Olympiad problems

2013 Princeton University Math Competition, 6

Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.

2023 Dutch BxMO TST, 5

Tags: modular arithmetic , diophantine equation , number theory

Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

1980 IMO, 13

Tags: modular arithmetic , number theory

Prove that the integer $145^{n} + 3114\cdot 138^{n}$ is divisible by $1981$ if $n=1981$, and that it is not divisible by $1981$ if $n=1980$.

1989 India National Olympiad, 1

Tags: polynomial , calculus , integration , quadratic , modular arithmetic , algebra

Prove that the Polynomial $ f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989$ can't be expressed as a product $ f(x) \equal{} p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.

2005 Junior Balkan Team Selection Tests - Romania, 12

Tags: modular arithmetic , number theory

Find all positive integers $n$ and $p$ if $p$ is prime and \[ n^8 - p^5 = n^2+p^2 . \] [i]Adrian Stoica[/i]

2014 USA Team Selection Test, 3

Tags: group theory , abstract algebra , modular arithmetic , algebra , polynomial , number theory

For a prime $p$, a subset $S$ of residues modulo $p$ is called a [i]sum-free multiplicative subgroup[/i] of $\mathbb F_p$ if $\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and $\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$. Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$. [i]Proposed by Noga Alon and Jean Bourgain[/i]

2009 IMO, 1

Tags: induction , modular arithmetic , number theory

Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$ [i]Proposed by Ross Atkins, Australia [/i]

2009 Croatia Team Selection Test, 4

Tags: modular arithmetic , number theory

Determine all triplets off positive integers $ (a,b,c)$ for which $ \mid2^a\minus{}b^c\mid\equal{}1$

2011 China Team Selection Test, 2

Tags: modular arithmetic , number theory

Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

2001 Stanford Mathematics Tournament, 3

Tags: college , modular arithmetic

Find the 2000th positive integer that is not the diﬀerence between any two integer squares.

2005 USAMO, 6

Tags: induction , modular arithmetic , ceiling function , floor function , pigeonhole principle , inequalities , logarithm

For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with \[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]

2008 Romania Team Selection Test, 3

Tags: quadratic residue , modular arithmetic , number theory , quadratic reciprocity

Let $ m,\ n \geq 3$ be positive odd integers. Prove that $ 2^{m}\minus{}1$ doesn't divide $ 3^{n}\minus{}1$.

2006 Baltic Way, 19

Tags: modular arithmetic , induction , number theory

Does there exist a sequence $a_1,a_2,a_3,\ldots $ of positive integers such that the sum of every $n$ consecutive elements is divisible by $n^2$ for every positive integer $n$?

2014 NIMO Problems, 3

Tags: modular arithmetic , algebra , polynomial , complex number

Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is divided by $1000$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in a set $X$.) [i]Proposed by David Altizio[/i]

2010 Vietnam Team Selection Test, 3

Tags: algebra , polynomial , modular arithmetic , number theory

Let $S_n $ be sum of squares of the coefficient of the polynomial $(1+x)^n$. Prove that $S_{2n} +1$ is not divisible by $3.$

2005 Irish Math Olympiad, 4

Tags: modular arithmetic , number theory

Find the first digit to the left and the first digit to the right of the decimal point in the expansion of $ (\sqrt{2}\plus{}\sqrt{5})^{2000}.$

2014 Romania Team Selection Test, 4

Tags: induction , modular arithmetic , relatively prime , prime factorization , number theory

Let $k$ be a nonzero natural number and $m$ an odd natural number . Prove that there exist a natural number $n$ such that the number $m^n+n^m$ has at least $k$ distinct prime factors.

2011 China National Olympiad, 3

Tags: modular arithmetic , algorithm , prime number , divisibility , number theory

Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]

2012 JBMO ShortLists, 5

Tags: modular arithmetic , number theory , greatest common divisor , algebra

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

2017 South East Mathematical Olympiad, 3

Tags: modular arithmetic , number theory

For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set $$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$ where $i = 1, 2$. Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$.

2010 Portugal MO, 3

Tags: analytic geometry , modular arithmetic , number theory

Consider a square $(p-1)\times(p-1)$, where $p$ is a prime number, which is divided by squares $1\times 1$ whose sides are parallel to the initial square's sides. Show that it is possible to select $p$ vertices such that there are no three collinear vertices.

2008 Grigore Moisil Intercounty, 2

Tags: modular arithmetic , number theory

Determine the natural numbers a, b, c s.t. : $ \frac{3a+2b}{6a}=\frac{8b+c}{10b}=\frac{3a+2c}{3c} $ and $ a^{2}+b^{2}+c^{2}=975 $ The challenge here is to come up with as basic solution as possible.

1978 IMO Longlists, 52

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

Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way: $(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$ $(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set. Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$

2001 Irish Math Olympiad, 1

Tags: modular arithmetic , number theory

Find all positive integer solutions $ (a,b,c,n)$ of the equation: $ 2^n\equal{}a!\plus{}b!\plus{}c!$.

2011 Math Prize for Girls Olympiad, 3

Tags: modular arithmetic , number theory , prime number

Let $n$ be a positive integer such that $n + 1$ is divisible by 24. Prove that the sum of all the positive divisors of $n$ is divisible by 24.