MathDB

3

Part of 2014 Dutch IMO TST

Problems(2)

Prove expression is rational

Source: Dutch IMO TST I Problem 3

7/17/2014
Let aa, bb and cc be rational numbers for which a+bca+bc, b+acb+ac and a+ba+b are all non-zero and for which we have 1a+bc+1b+ac=1a+b.\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}. Prove that (c−3)(c+1)\sqrt{(c-3)(c+1)} is rational.
algebra proposedalgebra
Orthocenter of an acute triangle

Source: Dutch IMO TST II Problem 3

7/17/2014
Let HH be the orthocentre of an acute triangle ABCABC. The line through AA perpendicular to ACAC and the line through BB perpendicular to BCBC intersect in DD. The circle with centre CC through HH intersects the circumcircle of triangle ABCABC in the points EE and FF. Prove that ∣DE∣=∣DF∣=∣AB∣|DE| = |DF| = |AB|.
geometrycircumcirclegeometric transformationreflectiontrapezoidgeometry unsolved