2023新加坡数学竞赛SMO（含答案）Open组第二轮.pdf

SMO Open 2023 Rd.2 1. In a scalene triangle 戆错暗戆 with centroid 戆 and circumcircle 饊寯 centred at 肀? the extension of 戆错昂 meets 饊寯 at 饊亐, lines 戆错暗 and 戆娥€亐 intersect at 肀傷 and lines 戆错岸 and 戆叼€亐 intersect at 肀? Suppose the circumcentre 肀 of the triangle 戆错眱肀 lies on 饊寯 and 戆? 肀? 肀 are collinear. Prove that 鈭犿按戆喉眰 = 90掳. Solution: Consider the homothety 鈩 centred at 戆 which sends 肀 to 肀 and 饊寯 to the circumcentre of 鈭嗧按肀冺眲 . Since 戆错眴 = 2戆错眰, 鈩 has scale factor 2, so 戆? 戆 are midpoints of 戆错眱, 戆错眲 respectively. This implies that 饊亐 is the centroid of 鈭嗧按肀冺眲 . Thus, 鈩?戆? = 饊亐, which implies 戆喉眰 ||饊亐肀? Thus 鈭犿按戆喉眰 = 鈭犿按饊亐肀 = 90掳. Remark: A homothety is just a scaling with respect to a particular point (A, in this case). If one is unfamiliar with the concept, the statements can easily be replaced by a series of claims that use similar triangles (or the midpoint theorem, in this case.) Solutions by Way Tan 2. A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset 肀 of dominoes is chosen. Is it true that at least one of the following 2 statements is false? (1) There are 2022 more horizontal dominoes than vertical dominoes in 肀 . (2) The cells covered by the dominoes in 肀 can be tiled completely and exactly by the tetrominoes shown below (including rotations reflections): Solution: Yes, at least one of the statements is false. Suppose otherwise that both statements (1) and (2) are true, and let 肀 contain 肀 vertical and 肀 + 2022 horizontal dominoes. Then there would be 肀 + 1011 L-shapes. Shade the grid such that the rows are alternately shaded and unshaded as shown above. Note that: 鈥 A horizontal domino has 2 shaded or 2 unshaded squares. 鈥 A vertical domino has 1 shaded and 1 unshaded square. 鈥 An L-shape has 3 shaded and 1 unshaded square, or vice versa. This mea