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Thread: competition practice Qs: length of strip forming octagon; time to paint room;...

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    competition practice Qs: length of strip forming octagon; time to paint room;...

    Hello, I've been getting ready for the Australian Mathematics competition and I have a few questions on 6 questions I could not solve. I can';t insert photos for some reason but I've attached them below. They are of intermediate level (I'm in Year 9)

    If anyone could help me out I would be extremely grateful!!
    Attached Images Attached Images

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    Quote Originally Posted by Rachelyoonji View Post
    Hello, I've been getting ready for the Australian Mathematics competition and I have a few questions on 6 questions I could not solve. I can';t insert photos for some reason but I've attached them below. They are of intermediate level (I'm in Year 9)

    If anyone could help me out I would be extremely grateful!!
    18) What is the definition of regular polygon?
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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    Elite Member mmm4444bot's Avatar
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    28) Let x = rectangle width

    Let y = rectangle height

    The area is xy.

    Can you write an equation modeling the fact that xy is tripled, when x and y increase by the given amounts?

    I solved mine for y as a function of x, and analyzing that Rational Function showed that three such rectangles exist.

    PS: In the future, please start separate threads for separate exercises. (Also, read the forum guidelines.)
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

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    Question

    Quote Originally Posted by Subhotosh Khan View Post
    18) What is the definition of regular polygon?
    a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral...

    but I don't see how this leads into anything.

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    Elite Member mmm4444bot's Avatar
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    I was pondering (30), at the bar last night. Seems like the maximum area will happen when AP and BQ are as short as possible. I'm not 100% sure of this, but a bit o' napkin math looked promising.
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

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    Quote Originally Posted by Rachelyoonji View Post
    a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral...

    but I don't see how this leads into anything.
    Then each side of the octagon = √2 in
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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    Elite Member stapel's Avatar
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    Cool

    For other readers: The sideways questions appear to read as follows:



    27. How many positive integers n less than 2015 have the property that [tex]\frac{1}{3}\, +\, \frac{1}{n}[/tex] can be simplified to a fraction with a denominator less than n?

    28. A rectangle has all sides of integer length. When 3 units are added to the height and 2 units to the width, the area of the rectangle is tripled. What is the sum of the areas of all such rectangles?

    29. At Berracan station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. Each day, I walk to Berracan station at a random time in the afternoon and wait for the first train in either direction. On average, how many seconds should I expect to wait?

    30. In a 14-by-18 rectangle ABCD, points P, Q, R, and S are chosen, on each side of ABCD as pictured.

    quad.jpg

    The lengths AP, PB, BQ, QC, RD, DS, and SA are all positive integers and PQRS is a rectangle. What is the largest possible area that PQRS can have?

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    Elite Member stapel's Avatar
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    Quote Originally Posted by Rachelyoonji View Post
    I have a few questions on 6 questions I could not solve.

    27. How many positive integers n less than 2015 have the property that [tex]\frac{1}{3}\, +\, \frac{1}{n}[/tex] can be simplified to a fraction with a denominator less than n?
    What have you tried? For instance, you started with the given sum, and simplified:

    . . . . .[tex]\dfrac{1}{3}\, +\, \dfrac{1}{n}\, =\, \dfrac{n\, +\, 3}{3n}[/tex]

    For this to simplify to something with a denominator smaller than n, then we'll need to cancel something. For something to factor from n + 3, we must have a value of n which is a multiple of 3. Then n = 3k for some integer k. This gives us:

    . . . . .[tex]\dfrac{3k\, +\, 3}{(3k)(3)}\, =\, \dfrac{k\, +\, 1}{3k}[/tex]

    What can you say about k and k + 1? What does this tell you about k + 1? And so forth.


    Quote Originally Posted by Rachelyoonji View Post
    28. A rectangle has all sides of integer length. When 3 units are added to the height and 2 units to the width, the area of the rectangle is tripled. What is the sum of the areas of all such rectangles?
    What have you tried? For instance, you started with the basic geometry and algebra, which gave you the following:

    . . . . .[tex](h\, +\, 3)(w\, +\, 2)\, =\, 3hw[/tex]

    . . . . .[tex]hw\, +\, 3w\, +\, 2h\, +\, 6\, =\, 3hw[/tex]

    . . . . .[tex]3w\, +\, 2h\, +\, 6\,=\, 2hw[/tex]

    . . . . .[tex]6\, =\, 2hw\, -\, 3w\, -\, 2h[/tex]

    To make thiss factorable, we rearrange and add:

    . . . . .[tex]6\, +\, 3\, =\, 2hw\, -\, 2h\, -\, 3w\, +\, 3[/tex]

    . . . . .[tex]9\, =\, 2h(w\, -\, 1)\, -\, 3(w\, -\, 1)[/tex]

    And so forth. Using the fact that the sides have integer lengths, where does this lead?


    Quote Originally Posted by Rachelyoonji View Post
    29. At Berracan station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. Each day, I walk to Berracan station at a random time in the afternoon and wait for the first train in either direction. On average, how many seconds should I expect to wait?
    Stand in the station and start monitoring arrival times. What happens every fifteen minutes?

    Quote Originally Posted by Rachelyoonji View Post
    30. In a 14-by-18 rectangle ABCD, points P, Q, R, and S are chosen, on each side of ABCD as pictured.

    quad.jpg

    The lengths AP, PB, BQ, QC, RD, DS, and SA are all positive integers and PQRS is a rectangle. What is the largest possible area that PQRS can have?
    Have you tried anything with right triangles?

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