MathDB

Round 1
p1. Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.)
p2. A

3\times 3

square grid is filled with positive numbers so that (a) the product of the numbers in every row is

1

, (b) the product of the numbers in every column is

1

, (c) the product of the numbers in any of the four

2\times 2

squares is

2

. What is the middle number in the grid? Find all possible answers and show that there are no others.
p3. Each letter in

HAGRID

's name represents a distinct digit between

0

and

9

. Show that

HAGRID \times H \times A\times G\times R\times I\times D

is divisible by

3

. (For example, if

H=1

A=2

G=3

R = 4

I = 5

D = 64

, then

HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6

).
p4. You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins?
p5. Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of

10

games, Bob played

15

games, and Chad played

17

games. Who lost the second game?
Round 2
p6. After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.)
p7. You have a deck of

50

cards, each of which is labeled with a number between

1

and

25

. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to

25

students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement