IMC_2012_web_solutions

The squares of the even numbers have the form 2 2 n

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The squares of the even numbers have the form 2 ) 2 ( n , that is, 2 4 n .We see that the number 1 4 2 n occurs to the left of the cell containing 2 4 n . Below 1 4 2 n there occur the numbers 1 2 4 ,.., 3 4 , 2 4 2 2 2 n n n n , and then in the cells to the right of the cell containing 1 2 4 2 n n , there occur the numbers 1 4 4 ,..., 3 2 4 , 2 2 4 2 2 2 n n n n n n 2 ) 1 2 ( n . Now 1936 44 2 and 2025 45 2 . Thus 2011 is in the same row as 2025 and to the left of it, in the sequence 1981,….,2012,…,2025, and below these occur the numbers 2163,…., 2 47 2209 , with 2208 below 2025 as shown below. It follows that 2195 is the number below 2012. In the diagram below the square numbers are shown in bold. 32 31 17 16 15 14 13 30 18 5 4 3 12 29 19 6 1 2 11 28 20 7 8 9 10 27 21 22 23 24 25 26 1937 1936 32 31 17 16 15 14 13 30 18 5 4 3 12 29 19 6 1 2 11 28 20 7 8 9 10 27 21 22 23 24 25 26 1981 1982 2012 2024 2025 2164 2165 2195 2207 2208 2209
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14 25. The diagram shows a ceramic design by the Catalan architect Antoni Gaudi. It is formed by drawing eight lines connecting points which divide the edges of the outer regular octagon into three equal parts, as shown. What fraction of the octagon is shaded? A 5 1 B 9 2 C 4 1 D 10 3 E 16 5 Solution: B We consider the triangular segment of the octagon formed by joining two adjacent vertices, P and Q to the centre, O. For convenience, we show this segment, drawn on a larger scale, on the left, where we have added the lines RW , ST , TW and UV. These lines are parallel to the edges of the triangle POQ , as shown and together with the lines RU and SV they divide the triangle OPQ into 9 congruent triangles, of which 2 are shaded. Thus 9 2 of the segment is shaded. The same holds for all the other congruent segments of the octagon. So 9 2 of the whole octagon is shaded. Extension Problem 25.1 In the solution we have said that the triangle OPQ is divided into 9 congruent triangles, but we have not justified the claim that the triangles are congruent. Complete the argument by giving a proof that these triangles are congruent. P Q O P Q O R S T U V W
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