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Algebra Difficulty 4.8 AIME Prove it South Africa

Some balls were distributed into 20152015 boxes which were arranged in a row as indicated below. Any four consecutive boxes always had a total of 3030 balls. How many balls were there in the 20152015th box?

Figure 1

Solution

If the number of balls in the boxes are denoted x1,x2,…x_1, x_2, \dots, then

xn+xn+1+xn+2+xn+3=30=xn+1+xn+2+xn+3+xn+4. x_n + x_{n+1} + x_{n+2} + x_{n+3} = 30 = x_{n+1} + x_{n+2} + x_{n+3} + x_{n+4}.

Thus xn=xn+4x_n = x_{n+4} for all nn, which means that the numbers recur in cycles of length four. Since 2015=4×503+32015 = 4 \times 503 + 3, it follows that x2015=x3=7x_{2015} = x_3 = 7.

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