Olympiad problems

1964 AMC 12/AHSME, 37

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

PEN A Problems, 18

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

Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.

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} $

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$

2012 Pan African, 1

Tags: induction , sfft , geometry , 3d geometry , invariant , special factorizations , combinatorics

The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?

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

1989 Putnam, A1

Tags: algebra , difference of squares , special factorizations

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

1978 IMO Longlists, 22

Tags: special factorizations , number theory

Let $x$ and $y$ be two integers not equal to $0$ such that $x+y$ is a divisor of $x^2+y^2$. And let $\frac{x^2+y^2}{x+y}$ be a divisor of $1978$. Prove that $x = y$. [i]German IMO Selection Test 1979, problem 2[/i]

2001 Slovenia National Olympiad, Problem 1

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

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 $5$.

MBMT Team Rounds, 2020.18

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]

1983 AMC 12/AHSME, 21

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

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

2012 Today's Calculation Of Integral, 843

Tags: calculus , integration , function , inequalities , special factorizations , calculus computations

Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.

2009 India National Olympiad, 3

Tags: function , floor function , special factorizations , algebra

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$.)

2012 Kazakhstan National Olympiad, 1

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

Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.

2014 HMNT, 2

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

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

1977 AMC 12/AHSME, 21

Tags: quadratic , special factorizations

For how many values of the coeﬃcient $a$ do the equations \begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*} have a common real solution? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

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$.

1997 India National Olympiad, 2

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

Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]

2000 AIME Problems, 2

Tags: hyperbola , analytic geometry , conic , algebra , difference of squares , special factorizations

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2-y^2=2000^2.$

2024 AMC 10, 15

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 $

2008 AMC 10, 7

Tags: algebra , difference of squares , special factorizations

The fraction \[\frac {(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2}\] simplifies to which of the following? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac {9}{4} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac {9}{2} \qquad \textbf{(E)}\ 9$

1989 AIME Problems, 1

Tags: algebra , difference of squares , special factorizations

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

2002 AMC 12/AHSME, 25

Tags: parabola , inequalities , analytic geometry , geometry , special factorizations , conic

Let $ f(x)\equal{}x^2\plus{}6x\plus{}1$, and let $ R$ denote the set of points $ (x,y)$ in the coordinate plane such that \[ f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0 \]The area of $ R$ is closest to $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 22 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 25$

2004 Iran Team Selection Test, 1

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

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

1993 AMC 12/AHSME, 19

Tags: calculus , integration , sfft , special factorizations

How many ordered pairs $(m,n)$ of positive integers are solutions to $\frac{4}{m}+\frac{2}{n}=1$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $