MathDB

2017 Team #10: Line passes through fixed point

Source:

2/19/2017

Let

LBC

be a fixed triangle with

LB = LC

, and let

A

be a variable point on arc

LB

of its circumcircle. Let

I

be the incenter of

\triangle ABC

and

\overline{AK}

the altitude from

A

. The circumcircle of

\triangle IKL

intersects lines

KA

and

BC

again at

U \neq K

and

V \neq K

. Finally, let

T

be the projection of

I

onto line

UV

. Prove that the line through

T

and the midpoint of

\overline{IK}

passes through a fixed point as

A

varies.

geometry

2017 Algebra/NT #10: Number of Functions

Source:

2/19/2017

Let

\mathbb{N}

denote the natural numbers. Compute the number of functions

f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}

such that

f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}

for all integers

x\ge 1

.

functionmodular arithmetic

2017 Geometry #10: Midpoints of Quadrilateral

Source:

2/20/2017

Let

ABCD

be a quadrilateral with an inscribed circle

\omega

. Let

I

be the center of

\omega

, and let

IA=12,

IB=16,

IC=14,

and

ID=11

. Let

M

be the midpoint of segment

AC

. Compute the ratio

\frac{IM}{IN}

, where

N

is the midpoint of segment

BD

.

geometrytangential quadrilateralratio

2017 Combinatorics #10: Number of Words

Source:

2/20/2017

Compute the number of possible words

w=w_1w_2\dots w_{100}

satisfying:

\bullet

w

has exactly

50

A

's and

50

B

's (and no other letter).

\bullet

For

i=1,2,\dots,100

, the number of

A

's among

w_1, w_2, \dots, w_i

is at most the number of

B

's among

w_1, w_2, \dots, w_i

.

\bullet

For all

i=44,45,\dots,57

, if

w_i

is a

B

, then

w_{i+1}

must be a

B

.

2017 Theme #10

Source:

5/8/2018

Denote

\phi=\frac{\sqrt{5}+1}{2}

and consider the set of all finite binary strings without leading zeroes. Each string

S

has a “base-

\phi

” value

p(S)

. For example,

p(1101)=\phi^3+\phi^2+1

. For any positive integer n, let

f(n)

be the number of such strings S that satisfy

p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}

. The sequence of fractions

\frac{f(n+1)}{f(n)}

approaches a real number

c

as

n

goes to infinity. Determine the value of

c

.

algebra

2017 General #10

Source:

5/8/2018

Five equally skilled tennis players named Allen, Bob, Catheryn, David, and Evan play in a round robin tournament, such that each pair of people play exactly once, and there are no ties. In each of the ten games, the two players both have a 50% chance of winning, and the results of the games are independent. Compute the probability that there exist four distinct players

P_1

,

P_2

,

P_3

,

P_4

such that

P_i

beats

P_{i+1}

for

i=1, 2, 3, 4

. (We denote

P_5=P_1

).

combinatorics

2017 Guts #10: Weird limit

Source:

2/21/2017

Let

ABC

be a triangle in the plane with

AB = 13

,

BC = 14

,

AC = 15

. Let

M_n

denote the smallest possible value of

(AP^n + BP^n + CP^n)^{\frac{1}{n}}

over all points

P

in the plane. Find

\lim_{n \to \infty} M_n

.

geometry

10

Problems(7)