Difference between revisions of "2004 Pan African MO Problems/Problem 3"

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[[Category:Introductory Number Theory Problems]]
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[[Category:Intermediate Number Theory Problems]]

Latest revision as of 10:46, 27 March 2020

Problem

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.

Solution

Let $n_1, n_2, n_3, \cdots n_{267}, n_0$ be the 268 numbers. First, we will prove that a number twenty positions after said number would be the same number.

Lemma: $n_{i \pmod{268}} = n_{i+20 \pmod{268}}$ for integers $1 \le i \le 268$

We know that $\sum_{k=1}^{20} n_{a_k} = 75$, where $a_k = i+k-1 \pmod{268}$. By subtracting $n_{a_1}$ and adding $n_{a_{21}}$ from both sides, we must have $75 - n_{a_1} + n_{a_{21}} = 75$. Therefore, $n_{a_1} = n_{a_{21}}$, so $n_{i \pmod{268}} = n_{i+20 \pmod{268}}$. $\blacktriangleright$


Thus, we know that $n_{i \pmod{268}} = n_{i+20m \pmod{268}}$, where $m$ is an integer. If $m = 14m_1$, then $i + 20m \equiv i + 280m_1 \equiv i + 12m_i \pmod{268}$. If $m_1 = 22m_2$, then $i + 12m_1 \equiv i + 264m_2 \equiv i - 4m_2 \pmod{268}$. Note that $4$ is a factor of $268$, so we can know that if $a \equiv b \pmod{4}$, then $n_a = n_b$.


With this in mind, \begin{align*} \sum_{k=1}^{20} n_k &= 75 \\ 5(n_1 + n_2 + n_3 + n_4) &= 75 \\ 5(3 + n_2 + 4 + 9) &= 75 \\ 5(16 + n_2) &= 75 \\ 16 + n_2 &= 15 \\ n_2 &= -1. \end{align*} Thus, since $n_2 = n_{210}$, we must have $n_{210} = \boxed{-1}$.

See Also

2004 Pan African MO (Problems)
Preceded by
Problem 2
1 2 3 4 5 6 Followed by
Problem 4
All Pan African MO Problems and Solutions
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