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: 73

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]

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

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$.

1990 Baltic Way, 15
Tags: geometry , 3d geometry , modular arithmetic , algebra , 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.

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.

2024 AMC 12/AHSME, 9
Tags: algebra , special factorizations
Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

2007 AMC 12/AHSME, 23
Tags: geometry , perimeter , inradius , 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$

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.

2015 AMC 12/AHSME, 10
Tags: sfft , special factorizations
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$? $\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }26$

1953 AMC 12/AHSME, 3
Tags: algebra , difference of squares , special factorizations
The factors of the expression $ x^2\plus{}y^2$ are: $ \textbf{(A)}\ (x\plus{}y)(x\minus{}y) \qquad\textbf{(B)}\ (x\plus{}y)^2 \qquad\textbf{(C)}\ (x^{\frac{2}{3}}\plus{}y^{\frac{2}{3}})(x^{\frac{4}{3}}\plus{}y^{\frac{4}{3}}) \\ \textbf{(D)}\ (x\plus{}iy)(x\minus{}iy) \qquad\textbf{(E)}\ \text{none of these}$

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

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

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]

2023 AMC 10, 9
Tags: algebra , difference of squares , special factorizations
The numbers $16$ and $25$ are a pair of consecutive perfect squares whose difference is $9$. How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$? $\textbf{(A) } 674 \qquad \textbf{(B) } 1011 \qquad \textbf{(C) } 1010 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2017$

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$

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 $2013$.

1999 Romania Team Selection Test, 3
Tags: binomial coefficient , quadratic , modular arithmetic , induction , floor function , algebra , special factorizations
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]

2020 MBMT, 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]

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 AMC 12/AHSME, 6
Tags: quadratic , special factorizations , algebra , quadratic formula
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$? $ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $

2009 Indonesia MO, 1
Tags: probability , complementary counting , special factorizations , combinatorics
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?

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?

1996 Hungary-Israel Binational, 2
Tags: geometry , 3d geometry , modular arithmetic , algebra , factorization , difference of cubes , special factorizations
$ 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.

2013 AIME Problems, 5
Tags: algebra , polynomial , geometry , 3d geometry , difference of squares , special factorizations , factorization
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 $a+b+c$.