MathDB

10.5

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

\sqrt{2+\sqrt{2-\sqrt{2+x}}}=x (V Soros Olympiad 1998-99 Round 1 10.5)

Source:

5/25/2024
Solve the equation 2+2−2+x=x.\sqrt{2+\sqrt{2-\sqrt{2+x}}}=x.
algebraRadicals
locus, <MAC + <CMB = 90^o (V Soros Olympiad 1998-99 Round 2 10.5)

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5/25/2024
An isosceles triangle ABCABC (AB=BCAB = BC) is given on the plane. Find the locus of points MM of the plane such that ABCMABCM is a convex quadrilateral and ∠MAC+∠CMB=90o\angle MAC + \angle CMB = 90^o.
geometryLocus
sum of inradii of AMB , AMC (V Soros Olympiad 1998-99 Round 3 10.5)

Source:

5/26/2024
The radius of the circle inscribed in triangle ABCABC is equal to rr. This circle is tangent to BCBC at point MM and divides the segment AMAM in ratio kk (starting from vertex AA). Find the sum of the radii of the circles inscribed in triangles AMBAMB and AMCAMC.
geometryinradius