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ma017-2010-sol2

# ma017-2010-sol2 - MA017 2010 HOMEWORK 2(1 Write n ones on...

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MA017 2010 HOMEWORK 2 (1) Write n ones on the board, and repeat the process “erase any two numbers a, b and write a + b 4 ” until one number x remains. Show that x 1 /n . Solution. The sum 1 /a n , taken over the numbers a on the board, is non-increasing: for any two positive numbers a, b , by rearranging ( a - b ) 2 0 we find that 4 a + b 1 /a + 1 /b. Thus 1 /x 1 / 1 + · · · + 1 / 1 = n . (2) Given finitely many points in the plane, any three of which form a triangle of area at most one, show that all of them lie inside some triangle of area four, including its boundary. Solution. Amongst the finitely many triangles formed by three such points, choose ABC of maximal area. Let a, b, c be the lines parallel to BC, CA, AB passing through A, B, C , respectively. They bound a triangle Δ of area at most four. If any point P in our set does not lie in Δ, it lies on the other side of a, b or c , and we may assume without loss of generality that it lies on the other side of a . Then P is farther from the line BC than A is, so PBC has greater area than ABC , contradiction. (3) Starting with the number 7 2008 , cross out the first digit and add it to the remaining number (e.g., 325 7→ 28 ). You repeat this until a ten-digit number x remains. Show that x has two equal digits. Solution. The invariant is n (mod 9): if n has e digits and first digit n 0 , then we replace it with n - 10 e n 0 + n 0 = n - (10 e - 1) n 0 n (mod 9). If x does not have two equal digits, its digits are a permutation of 0 , 1 , 2 , . . . , 9, so x = 9 i =0 i · 10 e i 9 i =0 i 0 (mod 9). On the other hand 9 does

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