MathDB

p1. A robot is at position

0

on a number line. Each second, it randomly moves either one unit in the positive direction or one unit in the negative direction, with probability

\frac12

of doing each. Find the probability that after

4

seconds, the robot has returned to position

0

.
p2. How many positive integers

n \le 20

are such that the greatest common divisor of

n

and

20

is a prime number?
p3. A sequence of points

A_1

A_2

A_3

...

A_7

is shown in the diagram below, with

A_1A_2

parallel to

A_6A_7

. We have

\angle A_2A_3A_4 = 113^o

\angle A_3A_4A_5 = 100^o

, and

\angle A_4A_5A_6 = 122^o

. Find the degree measure of

\angle A_1A_2A_3 + \angle A_5A_6A_7

. https://cdn.artofproblemsolving.com/attachments/d/a/75b06a6663b2f4258e35ef0f68fcfbfaa903f7.png
p4. Compute

\log_3 \left( \frac{\log_3 3^{3^{3^3}}}{\log_{3^3} 3^{3^3}} \right)

p5. In an

8\times 8

chessboard, a pawn has been placed on the third column and fourth row, and all the other squares are empty. It is possible to place nine rooks on this board such that no two rooks attack each other. How many ways can this be done? (Recall that a rook can attack any square in its row or column provided all the squares in between are empty.)
p6. Suppose that

a, b

are positive real numbers with

a > b

and

ab = 8

. Find the minimum value of

\frac{a^2+b^2}{a-b}

.
p7. A cone of radius

4

and height

7

has

A

as its apex and

B

as the center of its base. A second cone of radius

3

and height

7

has

B

as its apex and

A

as the center of its base. What is the volume of the region contained in both cones?
p8. Let

a_1

a_2

a_3

a_4

a_5

a_6

be a permutation of the numbers

1

2

3

4

5

6

. We say

a_i

is visible if

a_i

is greater than any number that comes before it; that is,

a_j < a_i

for all

j < i

. For example, the permutation

2

4

1

3

6

5

has three visible elements:

2

4

6

. How many such permutations have exactly two visible elements?
p9. Let

f(x) = x+2x^2 +3x^3 +4x^4 +5x^5 +6x^6

, and let

S = [f(6)]^5 +[f(10)]^3 +[f(15)]^2

. Compute the remainder when

S

is divided by

30

.
p10. In triangle

ABC

, the angle bisector from

A

and the perpendicular bisector of

BC

meet at point

D

, the angle bisector from

B

and the perpendicular bisector of

AC

meet at point

E

, and the perpendicular bisectors of

BC

and

AC

meet at point

F

. Given that

\angle ADF = 5^o

\angle BEF = 10^o

, and

AC = 3

, find the length of

DF

. https://cdn.artofproblemsolving.com/attachments/6/d/6bb8409678a4c44135d393b9b942f8defb198e.png
p11. Let

F_0 = 0

F_1 = 1

, and

F_n = F_{n-1} + F_{n-2}

. How many subsets

S

\{1, 2,..., 2011\}

are there such that

F_{2012} - 1 =\sum_{i \in S}F_i?

p12. Let

a_k

be the number of perfect squares

m

such that

k^3 \le m < (k + 1)^3

. For example,

a_2 = 3

since three squares

m

satisfy

2^3 \le m < 3^3

, namely

9

16

, and

25

. Compute

\sum^{99}_{k=0} \lfloor \sqrt{k}\rfloor a_k,

where

\lfloor x\rfloor

denotes the largest integer less than or equal to

x

.
p13. Suppose that

a, b, c, d, e, f

are real numbers such that

a + b + c + d + e + f = 0,

a + 2b + 3c + 4d + 2e + 2f = 0,

a + 3b + 6c + 9d + 4e + 6f = 0,

a + 4b + 10c + 16d + 8e + 24f = 0,

a + 5b + 15c + 25d + 16e + 120f = 42.

Compute

a + 6b + 21c + 36d + 32e + 720f.

p14. In Cartesian space, three spheres centered at

(-2, 5, 4)

(2, 1, 4)

, and

(4, 7, 5)

are all tangent to the

xy

-plane. The

xy

-plane is one of two planes tangent to all three spheres; the second plane can be written as the equation

ax + by + cz = d

for some real numbers

a

b

c

d

. Find

\frac{c}{a}

.
p15. Find the number of pairs of positive integers

a

b

, with

a \le 125

and

b \le 100

, such that

a^b - 1

is divisible by

125

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

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