# Singapore Mathematical Olympiad

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.Objectives Prepare students keen in Mathematics for the SMO (Senior) Enhance students’ Mathematical knowledge and sharpen their ingenious problem solving skill Strengthen.solution singapore mathematical olympiads 1994-1997

Now suppose that n is an integer greater than 3, and that it has been proven that f(m) is the reversal of the binary digits of m for every positive integer m <Looking at the definition of j, we see that 3 cases arise, depending on whether n has the form 2m, 4m + 1, or 4m +Let us introduce that notation A to denote the reversal of the digits in A, if A is a string of binary digits (i.e., O's and l's). If n has the form 2m, then the binary representation of n has the form A0 2, where the string A 2 is .singapore international mathematical olympiad national team

Suppose that the altitude of ABC at B intersects the circle with diameter AC at P and Q, and the altitude at C intersects the circle with diameter AB at M and N . Prove that P, Q, M and N lie on a circle. Show that a path on a rectangular grid which starts at the northwest corner, goes through each point on the grid exactly once, and ends at the southeast corner divides the grid into two equal halves: (a) those regions opening north or east; and (b) those regions opening south or west.
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