This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2008

1998 Iran MO (3rd Round), 1
Tags: induction , modular arithmetic , number theory
Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$ \[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\] where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.

2008 Balkan MO Shortlist, N2
Tags: induction , algebra , polynomial , number theory , modular arithmetic , 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.

2008 USAMO, 1
Tags: modular arithmetic , number theory , induction
Prove that for each positive integer $ n$, there are pairwise relatively prime integers $ k_0,k_1,\ldots,k_n$, all strictly greater than $ 1$, such that $ k_0k_1\ldots k_n\minus{}1$ is the product of two consecutive integers.

2012 International Zhautykov Olympiad, 3
Tags: greatest common divisor , diophantine equation , number theory , modular arithmetic
Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.

1975 IMO, 2
Tags: sequence , additive number theory , modular arithmetic , number theory
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

Oliforum Contest II 2009, 1
Tags: function , modular arithmetic , prime factorization , number theory
Let $ \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0$ be the function from every positive integer $ n$ to the sum of divisors $ \sum_{d \mid n}{d}$ (i.e. $ \sigma(6) \equal{} 6 \plus{} 3 \plus{} 2 \plus{} 1$ and $ \sigma(8) \equal{} 8 \plus{} 4 \plus{} 2 \plus{} 1$). Find all primes $ p$ such that $ p \mid \sigma(p \minus{} 1)$. [i](Salvatore Tringali)[/i]

2007 USA Team Selection Test, 4
Tags: modular arithmetic , induction , number theory
Determine whether or not there exist positive integers $ a$ and $ b$ such that $ a$ does not divide $ b^n \minus{} n$ for all positive integers $ n$.

1970 IMO, 1
Tags: modular arithmetic , product , partition , number theory
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

2013 Saint Petersburg Mathematical Olympiad, 7
Tags: modular arithmetic , number theory
Given is a natural number $a$ with $54$ digits, each digit equal to $0$ or $1$. Prove the remainder of $a$ when divide by $ 33\cdot 34\cdots 39 $ is larger than $100000$. [hide](It's mean: $a \equiv r \pmod{33\cdot 34\cdots 39 }$ with $ 0<r<33\cdot 34\cdots 39 $ then prove that $r>100000$ )[/hide] M. Antipov

1966 IMO Shortlist, 54
Tags: decimal representation , digit , last digit , number theory , modular arithmetic
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

1995 Polish MO Finals, 3
Tags: number theory , induction , polynomial , algebra , modular arithmetic
Let $p$ be a prime number, and define a sequence by: $x_i=i$ for $i=,0,1,2...,p-1$ and $x_n=x_{n-1}+x_{n-p}$ for $n \geq p$ Find the remainder when $x_{p^3}$ is divided by $p$.

2016 Bangladesh Mathematical Olympiad, 2
Tags: modular arithmetic , divisibility , number theory
(a) How many positive integer factors does $6000$ have? (b) How many positive integer factors of $6000$ are not perfect squares?

2014 AMC 8, 21
Tags: modular arithmetic
The $7$-digit numbers $\underline{7}$ $ \underline{4}$ $ \underline{A}$ $ \underline{5}$ $ \underline{2}$ $ \underline{B}$ $ \underline{1}$ and $\underline{3}$ $ \underline{2}$ $ \underline{6}$ $ \underline{A}$ $ \underline{B}$ $ \underline{4}$ $ \underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad \textbf{(E) }8$

2018 China Girls Math Olympiad, 7
Tags: modular arithmetic , number theory , combinatorics
Given $2018 \times 4$ grids and tint them with red and blue. So that each row and each column has the same number of red and blue grids, respectively. Suppose there're $M$ ways to tint the grids with the mentioned requirement. Determine $M \pmod {2018}$.

2010 ELMO Shortlist, 2
Tags: modular arithmetic , difference of squares , special factorizations , number theory , 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]

1984 AIME Problems, 14
Tags: modular arithmetic
What is the largest even integer that cannot be written as the sum of two odd composite numbers?

2008 China Team Selection Test, 2
Tags: modular arithmetic , absolute value , polynomial , algebra
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that (1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$; (2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers $ x, |f(x)|$ isn't prime numbers.

1984 IMO Longlists, 66
Tags: number theory , modular arithmetic
Let $1=d_1<d_2<....<d_k=n$ be all different divisors of positive integer n written in ascending order. Determine all n such that: \[d_6^{2} +d_7^{2} - 1=n\]

2010 Contests, 1
Tags: modular arithmetic , number theory
Prove that the number of ordered triples $(x, y, z)$ such that $(x+y+z)^2 \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.

1985 IMO Longlists, 1
Tags: combinatorics , number theory , relatively prime , coloring , modular arithmetic
Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that: [i](i)[/i] $i$ and $n - i$ always receive the same color, and [i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$ Prove that all numbers in $N$ must receive the same color.

1999 IMO Shortlist, 3
Tags: number theory , integer sequence , divisibility , sequence , modular arithmetic
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

1987 Romania Team Selection Test, 2
Tags: number theory , ratio , modular arithmetic
Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \] where $m\geq n\geq p \geq 1$ are integer numbers. [i]Ioan Bogdan[/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$

1998 IMO Shortlist, 3
Tags: combinatorics , pigeonhole principle , divisibility , number theory , modular arithmetic
Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.

2007 Finnish National High School Mathematics Competition, 5
Tags: algebra , modular arithmetic , polynomial
Show that there exists a polynomial $P(x)$ with integer coefficients, such that the equation $P(x) = 0$ has no integer solutions, but for each positive integer $n$ there is an $x \in \Bbb{Z}$ such that $n \mid P(x).$