Difference between revisions of "2014 USAJMO Problems"
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===Problem 2=== | ===Problem 2=== | ||
− | Let <math>\triangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^\ | + | Let <math>\triangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math>\triangle{ABC}</math>, respectively. |
(a) Prove that line <math>OH</math> intersects both segments <math>AB</math> and <math>AC</math>. | (a) Prove that line <math>OH</math> intersects both segments <math>AB</math> and <math>AC</math>. |
Revision as of 20:44, 29 April 2014
Contents
Day 1
Problem 1
Let ,
,
be real numbers greater than or equal to
. Prove that
Solution
Problem 2
Let be a non-equilateral, acute triangle with
, and let
and
denote the circumcenter and orthocenter of
, respectively.
(a) Prove that line intersects both segments
and
.
(b) Line intersects segments
and
at
and
, respectively. Denote by
and
the respective areas of triangle
and quadrilateral
. Determine the range of possible values for
.
Problem 3
Let be the set of integers. Find all functions
such that
for all
with
.