Olympiad problems

1989 Putnam, A1

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

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$

2000 Junior Balkan MO, 2

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

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer. [i]Bulgaria[/i]

2001 Slovenia National Olympiad, Problem 1

Tags: difference of squares , special factorizations , divisibility , algebra , 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 $5$.

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 $24$.

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.

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 $a+b+c$.

2019 Azerbaijan Junior NMO, 3

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?

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

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}$

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$

1964 AMC 12/AHSME, 37

Tags: algebra , difference of squares , special factorizations

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 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.

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]

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

2000 AIME Problems, 2

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

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

2018 Malaysia National Olympiad, A5

Tags: number theory , difference of squares , perfect square

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

2014 Contests, 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.

1999 Finnish National High School Mathematics Competition, 3

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

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

2012 Kazakhstan National Olympiad, 1

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

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

1983 AMC 12/AHSME, 21

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

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

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 $n > N$: $$ d_2(S_n) > 4 \cdot d_3(S_n) $$

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 $x.$