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

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SMO Junior 2024 Rd.2 1. Let ��� be a n isosceles right -angled triangle of area 1. Find the length of the shortest segment that divides the triangle into 2 parts of equal area. 2. Let ���� be a parallelogram and the points �,� are in the exterior. If triangles ��� and ��� are similar, i.e. ∆��� ∼ ∆��� , prove that triangle ��� is similar to these two triangles. 3. Seven triangles of area 7 lie inside a square of area 27 . Prove that among the 7 tri angles there are 2 that intersect in a region of area not less than 1. 4. Suppose for some positive integer 𝑛, the numbers 2𝑛 and 5𝑛 hav e equal first digit. What are the possible values of this firs t digit? 5. Find all integer solutions of the equati on �2+ 2� = �4+ 20 �3+ 104 �2+ 40 �+ 2003 .