Olympiad problems

2016 Finnish National High School Mathematics Comp, 2

Suppose that $y$ is a positive integer written only with digit $1$, in base $9$ system. Prove that $y$ is a triangular number, that is, exists positive integer $n$ such that the number $y$ is the sum of the $n$ natural numbers from $1$ to $n$.

1988 ITAMO, 4

Tags: number theory , triangular number

Show that all terms of the sequence $1,11,111,1111,...$ in base $9$ are triangular numbers, i.e. of the form $\frac{m(m+1)}{2} $for an integer $m$

2020 Colombia National Olympiad, 3

Tags: triangular number , number theory

A number is said to be [i]triangular [/i] if it can be expressed in the form $1 + 2 +...+n$ for some positive integer $n$. We call a positive integer $a$ [i]retriangular [/i] if there exists a fixed positive integer $ b$ such that $aT +b$ is a triangular number whenever $T$ is a triangular number. Determine all retriangular numbers.