Difference between revisions of "2001 JBMO Problems"
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Solve the equation <math>a^3 + b^3 + c^3 = 2001</math> in positive integers. | Solve the equation <math>a^3 + b^3 + c^3 = 2001</math> in positive integers. | ||
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| + | [[2001 JBMO Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
Let <math>ABC</math> be a triangle with <math>\angle C = 90^\circ</math> and <math>CA \ne CB</math>. Let <math>CH</math> be an altitude and <math>CL</math> be an interior angle bisector. Show that for <math>X \ne C</math> on the line <math>CL,</math> we have <math>\angle XAC \ne \angle XBC</math>. Also show that for <math>Y \ne C</math> on the line <math>CH</math> we have <math>\angle XAC \ne \angle XBC</math>. | Let <math>ABC</math> be a triangle with <math>\angle C = 90^\circ</math> and <math>CA \ne CB</math>. Let <math>CH</math> be an altitude and <math>CL</math> be an interior angle bisector. Show that for <math>X \ne C</math> on the line <math>CL,</math> we have <math>\angle XAC \ne \angle XBC</math>. Also show that for <math>Y \ne C</math> on the line <math>CH</math> we have <math>\angle XAC \ne \angle XBC</math>. | ||
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| + | [[2001 JBMO Problems/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
Let <math>ABC</math> be an equilateral triangle and <math>D,E</math> on the sides <math>[AB]</math> and <math>[AC]</math> respectively. If <math>DF,EF</math> (with <math>F \in AE, G \in AD</math>) are the interior angle bisectors of the angles of the triangle <math>ADE</math>, prove that the sum of the areas of the triangles <math>DEF</math> and <math>DEG</math> is at most equal with the area of the triangle <math>ABC</math>. When does the equality hold? | Let <math>ABC</math> be an equilateral triangle and <math>D,E</math> on the sides <math>[AB]</math> and <math>[AC]</math> respectively. If <math>DF,EF</math> (with <math>F \in AE, G \in AD</math>) are the interior angle bisectors of the angles of the triangle <math>ADE</math>, prove that the sum of the areas of the triangles <math>DEF</math> and <math>DEG</math> is at most equal with the area of the triangle <math>ABC</math>. When does the equality hold? | ||
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| + | [[2001 JBMO Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
Let <math>N</math> be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of <math>N</math> which form a triangle of area smaller than 1. | Let <math>N</math> be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of <math>N</math> which form a triangle of area smaller than 1. | ||
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| + | [[2001 JBMO Problems/Problem 4|Solution]] | ||
==See Also== | ==See Also== | ||
{{JBMO box|year=2001|before=[[2000 JBMO]]|after=[[2002 JBMO]]|five=}} | {{JBMO box|year=2001|before=[[2000 JBMO]]|after=[[2002 JBMO]]|five=}} | ||
Revision as of 10:24, 11 August 2018
Problem 1
Solve the equation 
in positive integers.
Problem 2
Let 
be a triangle with 
and 
. Let 
be an altitude and 
be an interior angle bisector. Show that for 
on the line 
we have 
. Also show that for 
on the line we have .
Problem 3
Let be an equilateral triangle and 
on the sides ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
and ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively. If 
(with 
) are the interior angle bisectors of the angles of the triangle 
, prove that the sum of the areas of the triangles 
and 
is at most equal with the area of the triangle . When does the equality hold?
Problem 4
Let 
be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of which form a triangle of area smaller than 1.
See Also
| 2001 JBMO (Problems • Resources) | ||
| Preceded by 2000 JBMO |
Followed by 2002 JBMO | |
| 1 • 2 • 3 • 4 | ||
| All JBMO Problems and Solutions | ||
