## PROBLEM 4

Consider the number $N=a+3+15+b+35$, where the five terms of the sum are written in ascending order.

Determine $a$ and $b$ for which $N$ is a perfect square.

## Mathematical Gazette
```
If \(\mathrm{a}<3\) (1) and \(15<\mathrm{b}<35\) (2), then \(15<\mathrm{a}+\mathrm{b}<38\) (3).
.2 p

\(\mathrm{N}=\mathrm{a}+\mathrm{b}+53\)

By adding 53 to relation (3), we get \(68<\mathrm{a}+\mathrm{b}+53<91\), so \(68<\mathrm{N}<91\).

1 p
Since \(\mathrm{N}\) is a perfect square, we have \(\mathrm{N}=81\), so \(\mathrm{a}+\mathrm{b}=28\). ..... 1 p
From relations (1) and (2), we have the solutions
1) \(a=0\) and \(b=28\). ..... 1 p
2) \(a=1\) and \(b=27\)......... ..... 1 p
3) \(a=2\) and \(b=26\). ..... 1 p
```