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

Tags were heavily modified to better represent problems.

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Found problems: 73

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?

2004 Pan African, 2
Tags: special factorizations
Is: \[ 4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}} \] an integer?

2002 AMC 12/AHSME, 25
Tags: analytic geometry , parabola , inequalities , geometry , conic , special factorizations
Let $ f(x)\equal{}x^2\plus{}6x\plus{}1$, and let $ R$ denote the set of points $ (x,y)$ in the coordinate plane such that \[ f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0 \]The area of $ R$ is closest to $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 22 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 25$

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.

2009 Indonesia MO, 1
Tags: probability , combinatorics , complementary counting , special factorizations
In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?

2002 AMC 10, 14
Tags: number theory , algebra , quadratic , vieta , prime number , special factorizations
Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

2007 Harvard-MIT Mathematics Tournament, 3
Tags: trigonometry , special factorizations
The equation $x^2+2x=i$ has two complex solutions. Determine the product of their real parts.

2007 AMC 12/AHSME, 23
Tags: inradius , geometry , perimeter , sfft , special factorizations
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $ 3$ times their perimeters? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

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 $\textbf{(A) }61,63\qquad\textbf{(B) }61,65\qquad\textbf{(C) }63,65\qquad\textbf{(D) }63,67\qquad \textbf{(E) }67,69$

1978 IMO Longlists, 22
Tags: number theory , special factorizations
Let $x$ and $y$ be two integers not equal to $0$ such that $x+y$ is a divisor of $x^2+y^2$. And let $\frac{x^2+y^2}{x+y}$ be a divisor of $1978$. Prove that $x = y$. [i]German IMO Selection Test 1979, problem 2[/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$.

2007 Today's Calculation Of Integral, 204
Tags: trigonometry , integration , special factorizations , calculus , calculus computations
Evaluate \[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]

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 $

1990 Baltic Way, 15
Tags: geometry , modular arithmetic , algebra , 3d geometry , factorization , difference of cubes , special factorizations
Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.

2002 AIME Problems, 4
Tags: sfft , special factorizations
Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1.$ Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29,$ for positive integers $m$ and $n$ with $m<n$, find $m+n.$

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.

1996 Hungary-Israel Binational, 2
Tags: special factorizations , geometry , algebra , 3d geometry , factorization , difference of cubes , modular arithmetic
$ n>2$ is an integer such that $ n^2$ can be represented as a difference of cubes of 2 consecutive positive integers. Prove that $ n$ is a sum of 2 squares of positive integers, and that such $ n$ does exist.

1999 AIME Problems, 3
Tags: polynomial , algebra , vieta , quadratic , special factorizations
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.