2019 AMC 8 Problems/Problem 14
Problem 14
Isabella has coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every
days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the
dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
Monday
Tuesday
Wednesday
Thursday
Friday
Solution 1
Let to
denote a day where one coupon is redeemed and the day when the second coupon is redeemed.
If she starts on a she redeems her next coupon on
.
to $\text{Sunday$ (Error compiling LaTeX. Unknown error_msg)}.
Thus is incorrect.
If she starts on a she redeems her next coupon on
.
to
.
to
.
to
.
Thus is incorrect.
If she starts on a she redeems her next coupon on
.
to
.
to
.
to
.
to
.
And on she redeems her last coupon.
No sunday occured thus is correct.
Checking for the other options,
If she starts on a she redeems her next coupon on
.
Thus is incorrect.
If she starts on a she redeems her next coupon on
.
to
.
to
.
Checking for the other options gave us negative results, thus the answer is .
~phoenixfire
Solution 2
Let
Which clearly indicates if you start form a you will not get a
.
Any other starting value may lead to a .
Which means our answer is .
~phoenixfire
Solution 3
Like Solution 2, let the days of the week be numbers .
and
are coprime, so continuously adding
to a number
will cycle through all numbers from
to
. If a string of 6 numbers in this cycle does not contain
, then if you minus 3 from the first number of this cycle, it will always be
. So, the answer is
.
See Also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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 | ||
All AJHSME/AMC 8 Problems and Solutions |
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