2004 Pan African MO Problems/Problem 3
One writes 268 numbers around a circle, such that the sum of 20 consecutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.
Let be the 268 numbers. First, we will prove that a number twenty positions after said number would be the same number.
Lemma: for integers
We know that , where . By subtracting and adding from both sides, we must have . Therefore, , so .
Thus, we know that , where is an integer. If , then . If , then . Note that is a factor of , so we can know that if , then .
With this in mind, Thus, since , we must have .
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