Math
Puzzles #1 |

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!

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This page has been translated into the Belorussian language...

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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.