Difference between revisions of "2001 JBMO Problems"
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==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 | + | 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 YAC \ne \angle YBC</math>. |
[[2001 JBMO Problems/Problem 2|Solution]] | [[2001 JBMO Problems/Problem 2|Solution]] | ||
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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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+ | Bonus Question: | ||
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+ | In the above problem, prove that <math>DF/EG = AD/AE</math>. | ||
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+ | - Proposed by Kris17 | ||
[[2001 JBMO Problems/Problem 3|Solution]] | [[2001 JBMO Problems/Problem 3|Solution]] |
Latest revision as of 18:09, 8 December 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 and 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?
Bonus Question:
In the above problem, prove that .
- Proposed by Kris17
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 |