Difference between revisions of "2009 IMO Problems"
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− | Let <math>ABC</math> be a triangle with <math>AB=AC</math>. The angle bisectors of <math>\angle CAB</math> and <math>\angle | + | Let <math>ABC</math> be a triangle with <math>AB=AC</math>. The angle bisectors of <math>\angle CAB</math> and <math>\angle ABC</math> meet the sides <math>BC</math> and <math>CA</math> at <math>D</math> and <math>E</math>, respectively. Let <math>K</math> be the incentre of triangle <math>ADC</math>. Suppose that <math>\angle BEK=45^\circ</math>. Find all possible values of <math>\angle CAB</math>. |
''Authors: Jan Vonk and Peter Vandendriessche, Belgium, and Hojoo Lee, South Korea'' | ''Authors: Jan Vonk and Peter Vandendriessche, Belgium, and Hojoo Lee, South Korea'' | ||
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''Author: Dmitry Khramtsov, Russia'' | ''Author: Dmitry Khramtsov, Russia'' | ||
− | + | {{IMO box|year=2009|before=[[2008 IMO Problems]]|after=[[2010 IMO Problems]]}} |
Latest revision as of 09:23, 10 September 2020
Problems of the 50th IMO 2009 in Bremen, Germany.
Contents
Day I
Problem 1.
Let be a positive integer and let
be distinct integers in the set
such that
divides
for
. Prove that
doesn't divide
.
Author: Ross Atkins, Australia
Problem 2.
Let be a triangle with circumcentre
. The points
and
are interior points of the sides
and
respectively. Let
and
be the midpoints of the segments
and
, respectively, and let
be the circle passing through
and
. Suppose that the line
is tangent to the circle
. Prove that
.
Author: Sergei Berlov, Russia
Problem 3.
Suppose that is a strictly increasing sequence of positive integers such that the subsequences
![$s_{s_1},s_{s_2},s_{s_3},\ldots$](http://latex.artofproblemsolving.com/5/7/8/578b37a84533d816f7bb21a83536aa751620abc2.png)
![$s_{s_1+1},s_{s_2+1},s_{s_3+1},\ldots$](http://latex.artofproblemsolving.com/9/4/3/943bd637535b539c3df40146ad44f3ab658db56c.png)
are both arithmetic progressions. Prove that the sequence is itself an arithmetic progression.
Author: Gabriel Carroll, USA
Day 2
Problem 4.
Let be a triangle with
. The angle bisectors of
and
meet the sides
and
at
and
, respectively. Let
be the incentre of triangle
. Suppose that
. Find all possible values of
.
Authors: Jan Vonk and Peter Vandendriessche, Belgium, and Hojoo Lee, South Korea
Problem 5.
Determine all functions from the set of positive integers to the set of positive integers such that, for all positive integers
and
, there exists a non-degenerate triangle with sides of lengths
![$a,f(b)$](http://latex.artofproblemsolving.com/8/e/6/8e645a0393a2bb5633d525e276321203781753af.png)
![$f(b+f(a)-1)$](http://latex.artofproblemsolving.com/7/e/2/7e24db20845eba2e50f55b74a222d4ef4858c53a.png)
(A triangle is non-degenerate if its vertices are not collinear.)
Author: Bruno Le Floch, France
Problem 6.
Let be distinct positive integers and let
be a set of
positive integers not containing
. A grasshopper is to jump along the real axis, starting at the point
and making
jumps to the right with lengths
in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in
.
Author: Dmitry Khramtsov, Russia
2009 IMO (Problems) • Resources | ||
Preceded by 2008 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2010 IMO Problems |
All IMO Problems and Solutions |