Olympiad problems

1974 Bundeswettbewerb Mathematik, 4

Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

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 $

2004 IMO Shortlist, 6

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]

2006 Thailand Mathematical Olympiad, 10

Tags: number theory , factorial , remainder

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

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.

1946 Moscow Mathematical Olympiad, 108

Tags: number theory , 4-digit , remainder

Find a four-digit number such that the remainders after its division by $131$ and $132$ are $112$ and $98$, respectively.

2017 Dutch Mathematical Olympiad, 4

Tags: remainder , sum , number theory

If we divide the number $13$ by the three numbers $5, 7$, and $9$, then these divisions leave remainders: when dividing by $5$ the remainder is $3$, when dividing by $7$ the remainder is $6$, and when dividing by $9$ the remainder is 4. If we add these remainders, we obtain $3 + 6 + 4 = 13$, the original number. (a) Let $n$ be a positive integer and let $a$ and $b$ be two positive integers smaller than $n$. Prove: if you divide $n$ by $a$ and $b$, then the sum of the two remainders never equals $n$. (b) Determine all integers $n > 229$ having the property that if you divide $n$ by $99, 132$, and $229$, the sum of the three remainders is $n$.

2017 Junior Regional Olympiad - FBH, 4

Tags: equation , remainder

If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?

2020 New Zealand MO, 2

Tags: number theory , remainder

Find the smallest positive integer $N$ satisfying the following three properties. $\bullet$ N leaves a remainder of $5$ when divided by $7$. $\bullet$ N leaves a remainder of $6$ when divided by $ 8$. $\bullet$ N leaves a remainder of $7$ when divided by $9$.

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]

2011 Saudi Arabia BMO TST, 4

Tags: number theory , remainder

Let $p \ge 3$ be a prime. For $j = 1,2 ,... ,p - 1$, let $r_j$ be the remainder when the integer $\frac{j^{p-1}-1}{p}$ is divided by $p$. Prove that $$r_1 + 2r_2 + ... + (p - 1)r_{p-1} \equiv \frac{p+1}{2} (\mod p)$$

2014 Junior Regional Olympiad - FBH, 4

Tags: number theory , remainder

Positive integer $n$ when divided with number $3$ gives remainder $a$, when divided with $5$ has remainder $b$ and when divided with $7$ gives remainder $c$. Find remainder when dividing number $n$ with $105$ if $4a+3b+2c=30$

2022 Durer Math Competition Finals, 14

Tags: number theory , remainder

Benedek scripted a program which calculated the following sum: $1^1+2^2+3^3+. . .+2021^{2021}$. What is the remainder when the sum is divided by $35$?

2013 Saudi Arabia GMO TST, 4

Tags: fibonacci number , remainder , number theory

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 $m$ .

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 $35$.

1994 North Macedonia National Olympiad, 1

Tags: sum , number theory , remainder

Let $ a_1, a_2, ..., a_ {1994} $ be integers such that $ a_1 + a_2 + ... + a_{1994} = 1994 ^{1994} $ . Determine the remainder of the division of $ a ^ 3_1 + a ^ 3_2 + ... + a ^ 3_{1994} $ with $6$.

2014 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: remainder , number theory , combinatorics , set

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.

2018 Mathematical Talent Reward Programme, SAQ: P 5

Tags: fibonacci sequence , remainder , periodic sequence , polynomial , algebra

[list=1] [*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number. [*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime. [/list]

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 $n - 1$.

1947 Moscow Mathematical Olympiad, 123

Tags: remainder , polynomial , algebra

Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.

2016 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorial number theory , remainder , set , combinatorics

Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$

2000 Estonia National Olympiad, 1

Tags: number theory , power of prime , prime , remainder

Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$

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 $n$.

2017 Regional Olympiad of Mexico Northeast, 1

Tags: number theory , remainder

Let $n$ be a positive integer less than $1000$. The remainders obtained when dividing $n$ by $2, 2^2, 2^3, ... , 2^8$, and $2^9$ , are calculated. If the sum of all these remainders is $137$, what are all the possible values of $n$?

2024 Israel National Olympiad (Gillis), P2

Tags: remainder , number theory , fractional part

A positive integer $x$ satisfies the following: \[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\] Find all possible values of \[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\] where $\{y\}$ denotes the fractional part of $y$.