Olympiad problems

2018 Hanoi Open Mathematics Competitions, 4

A pyramid of non-negative integers is constructed as follows (a) The first row consists of only $0$, (b) The second row consists of $1$ and $1$, (c) The $n^{th}$ (for $n > 2$) is an array of $n$ integers among which the left most and right most elements are equal to $n - 1$ and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row (see Figure). Let $S_n$ be the sum of numbers in row $n^{th}$. Determine the remainder when dividing $S_{2018}$ by $2018$: A. $2$ B. $4$ C. $6$ D. $11$ E. $17$

2009 Postal Coaching, 2

Tags: number theory , remainder

Let $a > 2$ be a natural number. Show that there are infinitely many natural numbers n such that $a^n \equiv -1$ (mod $n^2$).

2016 Singapore MO Open, 3

Tags: number theory , permutation , remainder

Let $n$ be a prime number. Show that there is a permutation $a_1,a_2,...,a_n$ of $1,2,...,n$ so that $a_1,a_1a_2,...,a_1a_2...a_n$ leave distinct remainders when divided by $n$

1984 Spain Mathematical Olympiad, 8

Tags: determinant , remainder , polynomial , algebra

Find the remainder upon division by $x^2-1$ of the determinant $$\begin{vmatrix} x^3+3x & 2 & 1 & 0 \\ x^2+5x & 3 & 0 & 2 \\x^4 +x^2+1 & 2 & 1 & 3 \\x^5 +1 & 1 & 2 & 3 \\ \end{vmatrix}$$

2015 Saudi Arabia GMO TST, 4

Tags: binomial , remainder , number theory , combinatorics , inequalities

For each positive integer $n$, define $s(n) =\sum_{k=0}^n r_k$, where $r_k$ is the remainder when $n \choose k$ is divided by $3$. Find all positive integers $n$ such that $s(n) \ge n$. Malik Talbi

2002 Kazakhstan National Olympiad, 4

Tags: number theory , remainder

Prove that there is a set $ A $ consisting of $2002$ different natural numbers satisfying the condition: for each $ a \in A $, the product of all numbers from $ A $, except $ a $, when divided by $ a $ gives the remainder $1$.

2016 Singapore Senior Math Olympiad, 5

Tags: remainder , number theory

For each integer $n > 1$, find a set of $n$ integers $\{a_1, a_2,..., a_n\}$ such that the set of numbers $\{a_1+a_j | 1 \le i \le j \le n\}$ leave distinct remainders when divided by $n(n + 1)/2$. If such a set of integers does not exist, give a proof.

2020 Estonia Team Selection Test, 3

Tags: number theory , remainder

The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .

2021 Durer Math Competition Finals, 5

Tags: number theory , remainder , divisor

Let $n$ be a positive integer. Show that every divisors of $2n^2 - 1$ gives a different remainder after division by $2n$.

2005 Taiwan TST Round 3, 3

Tags: modular arithmetic , number theory , divisibility , remainder

Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

2013 Argentina National Olympiad, 3

Tags: digit , number theory , remainder

Find how many are the numbers of $2013$ digits $d_1d_2…d_{2013}$ with odd digits $d_1,d_2,…,d_{2013}$ such that the sum of $1809$ terms $$d_1 \cdot d_2+d_2\cdot d_3+…+d_{1809}\cdot d_{1810}$$ has remainder $1$ when divided by $4$ and the sum of $203$ terms $$d_{1810}\cdot d_{1811}+d_{1811}\cdot d_{1812}+…+d_{2012}\cdot d_{2013}$$ has remainder $1$ when dividing by $4$.

1999 Kazakhstan National Olympiad, 6

Tags: number theory with sequences , sequence , divisible , remainder , number theory

In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.

2014 Czech-Polish-Slovak Match, 5

Tags: number theory , modular arithmetic , remainder , binomial coefficient

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$, the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$. (Poland) PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

2008 JBMO Shortlist, 3

Tags: combinatorics , remainder

Integers $1,2, ...,2n$ are arbitrarily assigned to boxes labeled with numbers $1, 2,..., 2n$. Now, we add the number assigned to the box to the number on the box label. Show that two such sums give the same remainder modulo $2n$.

2007 JBMO Shortlist, 4

Tags: number theory , remainder

Let $a, b$ be two co-prime positive integers. A number is called [i]good [/i] if it can be written in the form $ax + by$ for non-negative integers $x, y$. Define the function $f : Z\to Z $as $f(n) = n - n_a - n_b$, where $s_t$ represents the remainder of $s$ upon division by $t$. Show that an integer $n$ is [i]good [/i]if and only if the infinite sequence $n, f(n), f(f(n)), ...$ contains only non-negative integers.

2022 IFYM, Sozopol, 2

Tags: number theory , remainder

Finding all quads of integers $(a, b, c, p)$ where $p \ge 5$ is prime number such that the remainders of the numbers $am^3 + bm^2 + cm$, $m = 0, 1, . . . , p - 1$, upon division of $p$ are two by two different..

2016 NIMO Problems, 1

Tags: remainder , number theory , divisor

Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$. [i] Proposed by Michael Ren [/i]

1941 Moscow Mathematical Olympiad, 085

Tags: prime , remainder , number theory

Prove that the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.

1984 Tournament Of Towns, (077) 2

Tags: number theory , combinatorics , remainder , sum , permutation

A set of numbers $a_1, a_2 , . . . , a_{100}$ is obtained by rearranging the numbers $1 , 2,..., 100$ . Form the numbers $b_1=a_1$ $b_2= a_1 + a_2$ $b_3=a_1 + a_2 + a_3$ ... $b_{100}=a_1 + a_2 + ...+a_{100}$ Prove that among the remainders on dividing the numbers by $100 , 11$ of them are different . ( L . D . Kurlyandchik , Leningrad)

2018 May Olympiad, 2

Tags: number theory , remainder

A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$. Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger?

1981 Kurschak Competition, 3

Tags: number theory , remainder

For a positive integer $n$, $r(n)$ denote the sum of the remainders when $n$ is divided by $1, 2,..., n$ respectively. Prove that $r(k) = r(k -1)$ for infinitely many positive integers $k$.

1977 IMO Longlists, 10

Tags: number theory , additive number theory , remainder , divisibility

Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

2024 Romanian Master of Mathematics, 4

Tags: number theory , remainder , size

Fix integers $a$ and $b$ greater than $1$. For any positive integer $n$, let $r_n$ be the (non-negative) remainder that $b^n$ leaves upon division by $a^n$. Assume there exists a positive integer $N$ such that $r_n < \frac{2^n}{n}$ for all integers $n\geq N$. Prove that $a$ divides $b$. [i]Pouria Mahmoudkhan Shirazi, Iran[/i]

2018 Ecuador Juniors, 5

Tags: number theory , remainder , digit

We call a positive integer [i]interesting [/i] if the number and the number with its digits written in reverse order both leave remainder $2$ in division by $4$. a) Determine if $2018$ is an interesting number. b) For every positive integer $n$, find how many interesting $n$-digit numbers there are.

2010 Contests, 2

Tags: number theory , remainder

Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.