Olympiad problems

1997 AIME Problems, 1

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

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

2001 Slovenia National Olympiad, Problem 1

Tags: difference of squares , divisibility , algebra , special factorizations , number theory

None of the positive integers $k,m,n$ are divisible by $5$. Prove that at least one of the numbers $k^2-m^2,m^2-n^2,n^2-k^2$ is divisible by $5$.

1989 Putnam, A1

Tags: difference of squares , special factorizations , algebra

How many base ten integers of the form 1010101...101 are prime?

1999 Finnish National High School Mathematics Competition, 3

Tags: special factorizations , difference of squares , number theory , geometric series , algebra , modular arithmetic

Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]

2009 Princeton University Math Competition, 8

Tags: relatively prime , euclidean algorithm , difference of squares , greatest common divisor , algebra , number theory , algorithm

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

1983 AMC 12/AHSME, 21

Tags: special factorizations , difference of squares , function , algebra , inequalities

Find the smallest positive number from the numbers below $\text{(A)} \ 10-3\sqrt{11} \qquad \text{(B)} \ 3\sqrt{11}-10 \qquad \text{(C)} \ 18-5\sqrt{13} \qquad \text{(D)} \ 51-10\sqrt{26} \qquad \text{(E)} \ 10\sqrt{26}-51$

2020 AMC 12/AHSME, 2

Tags: difference of squares , special factorizations , algebra

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} $

2023 AMC 10, 9

Tags: special factorizations , difference of squares , algebra

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

2019 Azerbaijan Junior NMO, 3

Tags: difference of squares , number theory

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: difference of squares , special factorizations , algebra , function

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$

1964 AMC 12/AHSME, 37

Tags: special factorizations , difference of squares , algebra

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: $\textbf{(A) }\dfrac{(b+a)^2}{ab}\qquad\textbf{(B) }\dfrac{(b+a)^2}{8b}\qquad\textbf{(C) }\dfrac{(b-a)^2}{ab}$ $\textbf{(D) }\dfrac{(b-a)^2}{8a}\qquad \textbf{(E) }\dfrac{(b-a)^2}{8b}$

2018 Malaysia National Olympiad, A5

Tags: perfect square , difference of squares , number theory

Determine the value of $(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.

2014 HMNT, 2

Tags: hmmt , difference of squares , special factorizations , algebra

Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$

2007 AMC 10, 23

Tags: difference of squares , special factorizations , algebra

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$

2005 AIME Problems, 4

Tags: algebra , quadratic formula , special factorizations , geometry , rectangle , quadratic , difference of squares

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.

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?

2009 Singapore Senior Math Olympiad, 2

Tags: difference of squares , number theory , modular arithmetic

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

1953 AMC 12/AHSME, 3

Tags: special factorizations , difference of squares , algebra

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}$

2012 China Team Selection Test, 2

Tags: floor function , algebra , function , inequalities , pigeonhole principle , difference of squares , ceiling function

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 $\alpha =\frac{|X|}{n}$.

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

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

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

2012 China Team Selection Test, 2

Tags: floor function , function , pigeonhole principle , difference of squares , inequalities , algebra , ceiling function

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 $\alpha =\frac{|X|}{n}$.

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$