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Algebra Difficulty 6.0 AIME, harder Prove it Ukraine

Three boys picked nuts. After discovering that 420420 nuts are picked in total, the boys decided to share them evenly. First, one of the boys gave each of two others one fourth of the nuts he picked and a nut. Then another boy gave each of two others one fourth of the nuts he collected (those he picked and those he obtained from the first boy) and a nut. After the last boy had made this operation, it turned out that, indeed, the nuts were distributed evenly. Determine, how many nuts each boy picked.

Solution

З рівностей
x3−14x3−14x3−2=140, x_3 - \frac{1}{4}x_3 - \frac{1}{4}x_3 - 2 = 140,
x2+14x3+1=140. x_2 + \frac{1}{4}x_3 + 1 = 140.
x1+14x3+1=140 x_1 + \frac{1}{4}x_3 + 1 = 140
знайдемо кількість горіхів у кожного з хлопчиків на передостанньому етапі. Аналогічно відновлюємо весь «ланцюжок»:
(140,140,140)←(68,68,284)←(32,140,248)←(68,122,230). (140, 140, 140) \leftarrow (68, 68, 284) \leftarrow (32, 140, 248) \leftarrow (68, 122, 230).
Відповідь: Перший хлопчик зібрав 6868 горіхів, другий хлопчик — 122122 горіхи, третій хлопчик — 230230 горіхів.

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