Hidden Telescoping Sum

Here’s an exercise I proposed in an online competition. You have the following sequence: Find the integer part of: Now, notice that Hence By substitution, we have that the sum telescopes and reduces to Since is a very large number (the sequence has a quadratic increasing rate), we have that the integer part of the […]

An Old Problem Came To Mind

Two years ago I participated in an online math competition and the problem which was considered as the most difficult was this one: Find Now the trick is finding upper and lower bounds for such quantity. Notice: Similarly If we sum up this relationship for all values we’ll found out it’s telescoping, so we’re left […]

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