Difference between revisions of "1997 JBMO Problems"
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==Problem 1== | ==Problem 1== | ||
− | + | Show that given any 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than <math>\frac{1}{8}</math>. | |
[[1997 JBMO Problems/Problem 1#Solution|Solution]] | [[1997 JBMO Problems/Problem 1#Solution|Solution]] | ||
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==Problem 2== | ==Problem 2== | ||
− | + | Let <math>\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k</math>. Compute the following expression in terms of <math>k</math>: | |
<cmath> E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}. </cmath> | <cmath> E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}. </cmath> | ||
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==Problem 3== | ==Problem 3== | ||
− | + | Let <math>ABC</math> be a triangle and let <math>I</math> be the incenter. Let <math>N</math>, <math>M</math> be the midpoints of the sides <math>AB</math> and <math>CA</math> respectively. The lines <math>BI</math> and <math>CI</math> meet <math>MN</math> at <math>K</math> and <math>L</math> respectively. Prove that <math>AI+BI+CI>BC+KL</math>. | |
[[1997 JBMO Problems/Problem 3#Solution|Solution]] | [[1997 JBMO Problems/Problem 3#Solution|Solution]] | ||
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==Problem 4== | ==Problem 4== | ||
− | + | Determine the triangle with sides <math>a,b,c</math> and circumradius <math>R</math> for which <math>R(b+c) = a\sqrt{bc}</math>. | |
[[1997 JBMO Problems/Problem 4#Solution|Solution]] | [[1997 JBMO Problems/Problem 4#Solution|Solution]] |
Latest revision as of 23:54, 3 August 2018
Problem 1
Show that given any 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than .
Problem 2
Let . Compute the following expression in terms of :
Problem 3
Let be a triangle and let be the incenter. Let , be the midpoints of the sides and respectively. The lines and meet at and respectively. Prove that .
Problem 4
Determine the triangle with sides and circumradius for which .
Problem 5
Let , , , be positive integers such that Show that at least two of the numbers are even.
See also
1997 JBMO (Problems • Resources) | ||
Preceded by First Olympiad |
Followed by 1998 JBMO Problems | |
1 • 2 • 3 • 4 • 5 | ||
All JBMO Problems and Solutions |