Found problems: 114
2018 Argentina National Olympiad, 1
Let $p$ a prime number and $r$ the remainder of the division of $p$ by $210$. It is known that $r$ is a composite number and can be written as the sum of two non-zero perfect squares. Find all primes less than $2018$ that satisfy these conditions.
2021 Canadian Junior Mathematical Olympiad, 2
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 $k$?
1997 Poland - Second Round, 5
We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.
2012 QEDMO 11th, 6
Let $p$ be an odd prime number. Prove that
$$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$
2001 Estonia National Olympiad, 2
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.
2006 Thailand Mathematical Olympiad, 11
Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$
2014 Rioplatense Mathematical Olympiad, Level 3, 1
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.
2011 Saudi Arabia IMO TST, 1
Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.
2013 Saudi Arabia Pre-TST, 2.1
Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.
2004 Thailand Mathematical Olympiad, 6
Let $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. Find the remainder when $f(x^7)$ is divided by $f(x)$.
2017 Regional Olympiad of Mexico Northeast, 1
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 $n$?
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 $35$?
1991 Greece National Olympiad, 4
If we divide number $1^{1990}+2^{1990}+3^{1990}+...+1990^{1990}$ with $10$, what remainder will we find?
2008 Postal Coaching, 6
Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.
2004 Thailand Mathematical Olympiad, 17
Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$.
2017 Junior Regional Olympiad - FBH, 4
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?
1994 Bulgaria National Olympiad, 5
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.
2019 May Olympiad, 1
A positive integer is called [i]piola [/i] if the $9$ is the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9$ and $10$ and it's digits are all different and nonzero. How many [i]piolas[/i] are there between $ 1$ and $100000$?
2024 Israel National Olympiad (Gillis), P2
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 $y$.
2019 Polish Junior MO First Round, 4
Positive integers $a, b, c$ have the property that:
$\bullet$ $a$ gives remainder $2$ when divided by $b$,
$\bullet$ $b$ gives remainder $2$ when divided by $c$,
$\bullet$ $c$ gives remainder $4$ when divided by $a$.
Prove that $c = 4$.
2023 AMC 12/AHSME, 24
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
2006 Thailand Mathematical Olympiad, 13
Compute the remainder when $\underbrace{\hbox{11...1}}_{\hbox{1862}}$ is divided by $2006$
2019 India PRMO, 21 incorrect
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 $E$
Kvant 2020, M2601
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
2016 Bosnia And Herzegovina - Regional Olympiad, 4
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 $2016$