Difference between revisions of "2024 IMO Problems/Problem 4"
Bobwang001 (talk | contribs) (→Video Solution) |
Codemaster11 (talk | contribs) |
||
Line 36: | Line 36: | ||
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=2024|num-b=2|num-a=4}} |
Revision as of 13:24, 30 July 2024
Let be a triangle with
. Let the incentre and incircle of triangle
be
and
, respectively. Let
be the point on line
different from
such that the line
through
parallel to
is tangent to
. Similarly, let
be the point on line
different from
such that the line through
parallel to
is tangent to
. Let
intersect the circumcircle of
triangle
again at
. Let
and
be the midpoints of
and
, respectively.
Prove that
.
Video Solution(In Chinese)
Video Solution
Video Solution
Part 1: Derive tangent values and
with trig values of angles
,
,
Part 2: Derive tangent values and
with side lengths
,
,
, where
is the midpoint of
Part 3: Prove that and
.
Comments: Although this is an IMO problem, the skills needed to solve this problem have all previously tested in AMC and its system math contests, such as HMMT.
Evidence 1: 2020 Spring HMMT Geometry Round Problem 8
I used the property that because point is on the angle bisector
,
is isosceles. This is a crucial step to analyze
. This technique was previously tested in this HMMT problem.
Evidence 2: 2022 AMC 12A Problem 25
The technique in this AMC problem can be easily and directly applied to this IMO problem to quickly determine the locations of points and
. If you read my solutions to both this AMC problem and this IMO problem, you will find that I simply took exactly the same approach to solve both.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2024 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |