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📅 July 7, 2026 👤 scraface

Jul 07, 2026

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📌 The shortest altitude corresponds to the longest side, which is 15 cm.

📌 Thus, the number of such triangles is the number of valid $ k $, which is $ 100 $.

📌 Therefore, each $ a = 7k $, $ k = 1 $ to $ 100 $, gives a valid triangle.

📌 Question: A palynologist observes that the number of pollen grains in successive layers of sediment follows a geometric sequence where the first term is $ 3 $ and the common ratio is a prime number. If the sum of the first $ 5 $ terms is divisible by $ 100 $, what is the smallest possible prime ratio?

📌 Let the prime ratio be $ p $. The sum of the first 5 terms of a geometric sequence is:

📌 Since $ 3 $ and $ 100 $ are coprime, divide both sides:

📌 $ p = 2 $: $ 16 + 8 + 4 + 2 + 1 = 31

📌 $ p = 3 $: $ 81 + 27 + 9 + 3 + 1 = 121

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