Math desu yo, Math desu yo!

October 17th, 2009 by ben Leave a reply »

I felt like it.

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

i^2

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:

Advertisement

10 comments

  1. k says:

    … at shirt.

  2. Benji says:

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

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

  3. Andy says:

    [spoiler]200[/spoiler]

    … Easy!! ^_^

  4. Benji says:

    NOW PROVE IT! >:]

  5. Benji says:

    Hold on, lemme enable LaTeX in comments.

  6. Andy says:

    Proof:
    [spoiler]
    200+56 = 256 = 16 squared
    200+124 = 324 = 18 squared
    [/spoiler]

  7. Benji says:

    Hank actually WANTED one of the Pikachu shirts.

  8. Benji says:

    Solution under the spoiler tag.

    [spoiler]

    Let x be 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]

  9. Andy says:

    ^_^ How to Solve ^_^
    ——————————-
    [spoiler]

    Uses a sequence of squares:

    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.

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.