Download [PDF] 2024 SMO Senior Section Round 2 Questions Singapore Mathematical Olympiad

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SMO Senior 202 4 Rd.2 1. In an acute triangle ��� , �� > �� , � is the point on �� such that �� = �� . Let ������1 be the circle through � tangent to �� at �, and ������2 the circle through � tangent to �� at �. Let ������(≠ �) be the second intersection of ������1 and ������2. Prove that ������ lies on �� . 2. Find a ll integer solutions of the equation �2+ 2� = �4+ 20 �3+ 104 �2+ 40 �+ 2003 . 3. Find the smallest po sitive integer � for which there exist integers �1< �2< ⋯ < �� such that every integer from 1000 to 2000 can be written as a sum of some of the inte gers from �1,�2,… ,�� without rep etition. 4. Suppose ������ is a prime number and �,�,� are integers satisfying 0< � < � < �< ������. If �3,�3,�3 have equal remainders when divid ed by ������, prove that �2+ �2+ �2 is divisible by �+ �+ �. 5. Let �1,�2,… be a sequence of positive numbers satisfying, for any positive integers �,�,� ,� such that �+ � = � + �, ��+ �� 1+ ���� = �� + �� 1+ ���� . Show that there exist positive numbers �,� so that � ≤ ��≤ � for any positive integer �.