Two congruent concentric equilateral triangles. Prove that the gold fraction is red/blue.
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The Christmas star
Category: Advanced
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Ruyles squares
There are four possible squares with a vertex at distance a from the centre and two vertices on the circle of radius r. Find their areas in terms of a and r.
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Square one
Three squares share a vertex. Two line segments pass through side midpoints. Prove that a : b = c : d.
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The inner parallel
A cyclic quadrilateral with its diagonals and two altitudes. Prove that AB is parallel to EF.
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The door stop
A triangle with a cevian. What is α?
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The oppressed minority
A square with two inscribed squares. What is the maximal proportion green : blue?
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Cherry picking
Two quarter circles and a red circle tangent to two chords. What fraction is red?
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Entanglement
Two circles with two common tangents. Three red lines through tangency points and centres. Prove that they are concurrent.
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The maxbox
A square containing a red square of variable size sharing a vertex with a blue rectangle. What is the maximal blue fraction?
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The grazing shot
A parabola and its directrix in red. Two tangents intersect at a point on the directrix. What’s the angle α?