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Geometry Difficulty 5.5 AIME, harder Prove it Ukraine

The twins Pete and Ostap had had an argue and started to go to school different ways. Pete goes 210210 meters to the South, then 7070 meters to the East and reaches the school. Ostap goes to the North for a while and then he head to the school directly. How many meters Ostap has to go to the North if both twins have equal speeds and come to the school simultaneously?

Solution

Let us denote the points: home - DD, school - SS, the point where Pete turns to the East - BB, the point where Ostap begins to walk straight to the school - NN (fig. 12). Then DB=210DB = 210, BS=70BS = 70, DN=xDN = x, NS=yNS = y. Also we have x+y=280x + y = 280.

Figure 1

By Pythagorean theorem: BN2+BS2=NS2BN^2 + BS^2 = NS^2 or
(210+x)2+702=y2⇒(210+x)2+702=(280−x)2. (210 + x)^2 + 70^2 = y^2 \Rightarrow (210 + x)^2 + 70^2 = (280 - x)^2.
44100+420x+x2+4900=78400−560x+x2, hence x=30. 44100 + 420x + x^2 + 4900 = 78400 - 560x + x^2, \text{ hence } x = 30.

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