Download 2009 Mississippi Mu Alpha Theta Inter-School Test

2009 Mississippi Mu Alpha Theta Inter-School Test
1. Let ABC be a triangle inscribed in a circle with center O. If the vertices of the triangle
partition the circle into three arcs of lengths 15, 20, 25, find the area of the triangle.
2. A number x is selected uniformly at random between 250 and 300. If [ x ] = 16, find the
probability that [ 100 x ] = 160. (Note: [y] is the greatest integer less than or equal to y.)
3. Find the sum of the solutions of tan 2 x – 9tan x + 1 = 0 that are in the interval (0, 2π) (the
endpoints of the interval are represented in terms of radians).
4. Find all primes p such that 31p + 4 is a perfect square.
5. Prove that any two consecutive integers are relatively prime, that is, gcd(a, a + 1) = 1 for any
integer a.
6. Find the number of proper divisors of 9878400.
7. Find the number of ways in which n boys and n girls can be seated in a row of 2n chairs if
boys and girls must alternate.
8. Use mathematical induction to prove that for each natural number n, 7 n  2 n is divisible by
5.
9. Let { Ai }iI denote the set of all sets Ai as i ranges over set I. Let A' denote the complement
of set A. Prove that (  Ai )' =  Ai ' , where
iI
iI
A
i
iI
is the union of all the sets Ai and  Ai is
iI
the intersection of all sets Ai .
10. Five married couples are standing is a room. If the 10 people are divided into 5 pairs, find the
probability p that (a) each pair is married, (b) each pair contains a male and a female (notice
that there are 5 males and 5 females).
n
) denote the reminder of n when divided by p. Find the
p
n
smallest number n that satisfies r( ) = p – 1, for p = 2, 3, 4, …, 10.
p
11. For natural numbers n and p, let r(
xz
xy
yz
= a,
= b, and
= c, where a, b, and c are real numbers different from
xz
x y
yz
zero, represent x in terms of a, b, and c.
12. If
13. Let P be a fixed external point to a circle with center O and radius r. Find the locus of the
midpoint M of segment PA as A moves around the circle.
14. A projectile, fired straight upward with initial velocity of 500 ft/sec, moves according to s(t)
= – 16 t 2 + 500t, where s is the distance above the ground after t seconds after being fired.
(a) Find the velocity and acceleration at the time the projectile hits the ground.
(b) Find the greatest height reached.
15. Use the theorem that
lim
x 0
sin x
= 1 to prove that
x
lim
x 0
1  cos x
= 0.
x
2009 Mississippi Mu Alpha Theta Inter-School Test
Tie breakers
1. Prove that there does not exist integers m, n, and p, except 0, 0, 0, for which m + n 2 +
p 3 = 0.
2. The sequence of natural numbers is partitioned as follows
1, (2, 3), (4, 5, 6), (7, 8, 9, 10), (11, 12, 13, 14, 15), …
Find the sum of the natural numbers in the nth group.
3. Let E, F, G, and H be the midpoints of the sides of a quadrilateral ABCD. If EFGH is a
square, what type of quadrilateral is ABCD?
C
F
E
D
G
A
H
B