Olympiad problems

2020-21 IOQM India, 28

A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \geq 2 $. Find the number of good numbers in the set $\{1,2 \dots , 100\}$.

2020-21 IOQM India, 25

Tags: IOQM , number theory , Olympiad , Olympiads

For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that $$ \langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N $$ find the sum of the squares of the digits of $N$.

2022 Nigerian Senior MO Round 2, Problem 5

Tags: combinatorics , Olympiads

For how many paths comsisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram below, is the word $\textup{OLYMPIADS}$ spelled out as the path is traversed from beginning to end? $\begin{tabular}{ccccccccccccccccc}& & & & & & & & O & & & & & & & &\\ & & & & & & & O & L & O & & & & & & &\\ & & & & & & O & L & Y & L & O & & & & & &\\ & & & & & O & L & Y & M & Y & L & O & & & & &\\ & & & & O & L & Y & M & P & M & Y & L & O & & & &\\ & & & O & L & Y & M & P & I & P & M & Y & L & O & & &\\ & & O & L & Y & M & P & I & A & I & P & M & Y & L & O & &\\ & O & L & Y & M & P & I & A & D & A & I & P & M & Y & L & O &\\ O & L & Y & M & P & I & A & D & S & D & A & I & P & M & Y & L & O \end{tabular}$

2020-21 IOQM India, 13

Tags: number theory , IOQM , Olympiad , Olympiads

Find the sum of all positive integers $n$ for which $\mid 2^n + 5^n - 65 \mid$ is a perfect square.