Difference between revisions of "2001 JBMO Problems"

Revision as of 09:22, 11 August 2018

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 $\angle XAC \ne \angle XBC$ .

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?

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.

Revision as of 09:22, 11 August 2018 (view source) Rockmanex3 (talk \| contribs) (2001 JBMO Problems are up!)		Revision as of 09:22, 11 August 2018 (view source) Rockmanex3 (talk \| contribs) m Newer edit →
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	==See Also==		==See Also==
−	{{JBMO box\|year=2001\|before=[[2000 JBMO]]\|~~num-a~~=[[2002 JBMO]]\|five=}}	+	{{JBMO box\|year=2001\|before=[[2000 JBMO]]\|after=[[2002 JBMO]]\|five=}}

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

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