MathDB

Round 9
p25. Define a sequence

\{a_n\}_{n \ge 1}

of positive real numbers by

a_1 = 2

and

a^2_n -2a_n +5 =4a_{n-1}

for

n \ge 2

. Suppose

k

is a positive real number such that

a_n <k

for all positive integers

n

. Find the minimum possible value of

k

.
p26. Let

\vartriangle ABC

be a triangle with

AB = 13

BC = 14

, and

C A = 15

. Suppose the incenter of

\vartriangle ABC

I

and the incircle is tangent to

BC

and

AB

D

and

E

, respectively. Line

\ell

passes through the midpoints of

BD

and

BE

and point

X

is on

\ell

such that

AX \parallel BC

. Find

X I

.
p27. Let

x, y, z

be positive real numbers such that

x y + yz +zx = 20

and

x^2yz +x y^2z +x yz^2 = 100

. Additionally, let

s = \max (x y, yz,xz)

and

m = \min(x, y, z)

. If

s

is maximal, find

m

.
Round 10
p28. Let

\omega_1

be a circle with center

O

and radius

1

that is internally tangent to a circle

\omega_2

with radius

2

T

. Let

R

be a point on

\omega_1

and let

N

be the projection of

R

onto line

TO

. Suppose that

O

lies on segment

NT

and

\frac{RN}{NO} = \frac4 3

. Additionally, let

S

be a point on

\omega_2

such that

T,R,S

are collinear. Tangents are drawn from

S

\omega_1

and touch

\omega_1

P

and

Q

. The tangent to

\omega_1

R

intersects

PQ

Z

. Find the area of triangle

\vartriangle ZRS

.
p29. Let

m

and

n

be positive integers such that

k =\frac{ m^2+n^2}{mn-1}

is also a positive integer. Find the sum of all possible values of

k

.
p30. Let

f_k (x) = k \cdot \ min (x,1-x)

. Find the maximum value of

k \le 2

for which the equation

f_k ( f_k ( f_k (x))) = x

has fewer than

8

solutions for

x

with

0 \le x \le 1

.
Round 11
In the following problems,

A

is the answer to Problem

31

B

is the answer to Problem

32

, and

C

is the answer to Problem

33

. For this set, you should find the values of

A

B

, and

C

and submit them as answers to problems

31

32

, and

33

, respectively. Although these answers depend on each other, each problem will be scored separately.
p31. Find

A \cdot B \cdot C + \dfrac{1}{B+ \dfrac{1}{C +\dfrac{1}{B+\dfrac{1}{...}}}}

p32. Let

D = 7 \cdot B \cdot C

. An ant begins at the bottom of a unit circle. Every turn, the ant moves a distance of

r

units clockwise along the circle, where

r

is picked uniformly at random from the interval

\left[ \frac{\pi}{2D} , \frac{\pi}{D} \right]

. Then, the entire unit circle is rotated

\frac{\pi}{4}

radians counterclockwise. The ant wins the game if it doesn’t get crushed between the circle and the

x

-axis for the first two turns. Find the probability that the ant wins the game.
p33. Let

m

and

n

be the two-digit numbers consisting of the products of the digits and the sum of the digits of the integer

2016 \cdot B

, respectively. Find

\frac{n^2}{m^2 - mn}

.
Round 12
p34. There are five regular platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For each of these solids, define its adjacency angle to be the dihedral angle formed between two adjacent faces. Estimate the sum of the adjacency angles of all five solids, in degrees. If your estimate is

E

and the correct answer is

A

, your score for this problem will be

\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).

p35. Estimate the value of

\log_{10} \left(\prod_{k|2016} k!\right),

where the product is taken over all positive divisors

k

2016

. If your estimate is

E

and the correct answer is

A

, your score for this problem will be

\max \left(0, \lceil 15 \cdot \min \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).

p36. Estimate the value of

\sqrt{2016}^{\sqrt[4]{2016}}

. If your estimate is

E

and the correct answer is

A

, your score for this problem will be

\max \left(0, \lceil 15 \cdot \min \left(\frac{\ln E}{\ln A}, 2- \frac{\ln E}{\ln A}\right) \rceil \right).

PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here and 5-8 [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement