Olympiad problems

2023 USAMO, 5

Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is [i]row-valid[/i] if the numbers in each row can be permuted to form an arithmetic progression, and [i]column-valid[/i] if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

2008 National Olympiad First Round, 2

Tags: modular arithmetic , quadratic

For which value of $A$, does the equation $3m^2n = n^3 + A$ have a solution in natural numbers? $ \textbf{(A)}\ 301 \qquad\textbf{(B)}\ 403 \qquad\textbf{(C)}\ 415 \qquad\textbf{(D)}\ 427 \qquad\textbf{(E)}\ 481 $

2005 Romania Team Selection Test, 3

Tags: functional equation , floor function , induction , algebra , modular arithmetic , quadratic , number theory

Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that \[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \] [i]Călin Popescu[/i]

2012 AMC 10, 22

Tags: system of equations , algebra , number theory , modular arithmetic , diophantine equation

The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$? $ \textbf{(A)}\ 255 \qquad\textbf{(B)}\ 256 \qquad\textbf{(C)}\ 257 \qquad\textbf{(D)}\ 258 \qquad\textbf{(E)}\ 259 $

PEN P Problems, 18

Tags: number theory , modular arithmetic , additive number theory

Let $p$ be a prime with $p \equiv 1 \pmod{4}$. Let $a$ be the unique integer such that \[p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2}\] Prove that \[\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right),\] where $\left(\frac{k}{p}\right)$ denotes the Legendre Symbol.

PEN D Problems, 14

Tags: modular arithmetic , congruence

Determine the number of integers $n \ge 2$ for which the congruence \[x^{25}\equiv x \; \pmod{n}\] is true for all integers $x$.

2013 All-Russian Olympiad, 1

Tags: number theory , modular arithmetic

Does exist natural $n$, such that for any non-zero digits $a$ and $b$ \[\overline {ab}\ |\ \overline {anb}\ ?\] (Here by $ \overline {x \ldots y} $ denotes the number obtained by concatenation decimal digits $x$, $\dots$, $y$.) [i]V. Senderov[/i]

2006 France Team Selection Test, 3

Tags: number theory , modular arithmetic , divisibility , chinese remainder theorem

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

2004 IberoAmerican, 3

Tags: function , induction , modular arithmetic , quadratic , number theory

Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$

2011 Math Prize for Girls Olympiad, 3

Tags: modular arithmetic , prime number , number theory

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.

1994 IMO Shortlist, 6

Tags: number theory , modular arithmetic , integer sequence , recurrence relation , divisibility

Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?

PEN A Problems, 96

Tags: number theory , modular arithmetic , integration , relatively prime , divisibility theory

Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.

1988 AIME Problems, 9

Tags: modular arithmetic , 3d geometry , geometry

Find the smallest positive integer whose cube ends in 888.

2006 Iran MO (3rd Round), 1

Tags: function , modular arithmetic , abstract algebra , least common multiple , group theory , number theory

$n$ is a natural number. $d$ is the least natural number that for each $a$ that $gcd(a,n)=1$ we know $a^{d}\equiv1\pmod{n}$. Prove that there exist a natural number that $\mbox{ord}_{n}b=d$

1984 IMO Longlists, 40

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

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

2002 IberoAmerican, 2

Tags: geometric transformation , geometry , modular arithmetic , integration , invariant , calculus , reflection

Given any set of $9$ points in the plane such that there is no $3$ of them collinear, show that for each point $P$ of the set, the number of triangles with its vertices on the other $8$ points and that contain $P$ on its interior is even.

2008 Harvard-MIT Mathematics Tournament, 22

Tags: function , modular arithmetic

For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.

2012 Regional Olympiad of Mexico Center Zone, 2

Tags: number theory , modular arithmetic

Let $m, n$ integers such that: $(n-1)^3+n^3+(n+1)^3=m^3$ Prove that 4 divides $n$

PEN M Problems, 15

Tags: logarithm , modular arithmetic , recursive sequences

For a given positive integer $k$ denote the square of the sum of its digits by $f_{1}(k)$ and let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of $f_{1991}(2^{1990})$.

2013 Iran Team Selection Test, 8

Tags: modular arithmetic , greatest common divisor , number theory

Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$: $a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$

PEN H Problems, 79

Tags: modular arithmetic , diophantine equation

Find all positive integers $m$ and $n$ for which \[1!+2!+3!+\cdots+n!=m^{2}\]

2010 Stanford Mathematics Tournament, 13

Tags: number theory , modular arithmetic

Find all the integers $x$ in $[20, 50]$ such that $6x+5\equiv 19 \mod 10$, that is, $10$ divides $(6x+15)+19$.

2010 IMO Shortlist, 7

Tags: combinatorics , modular arithmetic , arithmetic sequence , sequence , number theory

Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied: [b](i)[/b] each integer belongs to at least one of them; [b](ii)[/b] each progression contains a number which does not belong to other progressions. Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization. Prove that \[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\] [i]Proposed by Dierk Schleicher, Germany[/i]

2004 AMC 10, 21

Tags: modular arithmetic

Let $ 1,4,\cdots$ and $ 9,16,\cdots$ be two arithmetic progressions. The set $ S$ is the union of the first $ 2004$ terms of each sequence. How many distinct numbers are in $ S$? $ \textbf{(A)}\ 3722\qquad \textbf{(B)}\ 3732\qquad \textbf{(C)}\ 3914\qquad \textbf{(D)}\ 3924\qquad \textbf{(E)}\ 4007$

2014 Kyiv Mathematical Festival, 4a

Tags: least common multiple , modular arithmetic , number theory

a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$ b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$