All About Math Olympiad Training & Questions
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Absolutely Bo Chap:
...maths competitions were not prevalent during my \"era\" .. just pure passion for mathsThanks a lot, MathIzzzFun. Very nice solutions! Your skills are very impressive! Really appreciated your help.
Presumably, you have represented Singapore in the IMO before? Or you have at least some similar experience?
cheers. -
http://i47.tinypic.com/2ic9x6s.png\">
cheers. -
Nice one, MathIzzzFun!
I also have lots of interests in Mathematics, and also competitive maths and brainteaser in particular. They tend to stretch the mind.
After reading your solutions, I would have want to ask you to be my teacher so that I could learn from the Master if I am still in school.
Just out of curiousity, are you also a Maths Olympiad trainer, just like some of other people in this forum? (Even if you are not, you are definitely more than qualified to be one.) Or even a National Team trainer? -
Absolutely Bo Chap:
thanks... 帽子太大了Nice one, MathIzzzFun!
I also have lots of interests in Mathematics, and also competitive maths and brainteaser in particular. They tend to stretch the mind.
After reading your solutions, I would have want to ask you to be my teacher so that I could learn from the Master if I am still in school.
Just out of curiousity, are you also a Maths Olympiad trainer, just like some of other people in this forum? (Even if you are not, you are definitely more than qualified to be one.) Or even a National Team trainer?
..I do not conduct formal Maths Olympiad training but I incorporate MO techniques/strategies and MO-styled questions in my tuition sessions..
cheers. -
Hi all, pls help on the following 2 qns:
1) If x & y are positive integers which satisfy x² - 8x - 1001y² = 0, what is
the smallest possible value of x + y?
2a) If y = |x| + |x + 1| + |x - 1|, find the minimum value of y.
b) Find the area, in square units, enclosed by the figure whose boundary
points satisfy |x|+ |y|=4
TIA. -
1) If x & y are positive integers which satisfy x² - 8x - 1001y² = 0, what is
the smallest possible value of x + y?
From the question, we know that x² - 8x = 1001y²
So, let us substitute positive intigers into Y and find 1001y²
When Y = 1, 1001y² = 1001
When Y = 2, 1001y² = 4004
When Y = 3, 1001y² = 9009
When Y = 4, 1001y² = 16016
So on and so forth.
Next, we substitute values into x and try to find an x value that is equivalent to one of the 1001y² values. This process involves intelligent guessing
When x is 35, x² - 8x = 945 - not exact
When x is 36, x² - 8x = 1008 - not exact
When x is 67, x² - 8x = 3953 - not exact
When x is 68, x² - 8x = 4080 - not exact
When x is 98, x² - 8x = 8820 - not exact
When x is 99, x² - 8x = 9009 - exactly corresponding to y = 3
Hence, the smallest value is 3 + 99 = 102
2a) If y = |x| + |x + 1| + |x - 1|, find the minimum value of y.
To solve this question, we have to reduce one of the mods to 0.
So either x = 0, therefore |x| = 0
or x = -1, |x + 1| = 0
or x = 1, |x - 1| = 0
Any other substitution of X will produce a larger y.
When x is -1 or 1, y = 3
When x is 0, y is 2
Therefore the minimum value of y is 2.
b) Find the area, in square units, enclosed by the figure whose boundary
points satisfy |x|+ |y|=4
For this question, the minimum and maximum values of both X and Y are -4 and 4 respectively, because anything more or anything less will not satisfy the equation due to the mod signs.
I have plotted the graph on graphmatica, with the values for X and Y.
Apologise for the grainy image, but you should get the idea.
http://i47.tinypic.com/miie6d.png\">
Enjoy! -
Thank u v much, Maths Olympiad Tutor for yr clear explanation:)
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1) If x & y are positive integers which satisfy x² - 8x - 1001y² = 0, what is
the smallest possible value of x + y?
There is another way to solve this problem.
x² - 8x = x * ( x - 8 ) = 1001y²
Without loss of generality, we may suppose that x and y are coprime, that is, they do not have any common factors.
But we know that 1001 = 7 * 11 * 13, so there are only 4 ways to split 1001 into 2 different factors, namely,
a. 77 * 13
b. 91 * 11
c. 143 * 7
d. 1001 * 1
To get the corresponding value for y, it will be near the square root of the quotient of the 2 factors. The reason is that x * ( x - 8 ) is equal to ( x - 4 )² less 16.
a. about sqrt( 77 / 13 ) = 2 to 3
b. about sqrt( 91 / 11 ) = 2 to 3
c. about sqrt( 143 / 7 ) = 4 to 5
d. about sqrt( 1001 / 1 ) = 31 to 32
So we only need to check the following:
a. y = 2, 3 against x = 69 (=77 - 8), 85 (= 77 +
b. y = 2, 3 against x = 83 (= 91 - 8), 99 (91 +
c. y = 4, 5 against x = 135 (= 143 - 8), 151 (= 143 +
d. y = 31, 32 against x = 993 (= 1001 - 8), 1009 (= 1001 +
Out of the above 8 possibilities, only x = 99 against y = 3 produce an answer.
There are other possibilities when x has has a factor of 2, 4, or 8. I did not look in the details, since that would generate a higher value of x + y even if there are solutions. However, looking at the discriminant suggests that there are no positive integer solutions on x and y.
Hence, this may be the only solution to this Diophantine equation, so there is only 1 possible value of x + y if x and y are to be positive integers. -
Thx for d alternative solution, Absolutely Bo Chap.
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I am joining the math olympiad 2013 this year and can some people give me some tips on how to get into the next round? :xedfingers: