Maths Olympiad Prep

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Geometry Difficulty 5.4 AIME, harder Prove it Philippines

Problem:

Consider an equilateral triangle with side 700700. Suppose that one move consists of changing the length of any of the sides of a triangle such that the result will still be a triangle. Find the minimum number of moves to change the given triangle to an equilateral triangle with side 22.

Solution

Solution:

Work backwards by starting from (2,2,2)(2,2,2). The fastest way to do this is to lengthen the shortest side, making it as long as the sum of the other two sides. Denote k−k^{-} to be a real number arbitrarily close to kk, but less than kk.

First move: (2,2,4−)(2,2,4^{-})

Second move: (2,4−,6−)(2,4^{-}, 6^{-})

Third move: (4−,6−,10−)(4^{-}, 6^{-}, 10^{-})

and so on. Notice that it follows double of the Fibonacci sequence.

2,2,4,6,10,16,26,42,68,110,178,288,466 2,2,4,6,10,16,26,42,68,110,178,288,466

The side lengths after the 1111th move are (178−,288−,466−)(178^{-}, 288^{-}, 466^{-}). After 33 moves, we can change all the sides to 700700 as follows:

Step 12: (288,466,700)(288,466,700)

Step 13: (466,700,700)(466,700,700)

Step 14: (700,700,700)(700,700,700)

Thus, the minimum number of moves is 1414.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.