Olympiad problems

1991 Greece National Olympiad, 4

If we divide number $1^{1990}+2^{1990}+3^{1990}+...+1990^{1990}$ with $10$, what remainder will we find?

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.

2019 Saudi Arabia Pre-TST + Training Tests, 2.1

Tags: remainder , number theory

Let be given a positive integer $n \ge 3$. Consider integers $a_1,a_2,...,a_n >1$ with the product equals to $A$ such that: for each $k \in \{1, 2,..., n\}$ then the remainder when $\frac{A}{a_k}$ divided by $a_k$ are all equal to $r$. Prove that $r \le n- 2$

2019 Taiwan APMO Preliminary Test, P2

Tags: remainder , modulo , number theory

Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$.

1939 Moscow Mathematical Olympiad, 051

Tags: remainder , number theory

Find the remainder after division of $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ by $7$.

1977 IMO, 2

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

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 $q^2 + r = 1977.$

2020 Tournament Of Towns, 7

Tags: game , combinatorics , remainder

Gleb picked positive integers $N$ and $a$ ($a < N$). He wrote the number $a$ on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number $N$ by this last number with remainder and wrote the remainder onto the board. When he wrote the number $0$ onto the board, he stopped. Could he pick $N$ and $a$ such that the sum of the numbers on the blackboard would become greater than $100N$ ? Ivan Mitrofanov

1990 Bundeswettbewerb Mathematik, 1

Tags: number theory , remainder

Find all triples $(a,b,c)$ of positive integers such that the product of any two of them when divided by the third leaves the remainder $1$.

2012 QEDMO 11th, 6

Tags: number theory , remainder , sum of powers

Let $p$ be an odd prime number. Prove that $$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$

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?

2014 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: combinatorics , set , number theory , remainder

Let $n \ge 3$ be a positive integer. Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions: $\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap B = A \cap C= B \cap C= \emptyset$. $\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$. $\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$. Note: One or more of the sets may be empty.

2020 OMMock - Mexico National Olympiad Mock Exam, 2

Tags: permutation , remainder , number theory

We say that a permutation $(a_1, \dots, a_n)$ of $(1, 2, \dots, n)$ is good if the sums $a_1 + a_2 + \dots + a_i$ are all distinct modulo $n$. Prove that there exists a positive integer $n$ such that there are at least $2020$ good permutations of $(1, 2, \dots, n)$. [i]Proposed by Ariel García[/i]

2021 Canadian Junior Mathematical Olympiad, 2

Tags: number theory , remainder

How many ways are there to permute the first $n$ positive integers such that in the permutation, for each value of $k \le n$, the first $k$ elements of the permutation have distinct remainder mod $k$?

2013 Saudi Arabia Pre-TST, 2.1

Tags: number theory , remainder , divide , divisible

Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.

2024 AMC 10, 18

Tags: remainder

How many different remainders can result when the $100$th power of an integer is divided by $125$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }5 \qquad \textbf{(D) }25 \qquad \textbf{(E) }125 \qquad $

2015 Saudi Arabia GMO TST, 4

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

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

2009 Postal Coaching, 4

Tags: digit , remainder , number theory

A four - digit natural number which is divisible by $7$ is given. The number obtained by writing the digits in reverse order is also divisible by $7$. Furthermore, both the numbers leave the same remainder when divided by $37$. Find the 4-digit number.

2011 Saudi Arabia IMO TST, 1

Tags: factorial , number theory , remainder

Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.

2016 Singapore Senior Math Olympiad, 5

Tags: number theory , remainder

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.

2019 India PRMO, 21 incorrect

Tags: sum of digits , remainder , number theory

Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$, find the sum of digits of $E$

2005 Thailand Mathematical Olympiad, 10

Tags: remainder , sum , number theory

What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?

2013 Saudi Arabia GMO TST, 4

Tags: number theory , fibonacci number , remainder

Let $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$, for all positive integer $n$, be the Fibonacci sequence. Prove that for any positive integer $m$ there exist infinitely many positive integers $n$ such that $F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2}$ mod $m$ .

2006 Thailand Mathematical Olympiad, 10

Tags: number theory , factorial , remainder

Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.

2017 Finnish National High School Mathematics Comp, 1

Tags: number theory , remainder , euclidean algorithm , division

By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder. As the division of the remainder with $n$ continues, the new quotient is $0.4$ and the new remainder is $0.2$. Find $m$ and $n$.

1984 Tournament Of Towns, (077) 2

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

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)