The International Olympiad of Mathematics took place on July 8 and 9, and the six questions were made available the second day. This was a two day contest, three questions per day, and four and a half hours per day, giving a total of nine hours for the six questions. The international contest included questions on geometry, algebra, combinatorics, and number theory.
Contest questions may be downloaded here.
This year, it seems that problems 1 and 4 were the easiest (by comparison). I’ve only looked at problem 4 in detail (probably because it’s geometry).
Problem
Let P be a point inside the triangle . The lines
,
and
intersect the circumcircle
of triangle
again at the points K, L and M respectively. The tangent to
at C intersects the line
at S. Suppose that
. Prove that
.
Solution
As often with geometry problems, a variety of different solutions are possible. Some are simple, while others rely on homothetic transformations and inversions. I’m going to go with the method I came up with (albeit with a few peeks at the Art of Problem Solving forums).
From the Power of a Point theorem, we have
But it’s given that , so
Or,
This is enough to show that triangles and
are similar, since they also share an angle.
Therefore,
Or,
Since ,
Triangles and
are similar, since they share an angle and
. Thus,
Or,
From (1),
Or equivalently,
Notice that triangles and
are similar (
and
subtending the same arc); so,
Similarly, triangles and
are also similar, giving
From (2),
This shows that , as desired. QED.
hello,,, can you help me,,, do you have any books recommendation in preparing for the international mathematical olympiad,,,, maybe looking at your specialty,, books that are about imo level geometry?
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check imo copendium
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