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

2008 AMC 10, 15
Tags: algebra , difference of squares , special factorizations
How many right triangles have integer leg lengths $ a$ and $ b$ and a hypotenuse of length $ b\plus{}1$, where $ b<100$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

2008 AIME Problems, 4
Tags: trigonometry , quadratic , special factorizations , diophantine equation
There exist unique positive integers $ x$ and $ y$ that satisfy the equation $ x^2 \plus{} 84x \plus{} 2008 \equal{} y^2$. Find $ x \plus{} y$.

2009 India National Olympiad, 3
Tags: function , floor function , special factorizations , algebra
Find all real numbers $ x$ such that: $ [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x]$ (Here $ [x]$ denotes the largest integer not exceeding $ x$.)

MBMT Team Rounds, 2020.18
Tags: special factorizations
Let $w, x, y, z$ be integers from $0$ to $3$ inclusive. Find the number of ordered quadruples of $(w, x, y, z)$ such that $5x^2 + 5y^2 + 5z^2 - 6wx-6wy -6wz$ is divisible by $4$. [i]Proposed by Timothy Qian[/i]

1985 AMC 12/AHSME, 19
Tags: function , quadratic , special factorizations
Consider the graphs $ y \equal{} Ax^2$ and and $ y^2 \plus{} 3 \equal{} x^2 \plus{} 4y$, where $ A$ is a positive constant and $ x$ and $ y$ are real variables. In how many points do the two graphs intersect? $ \textbf{(A)}\ \text{exactly } 4 \qquad \textbf{(B)}\ \text{exactly } 2$ $ \textbf{(C)}\ \text{at least } 1, \text{ but the number varies for different positive values of } A$ $ \textbf{(D)}\ 0 \text{ for at least one positive value of } A \qquad \textbf{(E)}\ \text{none of these}$

2002 AMC 12/AHSME, 12
Tags: quadratic , vieta , algebra , number theory , 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}$

1964 AMC 12/AHSME, 37
Tags: algebra , difference of squares , special factorizations
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 L Problems, 13
Tags: induction , algebra , polynomial , 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.

2013 Princeton University Math Competition, 7
Tags: sfft , special factorizations
Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$.

1993 AMC 12/AHSME, 19
Tags: calculus , integration , sfft , special factorizations
How many ordered pairs $(m,n)$ of positive integers are solutions to $\frac{4}{m}+\frac{2}{n}=1$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $

1997 India National Olympiad, 2
Tags: quadratic , modular arithmetic , diophantine equation , special factorizations , number theory
Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]

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

2010 ELMO Shortlist, 2
Tags: modular arithmetic , algebra , difference of squares , special factorizations , number theory
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]

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

2002 AMC 10, 14
Tags: quadratic , vieta , number theory , prime number , special factorizations , algebra
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}$

1999 Romania Team Selection Test, 3
Tags: quadratic , modular arithmetic , induction , floor function , algebra , special factorizations , binomial coefficient
Prove that for any positive integer $n$, the number \[ S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n \] is the sum of two consecutive perfect squares. [i]Dorin Andrica[/i]

1999 Finnish National High School Mathematics Competition, 3
Tags: modular arithmetic , geometric series , difference of squares , special factorizations , number theory , algebra
Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]

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

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

1983 AMC 12/AHSME, 21
Tags: function , inequalities , algebra , difference of squares , special factorizations
Find the smallest positive number from the numbers below $\text{(A)} \ 10-3\sqrt{11} \qquad \text{(B)} \ 3\sqrt{11}-10 \qquad \text{(C)} \ 18-5\sqrt{13} \qquad \text{(D)} \ 51-10\sqrt{26} \qquad \text{(E)} \ 10\sqrt{26}-51$

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

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 $x.$

1978 IMO Longlists, 22
Tags: special factorizations , number theory
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]