A point is chosen on the side of a rectangle . From the vertex , the perpendicular is dropped to the segment . The segments and divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to are equal then the third circle is also equal to them.

In a rectangle be a rectangle and , let be the midpoint of and an arbitrary inner point of . Let and be the feet of perpendiculars drawn correspondingly from to and from to . Prove that the points are concyclic.

Consider decompositions of an chessboard into non-overlapping rectangles subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If is the number of white squares in the -th rectangle, then .
Find the maximum value of for which such a decomposition is possible. For this value of , determine all possible sequences .

A square is cut into several rectangles, none of which is a square ,so that the sides of each rectangles are parallel to the sides of a square .For each rectangle with sides a,b,a<b compute the ratio a/b Prove that the sum of these ratios is at least 1

Treasure was buried in a single cell of an (, ) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid with and . Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up detectors, which may only be run simultaneously after all detectors are ready.

In terms of and , what is the minimum required to gaurantee to determine the location of the treasure?

Let be a triangle with incenter , and suppose the incircle is tangent to and at and . Denote by and the reflections of and over . Let be the intersection of with , and let be the midpoint of . Prove that and are perpendicular.

Let be a triangle. Construct rectangles , , and outside such that . Let and intersect at and define similarly. Prove that line bisects .

Linus Tang

We have an board, filled with rectangles aligned to the grid. The rectangles cover all the board and are never superposed. Find, in terms of , the smallest value the sum of the diagonals of the rectangles can take.

Suppose is a cyclic quadrilateral inscribed in a circle of radius one unit. If , prove that is a square.

Let be an acute triangle. Point lies on side . Let be the circumcenters of triangles and , respectively. Suppose that the points lies on a circle centered at . Let be the orthocenter of triangle . Prove that .

Zuming Feng.