MathDB

2003 Italy TST

Part of Italy TST

Subcontests

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Trominoes on a white and black chessboard

For nn an odd positive integer, the unit squares of an n×nn\times n chessboard are coloured alternately black and white, with the four corners coloured black. A tromino is an LL-shape formed by three connected unit squares. (a)(a) For which values of nn is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard? (b)(b) When it is possible, find the minimum number of trominoes needed.

O lies on the circumcircle of ABC

Let B≠AB\not= A be a point on the tangent to circle S1S_1 through the point AA on the circle. A point CC outside the circle is chosen so that segment ACAC intersects the circle in two distinct points. Let S2S_2 be the circle tangent to ACAC at CC and to S1S_1 at some point DD, where DD and BB are on the opposite sides of the line ACAC. Let OO be the circumcentre of triangle BCDBCD. Show that OO lies on the circumcircle of triangle ABCABC.
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