MathDB

Round 5
p13. You have a

26 \times 26

grid of squares. Color each randomly with red, yellow, or blue. What is the expected number (to the nearest integer) of

2 \times 2

squares that are entirely red?
p14. Four snakes are boarding a plane with four seats. Each snake has been assigned to a different seat. The first snake sits in the wrong seat. Any subsequent snake will sit in their assigned seat if vacant, if not, they will choose a random seat that is available. What is the expected number of snakes who sit in their correct seats?
p15. Let

n \ge 1

be an integer and

a > 0

a real number. In terms of n, find the number of solutions

(x_1, ..., x_n)

of the equation

\sum^n_{i=1}(x^2_i + (a - x_i)^2) = na^2

such that

x_i

belongs to the interval

[0, a]

, for

i = 1, 2, . . . , n

.
Round 6
p16. All roots of

\prod^{25}_{n=1} \prod^{2n}_{k=0}(-1)^k \cdot x^k = 0

are written in the form

r(\cos \phi + i\sin \phi)

for

i^2 = -1

r > 0

, and

0 \le \phi < 2\pi

. What is the smallest positive value of

\phi

in radians?
p17. Find the sum of the distinct real roots of the equation

\sqrt[3]{x^2 - 2x + 1} + \sqrt[3]{x^2 - x - 6} = \sqrt[3]{2x^2 - 3x - 5}.

p18. If

a

and

b

satisfy the property that

a2^n + b

is a square for all positive integers

n

, find all possible value(s) of

a

.
Round 7
p19. Compute

(1 - \cot 19^o)(1 - \cot 26^o)

.
p20. Consider triangle

ABC

with

AB = 3

BC = 5

, and

\angle ABC = 120^o

. Let point

E

be any point inside

ABC

. The minimum of the sum of the squares of the distances from

E

to the three sides of

ABC

can be written in the form

a/b

, where a and b are natural numbers such that the greatest common divisor of

a

and

b

1

. Find

a + b

.
p21. Let

m \ne 1

be a square-free number (an integer – possibly negative – such that no square divides

m

). We denote

Q(\sqrt{m})

to be the set of all

a + b\sqrt{m}

where

a

and

b

are rational numbers. Now for a fixed

m

, let

S

be the set of all numbers

x

Q(\sqrt{m})

such that x is a solution to a polynomial of the form:

x^n + a_1x^{n-1} + .... + a_n = 0

, where

a_0

...

a_n

are integers. For many integers m,

S = Z[\frac{m}] = \{a + b\sqrt{m}\}

where

a

and

b

are integers. Give a classification of the integers for which this is not true. (Hint: It is true for

m = -1

and

2

.)
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782002p24434611]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement