2000 AMC 12 Problem 18

Question

In year

N,

the

300

th day of the year is a Tuesday. In year

N + 1,

the

200

th day is also a Tuesday. On what day of the week did the

100

th day of year

N - 1

occur?

Thursday

Friday

Saturday

Sunday

Monday

Answer: A

Concepts:date and time modular arithmetic

Difficulty rating: 1870

Small hint:

Two dates fall on the same weekday exactly when the number of days between them is a multiple of

7

Big hint:

Use the two given Tuesdays to decide which of

N - 1, N, N + 1

is a leap year

Solution:

From day

300

of year

N

to day

200

of year

N + 1,

the number of days is

365 - 300 + 200 = 265

if

N

is not a leap year. But

265 = 7 \cdot 37 + 6,

which would land on a Monday, not a Tuesday.

So year

N

is a leap year, and the gap is

266 = 7 \cdot 38

days, giving a Tuesday as stated. It follows that year

N - 1

is not a leap year.

The

100

th day of year

N - 1

precedes the Tuesday on day

300

of year

N

by

365 - 100 + 300 = 565

days. Since

565 = 7 \cdot 80 + 5,

that day is

5

weekdays before Tuesday, namely Thursday.

Thus, the correct answer is A.

2000 AMC 12 Problem 18

Answer: A

Difficulty rating: 1870

Small hint:

Big hint:

Solution:

Problem 18 in other years