Problem 4

Part of 2002 Federal Math Competition of S&M

Problems(3)

L-tiling 2001x2003 rectangle

Source: S&M 2002 1st Grade P4

Is it possible to cut a rectangle

2001\times2003

into pieces of the form https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNS82L2RjZTZjNzc0M2YxMzM1ZDIzZTY2Zjc2NGJlMWJlMWUwMmU2ZWRlLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNS0xMyBhdCAzLjQ2LjQ2IFBNLnBuZw== each consisting of three unit squares?

combinatoricsgeometryrectangle

ranking players from 1-50, ranges

Source: S&M 2002 2nd Grade P4

15

coaches ranked the

50

selected football players on the places from

1

50

. For each football player, the highest and lowest obtained ranks differ by at most

5

. For each of the players, the sum of the ranks he obtained is computed, and the sums are denoted by

S_1\le S_2\le\ldots\le S_{50}

. Find the largest possible value of

S_1

2002!\cdot k without certain digits

Source:

Is there a positive integer

k

such that none of the digits

3,4,5,6

appear in the decimal representation of the number

2002!\cdot k

number theory unsolvednumber theory