2004 AMC 10B Problems/Problem 13
In the United States, coins have the following thicknesses: penny, mm; nickel, mm; dime, mm; quarter, mm. If a stack of these coins is exactly mm high, how many coins are in the stack?
All numbers in this solution will be in hundredths of a millimeter.
The thinnest coin is the dime, with thickness . A stack of dimes has height .
The other three coin types have thicknesses , , and . By replacing some of the dimes in our stack by other, thicker coins, we can clearly create exactly all heights in the set .
If we take an odd , then all the possible heights will be odd, and thus none of them will be . Hence is even.
If the stack will be too low and if it will be too high. Thus we are left with cases and .
If the possible stack heights are , with the remaining ones exceeding .
Therefore there are coins in the stack.
Using the above observation we can easily construct such a stack. A stack of dimes would have height , thus we need to add . This can be done for example by replacing five dimes by nickels (for ), and one dime by a penny (for ).
Let , and be the number of pennies, nickels, dimes, and quarters used in the stack.
From the conditions above, we get the following equation:
Then we divide each side by five to get
Writing both sides in terms of mod 4, we have .
This means that the sum is divisible by 4. Therefore, the answer must be
We can easily add up and to get . We multiply that by to get . Since this works and it requires 8 coins, the answer is clearly .
Similarly, we can simply take quarters to get .
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