2001 JBMO Problems

Problem 1

Solve the equation $a^3 + b^3 + c^3 = 2001$ in positive integers.


Problem 2

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \ne CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \ne C$ on the line $CL,$ we have $\angle XAC \ne \angle XBC$. Also show that for $Y \ne C$ on the line $CH$ we have $\angle YAC \ne \angle YBC$.


Problem 3

Let $ABC$ be an equilateral triangle and $D,E$ on the sides $[AB]$ and $[AC]$ respectively. If $DF,EF$ (with $F \in AE, G \in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold?

Bonus Question:

In the above problem, prove that $DF/EG = AD/AE$.

- Proposed by Kris17


Problem 4

Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.


See Also

2001 JBMO (ProblemsResources)
Preceded by
2000 JBMO
Followed by
2002 JBMO
1 2 3 4
All JBMO Problems and Solutions