MathDB

p1. The following number is the product of the divisors of

n

2^63^3

What is

n

?
p2. Let a right triangle have the sides

AB =\sqrt3

BC =\sqrt2

, and

CA = 1

. Let

D

be a point such that

AD = BD = 1

. Let

E

be the point on line

BD

that is equidistant from

D

and

A

. Find the angle

\angle AEB

.
p3. There are twelve indistinguishable blackboards that are distributed to eight different schools. There must be at least one board for each school. How many ways are there of distributing the boards?
p4. A Nishop is a chess piece that moves like a knight on its first turn, like a bishop on its second turn, and in general like a knight on odd-numbered turns and like a bishop on even-numbered turns. A Nishop starts in the bottom-left square of a

3\times 3

-chessboard. How many ways can it travel to touch each square of the chessboard exactly once?
p5. Let a Fibonacci Spiral be a spiral constructed by the addition of quarter-circles of radius

n

, where each

n

is a term of the Fibonacci series:

1, 1, 2, 3, 5, 8,...

(Each term in this series is the sum of the two terms that precede it.) What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area

714

, whose side lengths are terms in the Fibonacci series?
p6. Suppose that

a_1 = 1

and

a_{n+1} = a_n -\frac{2}{n + 2}+\frac{4}{n + 1}-\frac{2}{n}

What is

a_{15}

?
p7. Consider

5

points in the plane, no three of which are collinear. Let

n

be the number of circles that can be drawn through at least three of the points. What are the possible values of

n

?
p8. Find the number of positive integers

n

satisfying

\lfloor n /2014 \rfloor =\lfloor n/2016 \rfloor

.
p9. Let

f

be a function taking real numbers to real numbers such that for all reals

x \ne 0, 1

, we have

f(x) + f \left( \frac{1}{1 - x}\right)= (2x - 1)^2 + f\left( 1 -\frac{1}{ x}\right)

Compute

f(3)

.
p10. Alice and Bob split

5

beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy?
p11. Triangle

ABC

is an equilateral triangle of side length

1

. Let point

M

be the midpoint of side

AC

. Another equilateral triangle

DEF

, also of side length

1

, is drawn such that the circumcenter of

DEF

M

, point

D

rests on side

AB

. The length of

AD

is of the form

\frac{a+\sqrt{b}}{c}

, where

b

is square free. What is

a + b + c

?
p12. Consider the function

f(x) = \max \{-11x- 37, x - 1, 9x + 3\}

defined for all real

x

. Let

p(x)

be a quadratic polynomial tangent to the graph of

f

at three distinct points with x values

t_1

t_2

and

t_3

Compute the maximum value of

t_1 + t_2 + t_3

over all possible

p

.
p13. Circle

J_1

of radius

77

is centered at point

X

and circle

J_2

of radius

39

is centered at point

Y

. Point

A

lies on

J1

and on line

XY

, such that A and Y are on opposite sides of

X

\Omega

is the unique circle simultaneously tangent to the tangent segments from point

A

J_2

and internally tangent to

J_1

. If

XY = 157

, what is the radius of

\Omega

?
p14. Find the smallest positive integer

n

so that for any integers

a_1, a_2,..., a_{527}

,the number

\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)

is divisible by

527

.
p15. A circle

\Omega

of unit radius is inscribed in the quadrilateral

ABCD

. Let circle

\omega_A

be the unique circle of radius

r_A

externally tangent to

\Omega

, and also tangent to segments

AB

and

DA

. Similarly define circles

\omega_B

\omega_C

, and

\omega_D

and radii

r_B

r_C

, and

r_D

. Compute the smallest positive real

\lambda

so that

r_C < \lambda

over all such configurations with

r_A > r_B > r_C > r_D

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

Problem Statement