Difference between revisions of "1999 JBMO Problems"
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==Problem 4== | ==Problem 4== | ||
− | Let <math> | + | Let <math>ABC</math> be a triangle with <math>AB=AC</math>. Also, let <math>D\in[BC]</math> be a point such that <math>BC>BD>DC>0</math>, and let <math>\mathcal{C}_1,\mathcal{C}_2</math> be the circumcircles of the triangles <math>ABD</math> and <math>ADC</math> respectively. Let <math>BB'</math> and <math>CC'</math> be diameters in the two circles, and let <math>M</math> be the midpoint of <math>B'C'</math>. Prove that the area of the triangle <math>MBC</math> is constant (i.e. it does not depend on the choice of the point <math>D</math>). |
[[1999 JBMO Problems/Problem 4|Solution]] | [[1999 JBMO Problems/Problem 4|Solution]] |
Latest revision as of 13:13, 25 August 2018
Problem 1
Let be five real numbers such that , and . If are all distinct numbers prove that their sum is zero.
Problem 2
For each nonnegative integer we define . Find the greatest common divisor of the numbers .
Problem 3
Let be a square with the side length 20 and let be the set of points formed with the vertices of and another 1999 points lying inside . Prove that there exists a triangle with vertices in and with area at most equal with .
Problem 4
Let be a triangle with . Also, let be a point such that , and let be the circumcircles of the triangles and respectively. Let and be diameters in the two circles, and let be the midpoint of . Prove that the area of the triangle is constant (i.e. it does not depend on the choice of the point ).
See Also
1999 JBMO (Problems • Resources) | ||
Preceded by 1998 JBMO Problems |
Followed by 2000 JBMO Problems | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |