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Froggy

Source: 2012 AIME I Problem 11

March 16, 2012
invariantmodular arithmeticanalytic geometrygraphing linessloperotationgeometry

Problem Statement

A frog begins at P0=(0,0)P_0 = (0,0) and makes a sequence of jumps according to the following rule: from Pn=(xn,yn)P_n=(x_n,y_n), the frog jumps to Pn+1P_{n+1}, which may be any of the points (xn+7,yn+2)(x_n+7, y_n+2), (xn+2,yn+7)(x_n+2,y_n+7), (xn−5,yn−10)(x_n-5, y_n-10), or (xn−10,yn−5)(x_n-10,y_n-5). There are MM points (x,y)(x,y) with ∣x∣+∣y∣≤100|x|+|y| \le 100 that can be reached by a sequence of such jumps. Find the remainder when MM is divided by 10001000.
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