2000 JBMO Problems

Problem 1

Let $x$ and $y$ be positive reals such that \[x^3 + y^3 + (x + y)^3 + 30xy = 2000.\] Show that $x + y = 10$.


Problem 2

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer.


Problem 3

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$.


Problem 4

At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is knowns that there were no draws. Find $n$.


See Also

2000 JBMO (ProblemsResources)
Preceded by
1999 JBMO
Followed by
2001 JBMO
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All JBMO Problems and Solutions