Problem 1

Three players

A

B

, and

C

play the following game. On each of three cards an integer is written; the three numbers

p, q, r

satisfy

0 < p < q < r

. The three cards are shuffled and one is dealt to each player, and each player receives as many counters as the number on the card they hold. The cards are shuffled and dealt again, and this is repeated for at least two rounds (counters from earlier rounds stay with the players). After the last round,

A

has

20

counters in all,

B

has

10

, and

C

has

9

. In the last round,

B

received

r

counters. Who received

q

counters on the first round?

Step 3 of 6: Squeezing the largest card

r

between

6

and

8

Previous step Next step

10-r\ge 2p\ge2 \implies r\le8,\qquad 3r\ge20 \implies r\ge7

Detailed analysis

Player $B$ 's total of $10$ over $3$ rounds includes one $r$ (received in the last round); the other two rounds give $B$ at least $2p$ more counters, so $10-r\ge2p\ge2$ , i.e. $r\le8$ . Player $A$ 's total of $20$ is a sum of three cards each at most $r$ , so $3r\ge20$ ; as $r$ is an integer this forces $r\ge7$ . So $r\in\{7,8\}$ .