Download 2005 Mississippi Mu Alpha Theta Inter-School Test

2006 Mississippi Mu Alpha Theta Inter-School Test
1. Let x, y, and z be three prime numbers such that x + y = z. If 1 < x < y, find x.
2. In a certain school, the ratio of girls to boys is 9 to 8. If the girls’ average age is 12 and the
boys’ average age is 11, find the average age of all children in the school.
3. Let a = xy, b = xz, and c = yz such that a ≠ 0, b ≠ 0, c ≠ 0. Represent x 2  y 2  z 2 in terms of
a, b, and c.
4. Let x, y, z and w be greater than 1. If log z w  36 , log y w  60 , log xyz w  9 , find log z w .
5. Find the product of the two real roots of the equation x 2  20 x  32 = 3 x 2  20 x  60
6. Find the value of cot ( cot 1 3  cot 1 7  cot 1 13  cot 1 21  cot 1 31 )
7. Let f(x) be a polynomial function such that f ( x 2  1) = x 4  6 x 2  1 , find f(x+1)
8. Let ABC be a right triangle with hypotenuse AB and side AC = 16. The altitude CD divides
AB into segments AD and DB such that DB = 24. Find the area of ΔABC.
9. If the roots of x 2  px  q  0 are sin  and sin  and the roots of x 2  rx  s  0 are csc
and csc  , represent rs in terms of p and q.
4
1
and P(B) = , find the smallest
5
3
interval containing the probability that both A and B occur.
10. Let P(E) the probability that an event E occurs. If P(A) =
11. Prove that the points of intersection of the four internal angle bisectors of a quadrilateral are,
in general, concyclic.
12. An urn contains 15 balls numbered 1, 2, 3, …,15. If 8 balls are drawn randomly and at the
same time, find the probability that the sum of the numbers on the drawn balls is even.
13. Find the least natural number for which
7 n  74
is a non-zero reducible fraction.
n  11
14. Find the equation of the circle that is tangent to the lines y = 2x – 8, x – 2y + 4 = 0, and whose
center is on the line x – 3y – 4 = 0.
15. Find the maximum volume of a right circular cylinder that can be inscribed in a right circular
cone of altitude 15 cm and radius 5 cm.
2006 Mississippi Mu Alpha Theta Inter-School Test
Tie breakers
1. Let P be one of the points of intersection of two circles with radii 12 and 9. Let Q and R
be two points each on one of the two circles such that P is on segment QR and PR = PQ.
If the distance between the centers of the two circles is 18, find PQ 2 .
2. Prove that the number of prime numbers is infinite
3. Let 2 be a divisor of n + 1.
a) Prove that
1
n
=
1
n 1
+
1
n(n  1)
b) Let n be a positive integer odd number. Prove that the fraction
3
n
can always be
written as a sum of three distinct unit fractions (that is, a fraction of the form
where n is a natural number)
1
n
,