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Omni-MATH / I have two cents and Bill has n cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other packa…

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problem

I have two cents and Bill has nn cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend all nn of his cents on some combination of pencil packages to get PP pencils. However, if I give my two cents to Bill, he then notes that he can instead spend all n+2n+2 of his cents on some combination of pencil packages to get fewer than PP pencils. What is the smallest value of nn for which this is possible?
Plain-text mathematical notation (without MathML)
I have two cents and Bill has n cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend all n of his cents on some combination of pencil packages to get P pencils. However, if I give my two cents to Bill, he then notes that he can instead spend all n+2 of his cents on some combination of pencil packages to get fewer than P pencils. What is the smallest value of n for which this is possible?
Original LaTeX notation
I have two cents and Bill has $n$ cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend all $n$ of his cents on some combination of pencil packages to get $P$ pencils. However, if I give my two cents to Bill, he then notes that he can instead spend all $n+2$ of his cents on some combination of pencil packages to get fewer than $P$ pencils. What is the smallest value of $n$ for which this is possible?

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Source and history

task_id
d4736454-b2bb-59d7-aba7-a21cfde4b815
task_key
test--d4736454-b2bb-59d7-aba7-a21cfde4b815
task_revision_id
1
config / split
/ test

Official source

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