Let AD, BE, and CF intersect the
circumcircle of triangle ABC again at D1, E1
and F1, respectively. Let I and I1 be the
incenters of triangles DEF and D1E1F1,
respectively. Prove that E, F, I, I1 are
concyclic. 08IMO6 Let ABCD be a convex
quadrilateral with BA 6= BC. Denote the
incircles of triangles ABC and ADC by 1 and
2 respectively. Suppose that there exists a
circle tangent to ray BA beyond A and to
the ray BC beyond C, which is also tangent
to the lines AD and CD. Prove that the
common external tangents to 1 and 2
intersect on . 20JMO2 Let be the incircle
of a fixed equilateral triangle ABC. Let ‘ be
a variable line that is tangent to and meets
the interior of segments BC and CA at
points P and Q, respectively. A point R is
chosen such that P R = P A and QR = QB.
Find all possible locations of the point R,
over all choices of ‘. 19USEMO1 Let ABCD
be a cyclic quadrilateral. A circle centered
at O passes th
Evan Chen
19 February 2025
1
Evan
is alsoChen — 19
tangent to February
the lines2025
AD and CD. Prove that the common external tangents to 1 and
2 intersect on . 20JMO2 Let be the incircle of a fixed equilateral triangle ABC. Let ‘ be a
variable line that is tangent to and meets the interior of segments BC and CA at points P
and Q, respectively. A point R is chosen such that P R = P A and QR = QB. Find all
possible locations of the point R, over all choices of ‘. 19USEMO1 Let ABCD be a cyclic
quadrilateral. A circle centered at O passes th
§1 Introduction
According to all known laws of aviation, there is no way a bee should be able to fly. Its
wings are too small to get its fat little body off the ground. The bee, of course, flies
anyway because bees don’t care what humans think is impossible. Yellow, black. Yellow,
black. Yellow, black. Yellow, black. Ooh, black and yellow! Let’s shake it up a little.
Barry! Breakfast is ready! Coming! Hang on a second. Hello? Barry? Adam? Can you
believe this is happening? I can’t. I’ll pick you up. Looking sharp. Use the stairs, Your
father paid good money for those. Sorry. I’m excited. Here’s the graduate. We’re very
proud of you, son.
Theorem 1.1
I might as well state one of the main results of this article up front, so the following
discussion seems a little more motivated.
Proof. According to all known laws of aviation, there is no way a bee should be able to
fly. Its wings are too small to get its fat little body off the ground. The bee, of course,
flies anyway because bees don’t care what humans think is impossible. Yellow, black.
Yellow, black. Yellow, black. Yellow, black. Ooh, black and yellow! Let’s shake it up a
little. Barry! Breakfast is ready! Coming! Hang on a second. Hello? Barry? Adam?
Can you believe this is happening? I can’t. I’ll pick you up. Looking sharp. Use the
stairs, Your father paid good money for those. Sorry. I’m excited. Here’s the graduate.
We’re very proud of you, son.
