2012 USAJMO Problems/Problem 5

Problem

For distinct positive integers $a$ , $b < 2012$ , define $f(a,b)$ to be the number of integers $k$ with $1 \le k < 2012$ such that the remainder when $ak$ divided by 2012 is greater than that of $bk$ divided by 2012. Let $S$ be the minimum value of $f(a,b)$ , where $a$ and $b$ range over all pairs of distinct positive integers less than 2012. Determine $S$ .

Solution

The key insight in this problem is noticing that when $ak$ is higher than $bk$ , $a(2012-k)$ is lower than $b(2012-k)$ , except at $2(mod 4)$ residues*. Also, they must be equal many times. $2012=2^2*503$ . We should have multiples of $503$ . After trying all three pairs and getting $503$ as our answer, we win. But look at the $2(mod 4)$ idea. What if we just took $2$ and plugged it in with $1006$ ? We get $502$ .

--Va2010 11:12, 28 April 2012 (EDT)va2010

Alternate Solution

Say that the problem is a race track with $2012$ spots. To intersect the most, we should get next to each other a lot so the negation is high. As $2012=2^2*503$ , we intersect at a lot of multiples of $503$ .

2012 USAJMO (Problems • Resources)
Preceded by Problem 4	Followed by Problem 6
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2012 USAJMO Problems/Problem 5

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