This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 48

2014 Contests, 3
Tags: limit , difference of squares , induction , inequalities , algebra
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.

2000 Junior Balkan MO, 2
Tags: inequalities , induction , number theory , difference of squares , special factorizations , algebra
Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer. [i]Bulgaria[/i]

2010 ELMO Shortlist, 2
Tags: modular arithmetic , difference of squares , special factorizations , number theory , algebra
Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

1995 All-Russian Olympiad, 5
Tags: number theory , algebra , difference of squares , special factorizations
Prove that for every natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_n$ such that $a^2_1+a^2_2+\cdots+a^2_k$ is divisible by $a_1+a_2+\cdots+a_k$ for all $k \geq 1$. [i]A. Golovanov[/i]

2000 AIME Problems, 2
Tags: analytic geometry , hyperbola , difference of squares , special factorizations , conic , algebra
A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2-y^2=2000^2.$

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.

1989 AIME Problems, 1
Tags: algebra , difference of squares , special factorizations
Compute $\sqrt{(31)(30)(29)(28)+1}$.

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 $x, y \in \mathbb{R}$. $$(x+y)f(x-y) = f(x^2-y^2).$$

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.

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$

2011 Math Prize For Girls Problems, 13
Tags: algebra , difference of squares , special factorizations
The number 104,060,465 is divisible by a five-digit prime number. What is that prime number?

PEN A Problems, 18
Tags: algebra , quadratic , 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 $24$.

2012 Kazakhstan National Olympiad, 1
Tags: number theory , difference of squares , special factorizations , algebra
Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.

2020 Spain Mathematical Olympiad, 6
Tags: number theory , difference of squares , analytic number theory
Let $S$ be a finite set of integers. We define $d_2(S)$ and $d_3(S)$ as: $\bullet$ $d_2(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^2-y^2 = a$ $\bullet$ $d_3(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^3-y^3 = a$ (a) Let $m$ be an integer and $S = \{m, m+1, \ldots, m+2019\}$. Prove: $$d_2(S) > \frac{13}{7} d_3(S)$$ (b) Let $S_n = \{1, 2, \ldots, n\}$ with $n$ a positive integer. Prove that there exists a $N$ so that for all $n > N$: $$ d_2(S_n) > 4 \cdot d_3(S_n) $$

1971 AMC 12/AHSME, 14
Tags: difference of squares , special factorizations , algebra
The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are $\textbf{(A) }61,63\qquad\textbf{(B) }61,65\qquad\textbf{(C) }63,65\qquad\textbf{(D) }63,67\qquad \textbf{(E) }67,69$

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$.

2006 Team Selection Test For CSMO, 1
Tags: relatively prime , modular arithmetic , number theory , quadratic , difference of squares , algebra
Find all the pairs of positive numbers such that the last digit of their sum is 3, their difference is a primer number and their product is a perfect square.

2013 AIME Problems, 5
Tags: polynomial , geometry , 3d geometry , difference of squares , special factorizations , factorization , algebra
The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

2006 Stanford Mathematics Tournament, 12
Tags: algebra , number theory , prime factorization , difference of squares , special factorizations
What is the largest prime factor of 8091?

2011 AMC 12/AHSME, 21
Tags: algebra , quadratic , difference of squares , special factorizations
The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70 $

PEN L Problems, 13
Tags: polynomial , algebra , induction , quadratic , difference of squares , special factorizations , linear recurrence
The sequence $\{x_{n}\}_{n \ge 1}$ is defined by \[x_{1}=x_{2}=1, \; x_{n+2}= 14x_{n+1}-x_{n}-4.\] Prove that $x_{n}$ is always a perfect square.

2010 Austria Beginners' Competition, 1
Tags: number theory , difference of squares , perfect square
Prove that $2010$ cannot be represented as the difference between two square numbers. (B. Schmidt, Graz University of Technology)

2014 Putnam, 3
Tags: algebra , limit , inequalities , induction , 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.