2008 iTest Problems/Problem 95

Revision as of 19:58, 2 January 2020 by Piphi (talk | contribs) (Created page)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Bored on their trip home, Joshua and Alexis decide to keep a tally of license plates they see in the other lanes: Joshua watches cars going the other way, and Alexis watches cars in the next lane.

After a few minutes, Wendy counts up the tallies and declares, "Joshua has counted $2008$ license plates, and there are $17$ license plate designs he's seen exactly $17$ times, but of Alexis's $2009$ license plates, there's none she's seen exactly $18$ times. Clearly, $17$ is the specialist number."

Michael, suspicious, pulls out the Almanac of American License Plates and notes, "According to confirmed demographic statistics, you'd only expect those numbers to be $5.4$ and $4.9$, respectively. But the $17^\text{th}$ state is weird: Joshua saw exactly $17$ of its license plates, which isn't what we'd expect."

Alexis asks, "How many Ohioan license plates did we expect to see?" and reaches for the Almanac to find out, but Michael snatches it away and says, "I'm not telling."

Alexis, disappointed, says, "I suppose that $17$ is my best guess," feeling that the answer must be pretty close to $17$.

Wendy smiles. "You can do better than that, actually. Given what Michael said and that we saw $17$ Ohioan license plates, we'd actually expect there to have been $\frac{a}{b}$ less than $17$."

Help Alexis. If $\frac{a}{b}$ is in lowest terms, find the product $ab$.

See Also

2008 iTest (Problems)
Preceded by:
Problem 94
Followed by:
Problem 96
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
Invalid username
Login to AoPS