Question

Each of the five numbers $1, 4, 7, 10,$ and $13$ is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is

$20$

$21$

$22$

$24$

$30$

Answer: D

Concepts:optimization double counting

Difficulty rating: 1140

Solution:

Let $x$ be the number in the center. The horizontal sum plus the vertical sum counts every number once, except the center number is counted twice.

So twice the common sum is $1+4+7+10+13+x$ $=35+x$ . This is largest when $x=13$ .

The largest common sum is therefore at most $\dfrac{35+13}{2}=24$ , and it is attainable: put $13$ in the center, $1$ and $10$ at the ends of one row, and $4$ and $7$ at the ends of the other. Both sums are $24$ .

Thus, D is the correct answer.

1999 AMC 8 Problem 11

Answer: D

Difficulty rating: 1140

Solution:

Problem 11 in Other Years