Problem #41

[This problem is worth 5 points.]

A fair coin is flipped 10 times. Find the probability that there are no consecutive tosses that come up heads.

Problem #40

[This problem is worth 4 points.]

Let k be an integer such that 36 + k, 300 + k, and 596 + k are the squares of three consecutive terms of an arithmetic sequence. Find k.

Problem #39

[This problem is worth 3 points.]

The polynomial x² + bx + c is a factor of both x⁴ + 6x² + 25 and 3x⁴ + 4x² + 28x + 5. What are b and c?

Problem #38

[This problem is worth 6 points.]

What is the area of the largest square in the Cartesian coordinate plane such that the interior of the square contains at most 3 points of the form (a,b), where a and b are both integers?

Problem #37

[This problem is worth 4 points.]

Find the area enclosed by the graph of |x-60| + |y| = |x/4|.

Problem #36

[This problem is worth 1 point.]

Without the use of a calculator, determine whether the following expression is positive, negative, or zero.


π² - 7π + 12

Problem #35

[This problem is worth 4 points.]

A bug is on the Cartesian coordinate plane at the point (0,0). Each second, the bug crawls either up, down, left, or right for 1 unit. What is the probability that the bug will be on the point (2,2) in 6 or fewer seconds?

Problem #34

[This problem is worth 5 points.]

Let a, b, and c be positive integers such that:

(i) a, b, c, is an increasing geometric sequence.
(ii) b-a is a perfect square.
(iii) log₆ a + log₆b + log₆c = 6

Find a + b + c

Problem #33

[This problem is worth 4 points.]

For how many real numbers x does (log₂ x)/5 = sin(5πx)?

Problem #32

[This problem is worth 5 points.]

PQR is a triangle with PQ = 4 and PR = 7. PM is the median to side QR, and PM = 7/2. What is QR?

Problem #31

[This problem is worth 2 points.]

Given a square ABCD of side length 3, a scalene triangle is constructed with AC as one of the sides. If the area of the triangle is the same as the area of the square, what is the length of the altitude of the triangle to the base AC?

Problem #30

[This problem is worth 5 points.]

Find all pairs (m,n) of integers such that m³ + 6m² + 5m = 27n³ + 9n² + 9n + 1.

Problem #29

[This problem is worth 2 points.]

Simplify as much as possible (i - i^-1)^-1.

Problem #28

[This problem is worth 3 points.]

A gecko is in a room that is 12 feet long, 10 feet wide, and 8 feet tall. The gecko is currently on a 10 x  8 side wall, one foot from the ceiling and one foot from the back 12 x 8 wall. The gecko spots a fly on the opposite side wall, one foot from the floor and one foot from the front wall. What is the length of the shortest path the gecko can take to reach the fly, assuming it can only walk across the ceiling and walls?

Problem #27

[This problem is worth 7 points.]

Define the function f(x) as follows:

(A) If x = 1, f(x) = 1
(B) If x is divisible by 10, f(x) = x/10
(C) Otherwise, f(x) = x+1

We can then use repeated application of f to generate a sequence from any starting number. So, for example, f(5) = 6, f(6) = 7, f(7) = 8, f(8) = 9, f(9) = 10, f(10) = 1, and f(1) = 1. This produces the sequence 5, 6, 7, 8, 9, 10, 1, 1, 1, ...

We then use this sequence to define a new function d(x). For any x, d(x) is the number of terms in the f-generated sequence until the first occurrence of 1 in the sequence. So from the previous example, we see that d(5) = 7, since the 7th term of the sequence 5, 6, 7, 8, 9, 10, 1, 1, ... is the first 1.

For how many values of x does d(x) = 20?

Problem #26

[This problem is worth 10 points.]

A and B play the following game:

Step 1: Two integers k and n are chosen, and revealed to both players.

Step 2: Player A selects two further integers x and N, with 1 <= x <= N. Player A tells N to B, but not x.

Step 3: Player B now asks A questions of the following form: "Does x belong to the set ... of positive integers", for any choice of set.
B may ask as many questions of this form as he wants.
A must give either a "yes" or a "no" answer to each question. The answers, however, do not have to be truthful.
The only constraint on A is that in any sequence of k+1 consecutive answers, at least one must be truthful.

Step 4: After B has asked as many questions as he wishes, he specifies a set X of at most n positive integers. If x is in X, B wins; otherwise, A wins.

Give a non-trivial condition on n, k, and N such that satisfaction of that condition guarantees that B can win the game, and prove that the condition is sufficient.

Problem #25

[This problem is worth 5 points.]

Find all ordered pairs (x,y) of positive integers such that 2/104729 can be written as 1/x + 1/y.

Problem #24

[This problem is worth 1 point.]

A bag contains 2 red, 3 blue, and 4 green chips. Chips are drawn from the bag with equal probability and without replacement. What is the probability that the chips are drawn in continuous color blocks (i.e., all the red, then all the blue, then all the green; or, all the blue, then all the green, then all the red; and so on)?

Problem #23

[This problem is worth 2 points.]

What is the product of the two smallest prime factors of 2¹⁰²⁴-1?

Problem #22

[This problem is worth 4 points.]

Two chords of a circle intersect perpendicularly. One chord is divided by the intersection into sections of length 3 and 4; the other chord is divided by the intersection into section of length 6 and 2. What is the diameter of the circle?