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 be the three real numbers, and from the initial statements, we know that
and
.
Expanding, substituting, and simplifying results in
By the Zero Product Property, either
or
.
If , then
. Since
,
, and
, we must have
in this case. Substituting the values back in satisfies both of the original equations.
If , then
, so by the Zero Product Property,
or
. If
, then we have
, so any real value of
can work. If
, then we have
, so any real value of
can work. Therefore
and
are also solutions (where
and
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 |