2010 USAMO Problems/Problem 4
Problem
Let be a triangle with . Points and lie on sides and , respectively, such that and . Segments and meet at . Determine whether or not it is possible for segments to all have integer lengths.
Solution
We know that angle , as the other two angles in triangle add to . Assume that only , and are integers. Using the Law of Cosines on triangle BIC,
. Observing that is an integer and that we have
and therefore,
The LHS () is irrational, while the RHS is the quotient of the division of two integers and thus is rational. Clearly, there is a contradiction. Therefore, it is impossible for , and to all be integers, which invalidates the original claim that all six lengths are integers, and we are done.
Solution 2
The answer is no.
Suppose otherwise. It is easy to see (through simple angle chasing) that . Also, since is the incenter, we have . Using the Law of Cosines, we have so that is irrational. But , thus , implying that is rational, contradiction.
Solution 3
The result can be also proved without direct appeal to trigonometry, via just the angle bisector theorem and the structure of Pythagorean triples. (This is a lot more work).
A triangle in which all the required lengths are integers exists if and only if there exists a triangle in which and are relatively-prime integers and the lengths of the segments are all rational (we divide all the lengths by the or conversely multiply all the lengths by the least common multiple of the denominators of the rational lengths).
Suppose there exists a triangle in which the lengths and are relatively-prime integers and the lengths are all rational.
Since is the bisector of , by the angle bisector theorem, the ratio , and since is the bisector of , . Therefore, . Now is by assumption rational, so is rational, but and are assumed integers so must also be rational. Since is the hypotenuse of a right-triangle, its length is the square root of an integer, and thus either an integer or irrational, so must be an integer.
With and relatively-prime, we conclude that the side lengths of must be a Pythagorean triple: , with relatively-prime positive integers and odd.
Without loss of generality, . By the angle bisector theorem,
Since is a right-triangle, we have:
and so is rational if and only if is a perfect square.
Also by the angle bisector theorem,
and therefore, since is a right-triangle, we have:
and so is rational if and only if is a perfect square.
Combining the conditions on and , we see that and must both be perfect squares. If it were so, their ratio, which is , would be the square of a rational number, but is irrational, and so the assumed triangle cannot exist.
Solution 4
We proceed by contradiction.
FTSOC, let have integer lengths. Then as well. By trigonometry, Rearranging we find . As so but there are no possible angles in the interval that result in a rational cosine by Niven's theorem. So we have contradiction.
~Aaryabhatta1.
Video Solution
https://www.youtube.com/watch?v=Dh9H_DNDMAg
See also
2010 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
2010 USAJMO (Problems • Resources) | ||
Preceded by Problem 5 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
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