Olympiad problems

2011 Albania Team Selection Test, 4

Find all prime numbers p such that $2^p+p^2 $ is also a prime number.

1999 Flanders Math Olympiad, 4

Tags: modular arithmetic , number theory

Let $a,b,m,n$ integers greater than 1. If $a^n-1$ and $b^m+1$ are both primes, give as much info as possible on $a,b,m,n$.

1996 India National Olympiad, 1

Tags: number theory , modular arithmetic

a) Given any positive integer $n$, show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$; b) If for some positive integers $x$ and $y$, $x+j$ divides $y+j$ for all positive integers $j$, prove that $x = y$.

2013 Waseda University Entrance Examination, 2

Tags: modular arithmetic , algebra , trigonometry , polynomial

For a complex number $z=1+2\sqrt{6}i$ and natural number $n=1,\ 2,\ 3,\ \cdots$, express the complex number $z^n$ in using real numbers $a_n,\ b_n$ as $z^n=a_n+b_ni$. Answer the following questions. (1) Show that $a_n^2+b_n^2=5^{2n}\ (n=1,\ 2,\ 3,\ \cdots).$ (2) Find the constants $p,\ q$ such that $a_{n+2}=pa_{n+1}+qa_n$ holds for all $n$. (3) Show that $a_n$ is not a multiple of $5$ for any $n$. (4) Show that $z^n\ (n=1,\ 2,\ 3,\ \cdots)$ is not a real number.

2003 USA Team Selection Test, 3

Tags: number theory , modular arithmetic , order

Find all ordered triples of primes $(p, q, r)$ such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \] [i]Reid Barton[/i]

2014 Contests, 3

Tags: modular arithmetic , number theory , greatest common divisor

For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers. (a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$. (b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.

2003 China Team Selection Test, 3

Tags: modular arithmetic , combinatorics

There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of $n$.

2008 USA Team Selection Test, 9

Tags: algorithm , algebra , function , geometry , modular arithmetic , 3d geometry , polynomial

Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if a) $ n$ is a positive integer not divisible by the square of a prime. b) $ n$ is a positive integer not divisible by the cube of a prime.

1995 Vietnam Team Selection Test, 2

Tags: number theory , modular arithmetic

For any nonnegative integer $ n$, let $ f(n)$ be the greatest integer such that $ 2^{f(n)} | n \plus{} 1$. A pair $ (n, p)$ of nonnegative integers is called nice if $ 2^{f(n)} > p$. Find all triples $ (n, p, q)$ of nonnegative integers such that the pairs $ (n, p)$, $ (p, q)$ and $ (n \plus{} p \plus{} q, n)$ are all nice.

2001 India National Olympiad, 2

Tags: modular arithmetic , algebra , number theory , arithmetic sequence , polynomial

Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.

1999 Irish Math Olympiad, 2

Tags: number theory , modular arithmetic

Show that there is a positive number in the Fibonacci sequence which is divisible by $ 1000$.

2013 India IMO Training Camp, 1

Tags: number theory , modular arithmetic , diophantine equation

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

2008 Balkan MO, 4

Tags: number theory , polynomial , modular arithmetic , algebra , induction , inequalities , relatively prime

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

2000 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , modular arithmetic

Solve in natural the equation $9^x-3^x=y^4+2y^3+y^2+2y$ _____________________________ Azerbaijan Land of the Fire :lol:

2014 Math Hour Olympiad, 8-10.7

Tags: floor function , modular arithmetic

If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$. Show that the sequence $\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$ contains infinitely many odd numbers.

2008 Iran Team Selection Test, 11

Tags: relatively prime , function , modular arithmetic , number theory , functional equation

$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]

PEN S Problems, 16

Tags: modular arithmetic

Show that if $a$ and $b$ are positive integers, then \[\left( a+\frac{1}{2}\right)^{n}+\left( b+\frac{1}{2}\right)^{n}\] is an integer for only finitely many positive integer $n$.

2005 AMC 10, 16

Tags: modular arithmetic

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $ 6$. How many two-digit numbers have this property? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 19$

1989 IMO Longlists, 27

Tags: combinatorics , modular arithmetic , relatively prime , calculus , number theory , integration

Let $ L$ denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if one considers four points of $ L$ instead of three?

PEN M Problems, 12

Tags: modular arithmetic , recursive sequences , induction

Let $k$ be a fixed positive integer. The sequence $\{a_{n}\}_{n\ge1}$ is defined by \[a_{1}=k+1, a_{n+1}=a_{n}^{2}-ka_{n}+k.\] Show that if $m \neq n$, then the numbers $a_{m}$ and $a_{n}$ are relatively prime.

2012 Junior Balkan MO, 3

Tags: modular arithmetic , combinatorics , ceiling function

On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors. a) Can $n$ be $6$ ? b) Can $n$ be $7$ ?

2011 Albania Team Selection Test, 4

1999 Flanders Math Olympiad, 4

1996 India National Olympiad, 1

2013 Waseda University Entrance Examination, 2

2003 USA Team Selection Test, 3

2014 Contests, 3

2003 China Team Selection Test, 3

2008 USA Team Selection Test, 9

1995 Vietnam Team Selection Test, 2

2001 India National Olympiad, 2

1999 Irish Math Olympiad, 2

2013 India IMO Training Camp, 1

2008 Balkan MO, 4

2000 Junior Balkan Team Selection Tests - Romania, 1

2014 Math Hour Olympiad, 8-10.7

2008 Iran Team Selection Test, 11

PEN S Problems, 16

2005 AMC 10, 16

1989 IMO Longlists, 27

PEN M Problems, 12

2012 Junior Balkan MO, 3

1998 Bulgaria National Olympiad, 2

2003 National Olympiad First Round, 14

2006 Germany Team Selection Test, 2

2008 Putnam, B4