I felt like it.

Here’s the design I’m sponsoring for the 2009-10 Arcadia Math Team t-shirt.

keepin’ it real

Anyways, let’s go with a really easy problem this time, okay? Great!

Um, I actually have more trouble coming up with “easy” problems, compared to hard ones. Okay, I’ll go with this one; it’s not too hard.

A number *x* is 56 less than a perfect square, and 124 less than another perfect square. What is *x*?

from October 2009 ARML Practice

Please comment your answers. (Pfft. As if anybody cares about this post.)

### Related Posts:

… at shirt.

What about the shirt? I know, genius, huh?

At least it’s better than the Pikachu shirts from last year.

[spoiler]200[/spoiler]

… Easy!! ^_^

NOW PROVE IT! >:]

True.

Hold on, lemme enable LaTeX in comments.

Proof:

[spoiler]

200+56 = 256 = 16 squared

200+124 = 324 = 18 squared

[/spoiler]

Hank actually WANTED one of the Pikachu shirts.

Solution under the spoiler tag.

[spoiler]

xbe the number we are looking for. From the problem:$latex x + 56 = m^2 $

and

$latex x + 124 = n^2 $

Subtract the two equations from each other, and you get:

$latex 124 – 56 = n^2 – m^2 $

Factor the difference in squares.

$latex 68 = (n+m)(n-m) $

What are two divisors of 68 that can satisfy this? They are 2 and 34.

$latex n+m = 34 $

and

$latex n-m = 2 $

Solving this trivial system, we get n and m to be 16 and 18. Thus, x is 256 minus 56, or 200.

[/spoiler]

^_^ How to Solve ^_^

——————————-

[spoiler]

A = 1, 4, 9, 16….

The interval between each term is based off another series:

B = 3, 5, 7, 9

With the formula

b of n = 2n+1

b of n is the interval to the next square.

—————–

We are given that x is 56 from one square and 124 from another….

sooo…..

The distance between the two squares is:

124-56 = 68

Finding it as a sum of consecutive nums:

68 = 35+33

so there are 2 intervals between the squares…

33 is the first interval after the first square

From above

b of n = 2n +1

Solve!

2n = 32

n = 16

Sooo…..

16 is the first number to be squared: 256

x + 56 = 256

x = 200

[/spoiler]

Yay!!!

Life is good.