2000 USAMO Problems: Difference between revisions
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=== Problem 4 === | === Problem 4 === | ||
Find the smallest positive integer <math>n</math> such that if <math>n</math> squares of a <math>1000\times 1000</math> chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board. | |||
* [[2000 USAMO Problems/Problem 4 | Solution]] | * [[2000 USAMO Problems/Problem 4 | Solution]] | ||
Revision as of 14:53, 30 March 2009
Day 1
Problem 1
Call a real-valued function 
very convex if
![\[\frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y|\]](http://latex.artofproblemsolving.com/4/e/2/4e298e1ba811b64c1726bdb5ffab3dddb4f284e0.png)
holds for all real numbers 
and 
. Prove that no very convex function exists.
Problem 2
Let 
be the set of all triangles 
for which
![\[5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},\]](http://latex.artofproblemsolving.com/7/5/5/7550e0471c956b79aed504fbed336a25b8325d5d.png)
where 
is the inradius and 
are the points of tangency of the incircle with sides 
respectively. Prove that all triangles in are isosceles and similar to one another.
Problem 3
A game of solitaire is played with 
red cards, 
white cards, and 
blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
Day 2
Problem 4
Find the smallest positive integer 
such that if squares of a 
chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
