1998 USAMO Problems/Problem 6
Problem
Let be an integer. Find the largest integer
(as a function of
) such that there exists a convex
-gon
for which exactly
of the quadrilaterals
have an inscribed circle. (Here
.)
Solution
Lemma: If quadrilaterals and
are tangential, and
is the longest side quadrilateral
for all
, then quadrilateral
is not tangential.
Proof:
If quadrilaterals and
are tangential, then
must have side length of
, and
must have side length of
(One can see this from what is known as walk-around). Suppose quadrilateral
is tangential. Then, again, we see that
must have side length
. We assumed by lemma that
for all
, so we have
,
, and
. If we add up the side lengths
, we get:
However, by the lemma, we assumed that ,
, and
. Adding these up, we get:
which is a contradiction. Thus, quadrilateral is not tangential, proving the lemma.
By lemma, the maximum number of quadrilaterals in a -gon occurs when the tangential quadrilaterals alternate, giving us
.
See Also
1998 USAMO (Problems • Resources) | ||
Preceded by Problem 5 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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