2021 AMC 10B Problem 17

Problem 17 of 25Intermediate

Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given

2

cards out of a set of

10

cards numbered

1,

2,

3,

\dots,

10.

The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon--

11,

Oscar--

4,

Aditi--

7,

Tyrone--

16,

Kim--

17.

Which of the following statements is true?

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Solution

Oscar’s score of

4

forces his cards to be

1

and

3.

From the remaining cards, the only pair with sum

7

2

and

5,

so those are Aditi’s cards. The unused cards are now

4,6,7,8,9,10.

Tyrone’s score of

16

can come only from

6+10

7+9.

If Tyrone had

7

and

9,

then the remaining cards

4,6,8,10

could not be paired to give Ravon’s score

11.

Thus Tyrone has

6

and

10.

The remaining cards are

4,7,8,9.

Ravon must have

4

and

7

to score

11,

leaving

8

and

9

for Kim’s score of

17.

Therefore, Ravon was given card

4.

Thus, the answer is C .