Olympiad problems

2022 Durer Math Competition Finals, 14

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 $35$?

1994 Bulgaria National Olympiad, 5

Tags: periodic , remainder , number theory

Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$. Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.

2022 Switzerland - Final Round, 2

Tags: remainder , divide , number theory

Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$.

2009 Cuba MO, 1

Tags: remainder , prime , number theory

Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.

1949-56 Chisinau City MO, 6

Tags: perfect square , remainder , number theory

Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.

2022 JBMO Shortlist, N5

Tags: remainder , prime , shortlist , number theory

Find all pairs $(a, p)$ of positive integers, where $p$ is a prime, such that for any pair of positive integers $m$ and $n$ the remainder obtained when $a^{2^n}$ is divided by $p^n$ is non-zero and equals the remainder obtained when $a^{2^m}$ is divided by $p^m$.

2018 Mathematical Talent Reward Programme, SAQ: P 5

Tags: fibonacci sequence , periodic sequence , remainder , 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]

1946 Moscow Mathematical Olympiad, 108

Tags: 4-digit , remainder , number theory

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

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?

1981 Poland - Second Round, 4

Tags: remainder , number theory

The given natural numbers are $ k, n $. We inductively define two sequences of numbers $ (a_j) $ and $ (r_j) $ as follows: Step one: we divide $ k $ by $ n $ and get the quotient $ a_1 $ and the remainder $ r_i $, step j: we divide $ k+r_{j-1} $ by $ n $ and get the quotient $ a_j $ and the remainder $ r_j $. Calculate the sum of $ a_1 + \ldots + a_n $.

2001 Estonia National Olympiad, 2

Tags: digit , remainder , number theory , sum of digits

Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.

2020 Estonia Team Selection Test, 3

Tags: remainder , number theory

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 $q$) .

2010 Contests, 2

Tags: remainder , number theory

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

1990 ITAMO, 3

Tags: algebra , remainder , polynomial

Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$, respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$.

1963 Swedish Mathematical Competition., 3

Tags: number theory , remainder

What is the remainder on dividing $1234^{567} + 89^{1011}$ by $12$?

2011 Saudi Arabia BMO TST, 4

Tags: remainder , number theory

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 $$r_1 + 2r_2 + ... + (p - 1)r_{p-1} \equiv \frac{p+1}{2} (\mod p)$$

2009 Postal Coaching, 2

Tags: remainder , number theory

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

2014 Czech-Polish-Slovak Match, 5

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

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

1969 Dutch Mathematical Olympiad, 1

Tags: remainder , number theory

Determine the smallest $n$ such that $n \equiv (a - 1)$ mod $a$ for all $a \in \{2,3,..., 10\}$.

2015 Singapore Senior Math Olympiad, 3

Tags: remainder , number theory

Let $n \ge 3$ be an integer. Prove that there exist positive integers $\ge 2$, $a_1,a_2,..,a_n$, such that $a_1 a_2 ... \widehat{a_i}... a_n \equiv 1$ (mod $a_i$), for $i = 1,..., n$. Here $\widehat{a_i}$ means the term $a_i$ is omitted.

1947 Moscow Mathematical Olympiad, 123

Tags: polynomial , remainder , algebra

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

2021 Durer Math Competition Finals, 5

Tags: remainder , divisor , number theory

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

2015 Bosnia and Herzegovina Junior BMO TST, 4

Tags: positive integer , sum , remainder , combinatorics

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be positive integers from set $\{1, 2,..., n\}$ such that every number from this set occurs exactly once. Is it possible that numbers $a_1$, $a_1 + a_2 ,..., a_1 + a_2 + ... + a_n$ all have different remainders upon division by $n$, if: $a)$ $n=7$ $b)$ $n=8$

2015 Caucasus Mathematical Olympiad, 1

Tags: digit , divisible , remainder , number theory

Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?

1999 Estonia National Olympiad, 1

Tags: power of 2 , power of 3 , remainder , number theory

Let $a, b, c$ and $d$ be non-negative integers. Prove that the numbers $2^a7^b$ and $2^c7^d$ give the same remainder when divided by $15$ iff the numbers $3^a5^b$ and $3^c5^d$ give the same remainder when divided by $16$.