematical Olympiad Qualification, 3

Tags: algebra , functional equation , function , difference of squares , special factorizations

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$. $$(x+y)f(x-y) = f(x^2-y^2).$$

1989 Putnam, A1

Tags: algebra , difference of squares , special factorizations

How many base ten integers of the form 1010101...101 are prime?

2001 Slovenia National Olympiad, Problem 1

Tags: algebra , difference of squares , special factorizations , number theory , divisibility

None of the positive integers $k,m,n$ are divisible by $5$. Prove that at least one of the numbers $k^2-m^2,m^2-n^2,n^2-k^2$ is divisible by $5$.

1983 AMC 12/AHSME, 21

Tags: function , inequalities , algebra , difference of squares , special factorizations

Find the smallest positive number from the numbers below $\text{(A)} \ 10-3\sqrt{11} \qquad \text{(B)} \ 3\sqrt{11}-10 \qquad \text{(C)} \ 18-5\sqrt{13} \qquad \text{(D)} \ 51-10\sqrt{26} \qquad \text{(E)} \ 10\sqrt{26}-51$

2012 Kazakhstan National Olympiad, 1

Tags: algebra , difference of squares , special factorizations , number theory

Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.

2014 HMNT, 2

Tags: hmmt , algebra , difference of squares , special factorizations

Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$

2014 Contests, 3

Tags: limit , induction , inequalities , algebra , difference of squares

Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

2012 China Team Selection Test, 2

Tags: pigeonhole principle , floor function , ceiling function , function , algebra , difference of squares , inequalities

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

2000 AIME Problems, 2

Tags: hyperbola , analytic geometry , conic , algebra , difference of squares , special factorizations

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2-y^2=2000^2.$

2019 Azerbaijan Junior NMO, 3

Tags: number theory , difference of squares

A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

2018 Malaysia National Olympiad, A5

Tags: perfect square , number theory , difference of squares

Determine the value of $(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.

2008 AMC 10, 7

Tags: algebra , difference of squares , special factorizations

The fraction \[\frac {(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2}\] simplifies to which of the following? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac {9}{4} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac {9}{2} \qquad \textbf{(E)}\ 9$

1989 AIME Problems, 1

Tags: algebra , difference of squares , special factorizations

Compute $\sqrt{(31)(30)(29)(28)+1}$.

2000 Junior Balkan MO, 2

Tags: induction , inequalities , algebra , difference of squares , special factorizations , number theory

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer. [i]Bulgaria[/i]

2008 AIME Problems, 1

Tags: algebra , difference of squares , special factorizations , arithmetic series

Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.

2013 Pan African, 1

Tags: algebra , difference of squares , special factorizations , number theory

A positive integer $n$ is such that $n(n+2013)$ is a perfect square. a) Show that $n$ cannot be prime. b) Find a value of $n$ such that $n(n+2013)$ is a perfect square.

2019 India PRMO, 3

Tags: number theory , algebra , difference of squares , special factorizations

Find the number of positive integers less than 101 that [i]can not [/i] be written as the difference of two squares of integers.

1997 AIME Problems, 1

Tags: modular arithmetic , linear algebra , matrix , geometry , 3d geometry , algebra , difference of squares

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

2019 Azerbaijan Senior NMO, 2

Tags: number theory , difference of squares

A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

2005 AIME Problems, 4

Tags: geometry , rectangle , quadratic , special factorizations , algebra , difference of squares , quadratic formula

The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.