Olympiad problems

2009 Singapore Senior Math Olympiad, 2

Find all positive integers $ m,n $ that satisfy the equation \[ 3.2^m +1 = n^2 \]

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$.

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]

2007 AMC 10, 23

Tags: algebra , difference of squares , special factorizations

How many ordered pairs $ (m,n)$ of positive integers, with $ m > n$, have the property that their squares differ by $ 96$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$

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

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.

2010 Austria Beginners' Competition, 1

Tags: number theory , difference of squares , perfect square

Prove that $2010$ cannot be represented as the difference between two square numbers. (B. Schmidt, Graz University of Technology)

1997 AIME Problems, 1

Tags: modular arithmetic , linear algebra , matrix , geometry , 3d geometry , algebra , difference of squares

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

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

1989 AIME Problems, 1

Tags: algebra , difference of squares , special factorizations

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

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

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

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

2014 Putnam, 3

Tags: limit , induction , inequalities , algebra , difference of squares

Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

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.

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 $641|2^{32}+1$.

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.

2020 AMC 12/AHSME, 2

Tags: algebra , difference of squares , special factorizations

What is the value of the following expression? $$\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}$$ $\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{9950} \qquad \textbf{(C) } \frac{4780}{4779} \qquad \textbf{(D) } \frac{108}{107} \qquad \textbf{(E) } \frac{81}{80} $

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

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?

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]

2006 Stanford Mathematics Tournament, 12

Tags: number theory , prime factorization , algebra , difference of squares , special factorizations

What is the largest prime factor of 8091?

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 $

2019 Azerbaijan Senior NMO, 2

Tags: number theory , difference of squares

A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

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$