1963 IMO Problems/Problem 3
Contents
Problem
In an -gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation
![$a_1\ge a_2\ge \cdots \ge a_n$](http://latex.artofproblemsolving.com/4/8/b/48baac61191c39562168731a01e8e841b4d4756f.png)
Prove that .
Solution
Let ,
, etc.
Plot the -gon on the cartesian plane such that
is on the
-axis and the entire shape is above the
-axis. There are two cases: the number of sides is even, and the number of sides is odd:
In this case, the side with the topmost points will be . To obtain the
-coordinate of this top side, we can multiply the lengths of the sides
,
, ...
by the sine of the angle they make with the
-axis:
We can obtain the -coordinate of the top side in a different way by multiplying the lengths of the sides
,
, ...
by the sine of the angle they make with the
-axis to get the
-coordinate of the top side:
It must be true that .
Solution 2
Define the vector to equal
. Now rotate and translate the given polygon in the Cartesian Coordinate Plane so that the side with length
is parallel to
. We then have that
But for all
, so
for all . This shows that
, with equality when
. Therefore
There is equality only when for all
. This implies that
and
, so we have that
.
See Also
1963 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |