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 $\angle YAC \ne \angle YBC$ .

Solution

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:

Let $ABC$ be an equilateral triangle and $D,E$ on the sides $[AB]$ and $[AC]$ respectively. If $DF,EG$ (with $F \in AE, G \in AD$ ) are the interior angle bisectors of the angles of the triangle $ADE$ , prove that $DF/EG = AD/AE$ .

Proposed by Kris17

Solution

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.

Solution

2001 JBMO (Problems • Resources)
Preceded by 2000 JBMO	Followed by 2002 JBMO
1 • 2 • 3 • 4
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2001 JBMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

See Also