Found problems: 114
2010 District Olympiad, 2
Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.
2007 Thailand Mathematical Olympiad, 18
Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.
2013 Saudi Arabia IMO TST, 4
Determine whether it is possible to place the integers $1, 2,...,2012$ in a circle in such a way that the $2012$ products of adjacent pairs of numbers leave pairwise distinct remainders when divided by $2013$.
2018 Ecuador Juniors, 5
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.
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.
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.
2012 Bosnia And Herzegovina - Regional Olympiad, 3
Find remainder when dividing upon $2012$ number $$A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011$$
2014 Junior Regional Olympiad - FBH, 4
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 $4a+3b+2c=30$
1941 Moscow Mathematical Olympiad, 085
Prove that the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.
1984 Spain Mathematical Olympiad, 8
Find the remainder upon division by $x^2-1$ of the determinant
$$\begin{vmatrix}
x^3+3x & 2 & 1 & 0
\\ x^2+5x & 3 & 0 & 2
\\x^4 +x^2+1 & 2 & 1 & 3
\\x^5 +1 & 1 & 2 & 3
\\ \end{vmatrix}$$
2020 New Zealand MO, 2
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 $9$.
2009 Ukraine Team Selection Test, 6
Find all odd prime numbers $p$ for which there exists a natural number $g$ for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.
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 $2016$
2004 Thailand Mathematical Olympiad, 17
Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$.