# 1975 Canadian MO Problems/Problem 4

## Problem 4

For a positive number such as $3.27$, $3$ is referred to as the integral part of the number and $.27$ as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

## Solution

Let $a$ be the integer part and $b$ be the decimal part, thus, we have the G.P. $(b,a,a+b)$

So, $b(a+b)=a^2$ $ab+b^2=a^2$

There we have a quadratic equation. We must isolate $a$ or $b$. $a^2-ab-b^2=0$ $b=\frac{-a\pm a\sqrt{5}}{2}$

If the number must be positive, thus we'll consider only the solution $b=\frac{-a+a\sqrt{5}}{2}$

As $b$ is the decimal part, then it must be lower than 1 $b<1$ $\frac{-a+a\sqrt{5}}{2}<1$ $a<\frac{2}{\sqrt{5}-1}$ $a<\frac{1+\sqrt{5}}{2}$

This is the golden ratio and it's approximately $1.618$.

As $a$ must be an integer, thus $a=1$. Therefore $b=\frac{-a+a\sqrt{5}}{2}$ $b=\frac{-1+\sqrt{5}}{2}$

To find our number we must sum $a+b$, so $a+b$ $1+\frac{-1+\sqrt5}{2}$ $\boxed{\frac{1+\sqrt{5}}{2}}$

The number we're searching for is the golden ratio.

 1975 Canadian MO (Problems) Preceded byProblem 3 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 Followed byProblem 5