Competition problem — Algebra — Maths Olympiad Prep

Library / /10 of 15

Algebra Difficulty 5.4 AIME, harder Prove it Philippines

Problem:

A positive integer $T$ is said to be triangular if $T = 1 + 2 + \cdots + n$ for some positive integer $n$ . Find the smallest positive integer $k$ such that whenever $T$ is triangular, $81T + k$ is also triangular.

Solution

Solution:

Clearly, taking $T = 1$ , we must have $k \geq 10$ . We show that $k = 10$ indeed works. If $T$ is triangular, then $T = \frac{n(n+1)}{2}$ for some positive integer $n$ . Then, we have
$\begin{aligned} 81T + 10 & = \frac{81n(n+1)}{2} + 10 \\ & = \frac{81n^2 + 81n + 20}{2} \\ & = \frac{(9n + 4)(9n + 5)}{2} \end{aligned}$
which indeed is triangular as well.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.