1997 JBMO Problems

Revision as of 21:44, 14 September 2023 by Aryamathematics (talk | contribs) (Problem 1)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 $\frac{1}{8}$.


Hyy I am anshu Kumar anahu from https://t.me/AOPS01

Problem 2

Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$. Compute the following expression in terms of $k$: \[E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.\]


Problem 3

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$.


Problem 4

Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$.


Problem 5

Let $n_1$, $n_2$, $\ldots$, $n_{1998}$ be positive integers such that \[n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2.\] Show that at least two of the numbers are even.


See also

1997 JBMO (ProblemsResources)
Preceded by
First Olympiad
Followed by
1998 JBMO Problems
1 2 3 4 5
All JBMO Problems and Solutions