MathDB

Part 1
p1. Two kids

A

and

B

play a game as follows: From a box containing

n

marbles (

n > 1

), they alternately take some marbles for themselves, such that: 1.

A

goes first. 2. The number of marbles taken by

A

in his first turn, denoted by

k

, must be between

1

and

n

, inclusive. 3. The number of marbles taken in a turn by any player must be between

1

and

k

, inclusive. The winner is the one who takes the last marble. What is the sum of all

n

for which

B

has a winning strategy?
p2. How many ways can your rearrange the letters of "Alejandro" such that it contains exactly one pair of adjacent vowels?
p3. Assuming real values for

p, q, r

, and

s

, the equation

x^4 + px^3 + qx^2 + rx + s

has four non-real roots. The sum of two of these roots is

q + 6i

, and the product of the other two roots is

3 - 4i

. Find the smallest value of

q

.
p4. Lisa has a

3

D box that is

48

units long,

140

units high, and

126

units wide. She shines a laser beam into the box through one of the corners, at a

45^o

angle with respect to all of the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a

45^o

angle. Compute the distance the laser beam travels until it hits one of the eight corners of the box.
Part 2
p5. How many ways can you divide a heptagon into five non-overlapping triangles such that the vertices of the triangles are vertices of the heptagon?
p6. Let

a

be the greatest root of

y = x^3 + 7x^2 - 14x - 48

. Let

b

be the number of ways to pick a group of

a

people out of a collection of

a^2

people. Find

\frac{b}{2}

.
p7. Consider the equation

1 -\frac{1}{d}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},

with

a, b, c

, and

d

being positive integers. What is the largest value for

d

?
p8. The number of non-negative integers

x_1, x_2,..., x_{12}

such that

x_1 + x_2 + ... + x_{12} \le 17

can be expressed in the form

{a \choose b}

, where

2b \le a

. Find

a + b

.
Part 3
p9. In the diagram below,

AB

is tangent to circle

O

. Given that

AC = 15

AB = 27/2

, and

BD = 243/34

, compute the area of

\vartriangle ABC

. https://cdn.artofproblemsolving.com/attachments/b/f/b403e5e188916ac4fb1b0ba74adb7f1e50e86a.png
p10. If

\left[2^{\log x}\right]^{[x^{\log 2}]^{[2^{\log x}]...}}= 2,

where

\log x

is the base-

10

logarithm of

x

, then it follows that

x =\sqrt{n}

. Compute

n^2

.
p11.
p12. Find

n

in the equation

133^5 + 110^5 + 84^5 + 27^5 = n^5,

where

n

is an integer less than

170

.
Part 4
p13. Let

x

be the answer to number

14

, and

z

be the answer to number

16

. Define

f(n)

as the number of distinct two-digit integers that can be formed from digits in

n

. For example,

f(15) = 4

because the integers

11

15

51

55

can be formed from digits of

15

. Let

w

be such that

f(3xz - w) = w

. Find

w

.
p14. Let

w

be the answer to number

13

and

z

be the answer to number

16

. Let

x

be such that the coefficient of

a^xb^x

(a + b)^{2x}

5z^2 + 2w - 1

. Find

x

.
p15. Let

w

be the answer to number

13

x

be the answer to number

14

, and

z

be the answer to number

16

. Let

A

B

C

D

be points on a circle, in that order, such that

\overline{AD}

is a diameter of the circle. Let

E

be the intersection of

\overleftrightarrow{AB}

and

\overleftrightarrow{DC}

, let

F

be the intersection of

\overleftrightarrow{AC}

and

\overleftrightarrow{BD}

, and let

G

be the intersection of

\overleftrightarrow{EF}

and

\overleftrightarrow{AD}

. Now, let

AE = 3x

ED = w^2 - w + 1

, and

AD = 2z

. If

FG = y

, find

y

.
p16. Let

w

be the answer to number

13

, and

x

be the answer to number

16

. Let

z

be the number of integers

n

in the set

S = \{w,w + 1, ... ,16x - 1, 16x\}

such that

n^2 + n^3

is a perfect square. Find

z

.

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

Problem Statement