Difference between revisions of "1997 JBMO Problems"
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Determine the triangle with sides <math>a,b,c</math> and circumradius <math>R</math> for which <math>R(b+c) = a\sqrt{bc}</math>. | Determine the triangle with sides <math>a,b,c</math> and circumradius <math>R</math> for which <math>R(b+c) = a\sqrt{bc}</math>. | ||
| − | + | ''Romania'' | |
==Problem 5== | ==Problem 5== | ||
Let <math>n_1</math>, <math>n_2</math>, <math>\ldots</math>, <math>n_{1998}</math> be positive integers such that <cmath> n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2. </cmath> Show that at least two of the numbers are even | Let <math>n_1</math>, <math>n_2</math>, <math>\ldots</math>, <math>n_{1998}</math> be positive integers such that <cmath> n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2. </cmath> Show that at least two of the numbers are even | ||
Revision as of 16:15, 30 July 2015
Problem 1
Show that given any 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than 
Bulgaria
Problem 2
Let 
. Compute the following expression in terms of 
:
![\[E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.\]](http://latex.artofproblemsolving.com/c/9/9/c99e935bc847cdcaee76b4a8af84db262067eca7.png)
Ciprus
Problem 3
Let 
be a triangle and let 
be the incenter. Let 
, 
be the midpoints of the sides 
and 
respectively. The lines 
and 
meet 
at 
and 
respectively. Prove that 
.
Greece
Problem 4
Determine the triangle with sides 
and circumradius 
for which 
.
Romania
Problem 5
Let 
, 
, 
, 
be positive integers such that ![\[n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2.\]](http://latex.artofproblemsolving.com/2/7/9/27994bc5861679d68c68e81da1b333f9770e3752.png)
Show that at least two of the numbers are even
