Math Puzzles #1

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This section is for math teachers, math students, and lovers
of math puzzles. We hope that these puzzles, which will change
monthly, will keep you and/or your students busy for hours
and are more fun then planting around your garden shed or
jumping on a trampoline!

By the way, did you know, New Jersey was one of the first US states to
legalize and regulate online poker. For more information and to learn
about current legislation, read the article at
http://www.legalpokersites.com .

US poker has always been both popular and contentious. PokerSites.com has
a useful page summarizing the history of poker in the USA, which can be found
HERE.
It will be interesting to see what happens in the next few years in terms of legislation.

This page has been translated into the Belorussian language...
If you are interested, go HERE!

The answers to these will be posted when the succeeding set
of problems are posted. Good luck!

1) Find all solutions of this system of equations:

x + yz = 6
y + xz = 6
z + xy = 6

2) Find the ordered pair (x,y) that satisfy:

sqrt(21/4 + 3 * sqrt(3)) = x + sqrt(y)

where sqrt means square root.

3) Find all ordered pairs of real numbers (x,y) that satisfy:

2x^2 - 2xy + y^2 = 2
3x^2 + 2xy - y^2 = 3

4) Factor 5^1995 - 1 (that's five to the 1995th power) into a product
of three integers, such that each factor is greater than 5^100.

5) Find all ordered pairs of real numbers (a,b) for which:

3 * sqrt(x - 2y) + 3 / sqrt(x - 2y) = 10
x = ay + b

6) The points of intersection of the graphs of xy = 20 and x^2 + y^2 = 41 are joined to form a convex quadrilateral. Find the area of that quadrilateral.

7) Find all ordered triples of real numbers (x,y,z) that satisfy:

sqrt(x - y + z) = sqrt(x) - sqrt(y) + sqrt(z)
x + y + z = 8
x - y + z = 4

8) Find all ordered triples of real numbers (x,y,z) that satisfy:

xz + yz = 13
xy + xz = 25
xy + yz = 20

9) The expression sqrt(10 + sqrt(10 + sqrt(10 + sqrt(10 + sqrt(10 + ..... recursively can be expressed in the real number form (a + sqrt(b)) / c, where a, b, and c are integers, no two of which have a common prime factor. Find the ordered triple (x,y,z).

10) If a + b + c = 0 and a^3 + b^3 + c^3 = 216, find the value of abc.

11) Find all real x such that

sqrt ((x+4) / (x-1)) + sqrt ((x-1) / (x+4)) = 5/2

12) Express in simplest terms as a real number:

(5th root of (sqrt(18) + sqrt(2)))^2

13) The number sqrt(20 + sqrt(384)) can be expressed as sqrt(a) + sqrt(b), where a and b are both rational and a < b. Find (a,b).

14) If John gets a 97 on his next test, his average will be 90. If he gets a 73, his average will be 87. How many tests has John already taken?

15) The integer 999,999,995,904 may be factored as:

a^16 x b^2 x c x d x e x f

where a thru f are primes and a < b < c < d < e < f. Compute f.

16) The area common to the circles (x-2)^2 + (y-2)^2 =25 and (x-2)^2 + (y-6)^2 = 25 is divided into two equal parts by the line 14x + 3y = k. Find k.