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Omni-MATH / For distinct positive integers a , b<2012 , define f(a,b) to be the number of integers k with 1≤k<2012 such that the remainder when ak divided by 2012 is greater than that of bk di…

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problem

For distinct positive integers aa , b<2012b < 2012 , define f(a,b)f(a,b) to be the number of integers kk with 1≤k<20121 \le k < 2012 such that the remainder when akak divided by 2012 is greater than that of bkbk divided by 2012. Let SS be the minimum value of f(a,b)f(a,b) , where aa and bb range over all pairs of distinct positive integers less than 2012. Determine SS .
Plain-text mathematical notation (without MathML)
For distinct positive integers a , b<2012 , define f(a,b) to be the number of integers k with 1≤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 .
Original LaTeX notation
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 $S$ .

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Source and history

task_id
734b45af-3766-57b1-80a3-a0239d8304ff
task_key
test--734b45af-3766-57b1-80a3-a0239d8304ff
task_revision_id
1
config / split
/ test

Official source

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