Olympiad problems

1953 AMC 12/AHSME, 3

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}$

2008 Puerto Rico Team Selection Test, 4

Tags: special factorizations

If the sides of a triangle have lengths $ a, b, c$, such that $ a \plus{} b \minus{} c \equal{} 2$, and $ 2ab \minus{} c^{2} \equal{} 4$, prove that the triangle is equilateral.

2010 AIME Problems, 5

Tags: logarithm , sfft , special factorizations

Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.

1981 AMC 12/AHSME, 21

Tags: special factorizations , geometry , difference of squares , trigonometry , algebra

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 $\displaystyle \text{(A)} \ 15^\circ \qquad \text{(B)} \ 30^\circ \qquad \text{(C)} \ 45^\circ \qquad \text{(D)} \ 60^\circ \qquad \text{(E)} \ 150^\circ$

2009 Princeton University Math Competition, 1

Tags: special factorizations

Find the number of pairs of integers $x$ and $y$ such that $x^2 + xy + y^2 = 28$.

1999 Romania Team Selection Test, 3

Tags: modular arithmetic , algebra , binomial coefficient , quadratic , special factorizations , floor function , induction

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]

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 $2013$.

2010 ELMO Shortlist, 2

Tags: number theory , difference of squares , special factorizations , modular arithmetic , 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]

1981 AMC 12/AHSME, 14

Tags: geometric series , difference of squares , special factorizations , algebra , ratio , geometric sequence

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 $\text{(A)}\ 28 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 49 \qquad \text{(E)}\ 84$

2013 AMC 12/AHSME, 6

Tags: quadratic , special factorizations , quadratic formula , algebra

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 $

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 $\tfrac{P(1)}{P(-1)}$.

2011 USAMTS Problems, 2

Tags: special factorizations , inequalities

Find all integers $a$, $b$, $c$, $d$, and $e$ such that \begin{align*}a^2&=a+b-2c+2d+e-8,\\b^2&=-a-2b-c+2d+2e-6,\\c^2&=3a+2b+c+2d+2e-31,\\d^2&=2a+b+c+2d+2e-2,\\e^2&=a+2b+3c+2d+e-8.\end{align*}

2008 AMC 10, 15

Tags: difference of squares , special factorizations , algebra

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$

2009 India National Olympiad, 3

Tags: floor function , function , algebra , special factorizations

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$.)

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 $x^2-y^2=2000^2.$

2010 Purple Comet Problems, 24

Tags: sfft , special factorizations

Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m-10n = mn$.

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.

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$.

2004 Iran Team Selection Test, 1

Tags: modular arithmetic , number theory , quadratic , special factorizations

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

1989 AIME Problems, 1

Tags: algebra , difference of squares , special factorizations

Compute $\sqrt{(31)(30)(29)(28)+1}$.

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 $

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