2004 Pan African MO Problems/Problem 4

Problem

Three real numbers satisfy the following statements:

(1) the square of their sum equals to the sum their squares.

(2) the product of the first two numbers is equal to the square of the third number.

Find these numbers.

Solution

Let $a,b,c$ be the three real numbers, and from the initial statements, we know that $(a+b+c)^2 = a^2 + b^2 + c^2$ and $ab = c^2$.


Expanding, substituting, and simplifying results in \begin{align*} a^2 + b^2 + c^2 + 2ab + 2ac + 2bc &= a^2 + b^2 + c^2 \\ 2ab + 2ac + 2bc &= 0 \\ 2c^2 + 2ac + 2bc &= 0 \\ 2c(a+b+c) &= 0. \end{align*} By the Zero Product Property, either $c = 0$ or $a+b+c = 0$.


If $a+b+c = 0$, then $a^2 + b^2 + c^2 = 0$. Since $a^2 \ge 0$, $b^2 \ge 0$, and $c^2 \ge 0$, we must have $a = b = c = 0$ in this case. Substituting the values back in satisfies both of the original equations.


If $c = 0$, then $ab = 0$, so by the Zero Product Property, $a = 0$ or $b = 0$. If $a = c = 0$, then we have $(0+b+0)^2 = 0 + b^2 + 0 = b^2$, so any real value of $b$ can work. If $b = c = 0$, then we have $(a+0+0)^2 = a^2 + 0 + 0 = a^2$, so any real value of $a$ can work. Therefore $(a,0,0)$ and $(0,b,0)$ are also solutions (where $a$ and $b$ are reals), and substituting the values back in satisfies both of the original equations.


Therefore, the numbers that satisfies the two statements are either all zeroes or two zeroes and one non-zero number.

See Also

2004 Pan African MO (Problems)
Preceded by
Problem 3
1 2 3 4 5 6 Followed by
Problem 5
All Pan African MO Problems and Solutions
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