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Seven chords

A regular 7-gon inscribed in a circle (ABCDEFG), with some other point H on the circle, in the arc CD. Show that the length sum of the blue chords equals the length sum of the green chords.

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Advanced

The gondola lift

A tangent and its normal in an arbitrary point on an ellipse with eccentricity e. The focal points are also shown. What is the minimum value of blue : red in terms of e?

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Advanced

The grassy knoll

A triangle with three cevians that are concurrent in an arbitrary interior point. What is the maximum value of bdf/ace?

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Intermediate

The tipi entrance

A semicircle with three tangents. Prove that purple : green = red : yellow.

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Intermediate

Baby block II

A square containing another square with an extended side. What is tan(α) in terms of the red and green segment lengths?

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Beginner

Flying purple people eater

What is the area of the purple people eater, in terms of the lengths AD and BC?

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Intermediate

Mountain roads

A circular mountain peak has coordinates (0,0,3) and its base has towns C=(-4,0,0), E=( 4,0,0) on opposite sides. Town D=(0,4,0) is halfway between them, around the base. A road goes partially up and down the mountain connecting C and E. Another road connects C and D. Both roads are shortest possible. What is the ratio of the road lengths?

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Beginner

Mouth to mouth

What is the ratio of the blue and yellow lines?

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Beginner

Equal rights

Two squares sharing a vertex. Two vertices are connected by a line segment. What is yellow : green?

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Intermediate

Holes in my heart

The heart is divided into right triangles, each of which contains a blue circle. What is the total circumference of the blue circles in relation to the green perimeter and the lengths of the purple and orange lines in the centre?