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Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Shape and Territory

Age 16 to 18 Challenge Level:

If for a triangle $ABC$

$\tan(A - B) + \tan(B - C) + \tan(C - A) = 0$

what can you say about the triangle?