Math
Puzzles #1
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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! The answers to these will be posted when the succeeding set of problems are posted. Good luck! |
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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.