Difference between revisions of "1988 AIME Problems/Problem 6"

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Problem

It is possible to place positive integers into the vacant twenty-one squares of the $5 \times 5$ square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).

Solution

Solution 1 (specific)

Let the coordinates of the square at the bottom left be $(0,0)$ , the square to the right $(1,0)$ , etc.

Label the leftmost column (from bottom to top) $0, a, 2a, 3a, 4a$ and the bottom-most row (from left to right) $0, b, 2b, 3b, 4b$ . Our method will be to use the given numbers to set up equations to solve for $a$ and $b$ , and then calculate $(*)$ .

$\begin{tabular}[b]{|c|c|c|c|c|}\hline 4a & & & * & \\ \hline 3a & 74 & & & \\ \hline 2a & & & & 186 \\ \hline a & & 103 & & \\ \hline 0 & b & 2b & 3b & 4b \\ \hline \end{tabular}$

We can compute the squares at the intersections of two existing numbers in terms of $a$ and $b$ ; two such equations will give us the values of $a$ and $b$ . On the fourth row from the bottom, the common difference is $74 - 3a$ , so the square at $(2,3)$ has a value of $148 - 3a$ . On the third column from the left, the common difference is $103 - 2b$ , so that square also has a value of $2b + 3(103 - 2b) = 309 - 4b$ . Equating, we get $148 - 3a = 309 - 4b \Longrightarrow 4b - 3a = 161$ .

Now we compute the square $(2,2)$ . By rows, this value is simply the average of $2a$ and $186$ , so it is equal to $\frac{2a + 186}{2} = a + 93$ . By columns, the common difference is $103 - 2b$ , so our value is $206 - 2b$ . Equating, $a + 93 = 206 - 2b \Longrightarrow a + 2b = 113$ .

Solving $\begin{align*}4b - 3a &= 161\\ a + 2b &= 113 \end{align*}$

gives $a = 13$ , $b = 50$ . Now it is simple to calculate $(4,3)$ . One way to do it is to see that $(2,2)$ has $206 - 2b = 106$ and $(4,2)$ has $186$ , so $(3,2)$ has $\frac{106 + 186}{2} = 146$ . Now, $(3,0)$ has $3b = 150$ , so $(3,2) = \frac{(3,0) + (3,4)}{2} \Longrightarrow (3,4) = * = \boxed{142}$ .

Solution 2 (general)

First, let $a =$ the number to be placed in the first column, fourth row. Let $b =$ the number to be placed in the second column, fifth row. We can determine the entire first column and fifth row in terms of $a$ and $b$ :

$\begin{tabular}[b]{|c|c|c|c|c|}\hline 4a & & & & \\ \hline 3a & & & & \\ \hline 2a & & & & \\ \hline a & & & & \\ \hline 0 & b & 2b & 3b & 4b \\ \hline \end{tabular}$

Next, let $a + b + c =$ the number to be placed in the second column, fourth row. We can determine the entire second column and fourth row in terms of $a$ , $b$ , and $c$ :

$\begin{tabular}[b]{|c|c|c|c|c|}\hline 4a & 4a + b + 4c & & & \\ \hline 3a & 3a + b + 3c & & & \\ \hline 2a & 2a + b + 2c & & & \\ \hline a & a + b + c & a + 2b + 2c & a + 3b + 3c & a + 4b + 4c \\ \hline 0 & b & 2b & 3b & 4b \\ \hline \end{tabular}$

We have now determined at least two values in each row and column. We can finish the table without introducing any more variables:

$\begin{tabular}[b]{|c|c|c|c|c|}\hline 4a & 4a + b + 4c & 4a + 2b + 8c & 4a + 3b + 12c & 4a + 4b + 16c \\ \hline 3a & 3a + b + 3c & 3a + 2b + 6c & 3a + 3b + 9c & 3a + 4b + 12c \\ \hline 2a & 2a + b + 2c & 2a + 2b + 4c & 2a + 3b + 6c & 2a + 4b + 8c \\ \hline a & a + b + c & a + 2b + 2c & a + 3b + 3c & a + 4b + 4c \\ \hline 0 & b & 2b & 3b & 4b \\ \hline \end{tabular}$

We now have a system of equations.

$3a + b + 3c = 74$

$2a + 4b + 8c = 186$

$a + 2b + 2c = 103$

Solving, we find that $(a,b,c) = (13,50, - 5)$ . The number in the square marked by the asterisk is $4a + 3b + 12c = \boxed{142}$

@@ Line 66: / Line 66: @@
 Solving, we find that <math>(a,b,c) = (13,50, - 5)</math>.  The number in the square marked by the asterisk is <math>4a + 3b + 12c = \boxed{142}</math>
-=== Solution 3 (Calculation Based) ===
-Start by labeling the indicated square <math>x</math>:
-<math>\begin{tabular}[b]{|c|c|c|c|c|}\hline & & & & * & \\
-\hline & & 74 & & \\
-\hline & & & & 186 \\
-\hline & x & 103 & &\\
-\hline 0 & & & & \\
-\hline \end{tabular}</math>
-Now we can fill in a few sequences:
-<math>\begin{tabular}[b]{|c|c|c|c|c|}\hline & 111 - \frac {x}{2} & & & * & \\
-\hline & & 74 & & \\
-\hline & 37 + \frac {x}{2} & & & 186 \\
-\hline 2x-103 & x & 103 & 206 - x & 309 - 2x \\
-\hline 0 & & & & \\
-\hline \end{tabular}</math>
-This, in turn, opens up more rows and columns:
-<math>\begin{tabular}[b]{|c|c|c|c|c|}\hline 8x - 412 & 111 - \frac {x}{2} & & & * & 4x - 60 \\
-\hline 6x - 309 & & 74 & & 2x + 63 \\
-\hline 4x - 206 & 37 + \frac {x}{2} & & & 186 \\
-\hline 2x - 103 & x & 103 & 206 - x & 309 - 2x \\
-\hline 0 & & & & \\
-\hline \end{tabular}</math>
-Note that the difference between consecutive terms must be <math>x-88</math> so the square labeled <math>111-\frac{x}{2}</math> can also be labeled as <math>7x-324</math>. Solving for <math>x</math> gives <math>x=58</math>, and filling in the top row will reveal that the asterisk is  <math>5x-148=\boxed{142}</math>
 == See also ==
 {{AIME box|year=1988|num-b=5|num-a=7}}
 {{MAA Notice}}

1988 AIME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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