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

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}$

2009 Princeton University Math Competition, 8
Tags: number theory , greatest common divisor , algorithm , relatively prime , euclidean algorithm , algebra , difference of squares
Find the largest positive integer $k$ such that $\phi ( \sigma ( 2^k)) = 2^k$. ($\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$). As a hint, you are given that $641|2^{32}+1$.

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 $2023$? $\textbf{(A) } 674 \qquad \textbf{(B) } 1011 \qquad \textbf{(C) } 1010 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2017$

2020 Spain Mathematical Olympiad, 6
Tags: difference of squares , number theory , 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 $n > N$: $$ d_2(S_n) > 4 \cdot d_3(S_n) $$

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$

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.

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?

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 $a+b+c$.

1981 AMC 12/AHSME, 14
Tags: ratio , geometric sequence , geometric series , algebra , difference of squares , special factorizations
In a geometric sequence of real numbers, the sum of the first two terms is 7, and the sum of the first 6 terms is 91. The sum of the first 4 terms is $\text{(A)}\ 28 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 49 \qquad \text{(E)}\ 84$

2004 Harvard-MIT Mathematics Tournament, 10
Tags: algebra , polynomial , difference of squares , special factorizations
There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.

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]

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

2004 AMC 12/AHSME, 13
Tags: function , algebra , difference of squares , special factorizations
If $ f(x) \equal{} ax \plus{} b$ and $ f^{ \minus{} 1}(x) \equal{} bx \plus{} a$ with $ a$ and $ b$ real, what is the value of $ a \plus{} b$? $ \textbf{(A)} \minus{} \!2 \qquad \textbf{(B)} \minus{} \!1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

1995 All-Russian Olympiad, 5
Tags: algebra , difference of squares , special factorizations , number theory
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]

2011 AMC 12/AHSME, 21
Tags: quadratic , algebra , 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 $

2006 Team Selection Test For CSMO, 1
Tags: modular arithmetic , quadratic , number theory , relatively prime , algebra , difference of squares
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.

1981 AMC 12/AHSME, 21
Tags: trigonometry , algebra , difference of squares , special factorizations , geometry
In a triangle with sides of lengths $a,b,$ and $c,$ $(a+b+c)(a+b-c) = 3ab.$ The measure of the angle opposite the side length $c$ is $\displaystyle \text{(A)} \ 15^\circ \qquad \text{(B)} \ 30^\circ \qquad \text{(C)} \ 45^\circ \qquad \text{(D)} \ 60^\circ \qquad \text{(E)} \ 150^\circ$

2012 China Team Selection Test, 2
Tags: pigeonhole principle , floor function , ceiling function , inequalities , function , algebra , difference of squares
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 $\alpha =\frac{|X|}{n}$.

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

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

2009 Singapore Senior Math Olympiad, 2
Tags: modular arithmetic , difference of squares , number theory
Find all positive integers $ m,n $ that satisfy the equation \[ 3.2^m +1 = n^2 \]