Difference between revisions of "2000 Pan African MO Problems/Problem 1"
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Latest revision as of 05:07, 23 February 2023
Problem
Solve for :
Solution
Let and
. Substitution results in
By the Zero Product Property,
or
. We can divide into two cases.
Case 1:
Substituting back results in . Since both sides don't equal each other when
, we can divide both sides by
to get
. Thus,
, where
is an integer.
Case 2:
Substituting back and applying a Pythagorean Identity results in . Multiplying both sides by
and rearranging results in
From the equation,
or
, where
is an integer. However, both are extraneous solutions because both make either
or
undefined, so there are no solutions in the case.
Therefore the solutions to the equation are .
See Also
2000 Pan African MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All Pan African MO Problems and Solutions |
[[Category:Introductory Trigonometry Problems]