 Math Puzzles #1 (Advanced Algebra) This section is for math teachers, math students, and lovers of math puzzles. I hope that these puzzles will keep you and/or your students busy for hours!
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.