Difference between revisions of "1997 JBMO Problems"
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==Problem 2== | ==Problem 2== | ||
Latest revision as of 19:38, 27 February 2025
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 
.
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)
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 
.
Problem 4
Determine the triangle with sides 
and circumradius 
for which 
.
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.
See also
| 1997 JBMO (Problems • Resources) | ||
| Preceded by First Olympiad |
Followed by 1998 JBMO Problems | |
| 1 • 2 • 3 • 4 • 5 | ||
| All JBMO Problems and Solutions | ||
