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

2001 Manhattan Mathematical Olympiad, 3
Tags: modular arithmetic , algebra , polynomial
Let $x_1$ and $x_2$ be roots of the equation $x^2 - 6x + 1 = 0$. Prove that for any integer $n \ge 1$ the number $x_1^n + x_2^n$ is integer and is not divisible by $5$.

2012 BMT Spring, 8
Tags: modular arithmetic , euler , probability , expected value
You are tossing an unbiased coin. The last $ 28 $ consecutive flips have all resulted in heads. Let $ x $ be the expected number of additional tosses you must make before you get $ 60 $ consecutive heads. Find the sum of all distinct prime factors in $ x $.

1975 IMO Shortlist, 6
Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

2005 National Olympiad First Round, 10
Tags: modular arithmetic , euler
Which of the following does not divide $n^{2225}-n^{2005}$ for every integer value of $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 23 $

2006 Baltic Way, 18
Tags: modular arithmetic , number theory
For a positive integer $n$ let $a_n$ denote the last digit of $n^{(n^n)}$. Prove that the sequence $(a_n)$ is periodic and determine the length of the minimal period.

2002 Finnish National High School Mathematics Competition, 5
Tags: geometry , circumcircle , modular arithmetic , combinatorics
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$ What is the least number of colours which suffices?

2006 National Olympiad First Round, 26
Tags: modular arithmetic
For how many primes $p$, there exists an integr $m$ such that $m^3+3m-2 \equiv 0 \pmod p$ and $m^2+4m+5\equiv 0 \pmod p$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many} $

2011 Indonesia TST, 4
Tags: modular arithmetic , number theory
Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.

2023 Serbia Team Selection Test, P5
Tags: factorial , nt , weird , modular arithmetic , number theory
For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\] Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.

PEN H Problems, 13
Tags: diophantine equation , modular arithmetic , geometry , 3d geometry
Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]

2011 Postal Coaching, 1
Tags: modular arithmetic , number theory
Does the sequence \[11, 111, 1111, 11111, \ldots\] contain any fifth power of a positive integer? Justify your answer.

2021 Romania Team Selection Test, 1
Tags: modular arithmetic , number theory
Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.

2003 ITAMO, 1
Tags: modular arithmetic , number theory
Find all three digit numbers $n$ which are equal to the number formed by three last digit of $n^2$.

2002 All-Russian Olympiad Regional Round, 10.1
Tags: modular arithmetic , arithmetic sequence , algebra
What is the largest possible length of an arithmetic progression of positive integers $ a_{1}, a_{2},\cdots , a_{n}$ with difference $ 2$, such that $ {a_{k}}^{2}\plus{}1$ is prime for $ k \equal{} 1, 2, . . . , n$?

PEN H Problems, 65
Tags: diophantine equation , modular arithmetic , number theory
Determine all pairs $(x, y)$ of integers such that \[(19a+b)^{18}+(a+b)^{18}+(19b+a)^{18}\] is a nonzero perfect square.

2006 Purple Comet Problems, 11
Tags: modular arithmetic
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$.

1980 IMO Longlists, 6
Tags: modular arithmetic , combinatorics , decimal representation , digit
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

1985 IMO Longlists, 54
Tags: number theory , greatest common divisor , induction , algorithm , modular arithmetic , algebra , binomial theorem
Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

2013 Purple Comet Problems, 7
Tags: modular arithmetic
Find the least six-digit palindrome that is a multiple of $45$. Note that a palindrome is a number that reads the same forward and backwards such as $1441$ or $35253$.

2010 Contests, 1
Tags: modular arithmetic , number theory
Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$

2013 AMC 12/AHSME, 23
Tags: search , modular arithmetic
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? ${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $

2014 Middle European Mathematical Olympiad, 8
Tags: modular arithmetic , quadratic , inequalities , geometry , 3d geometry , number theory
Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

2010 Putnam, A4
Tags: modular arithmetic , number theory , greatest common divisor , putnam number theory , logarithm
Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.

2014 Contests, 1
Tags: modular arithmetic , number theory
Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.

1997 Romania Team Selection Test, 2
Tags: modular arithmetic , divisibility , multiplicative nt , multiplicative order , number theory
Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$. [i]Marcel Tena[/i]