MathDB

1

2017 BMT Individual 20

\sum^{15}_{k=0}\left(2^{560}(-1)^k \cos^{560}\left( \frac{k\pi}{16}\right)\right) \pmod{17}.

18

1

2017 BMT Individual 18

Consider the sequence

(k_n)

defined by

k_{n+1} = n(k_n + k_{n-1})

and

k_0 = 0

,

k_1 = 1

. What is

\lim _{n\to \infty} \frac{k_n}{n!}

?

14

2

2017 BMT Team 14

Suppose that there is a set of

2016

positive numbers, such that both their sum, and the sum of their reciprocals, are equal to

2017

. Let

x

be one of those numbers. Find the maximum possible value of

x +\frac{1}{x}

. .

2017 BMT Individual 14

Let

x

be the first term in the sequence

31, 331, 3331, . . .

which is divisible by

17

. How many digits long is

x

?

12

2

2017 BMT Team 12

A robot starts at the origin of the Cartesian plane. At each of

10

steps, he decides to move

1

unit in any of the following directions: left, right, up, or down, each with equal probability. After

10

steps, the probability that the robot is at the origin is

\frac{n}{4^{10}}

. Find

n

2017 BMT Individual 12

Square

S

is the unit square with vertices at

(0, 0)

,

(0, 1)

,

(1, 0)

and

(1, 1)

. We choose a random point

(x, y)

inside

S

and construct a rectangle with length

x

and width

y

. What is the average of

\lfloor p \rfloor

where

p

is the perimeter of the rectangle?

\lfloor x \rfloor

is the greatest integer less than or equal to

x

.

11

2

2017 BMT Team 11

Ben picks a positive number

n

less than

2017

uniformly at random. Then Rex, starting with the number

1

, repeatedly multiplies his number by

n

and then finds the remainder when dividing by

2017

. Rex does this until he gets back to the number

1

. What is the probability that, during this process, Rex reaches every positive number less than

2017

before returning back to

1

?

2017 BMT Individual 11

Naomi has a class of

100

students who will compete with each other in five teams. Once the teams are made, each student will shake hands with every other student, except the students in his or her own team. Naomi chooses to partition the students into teams so as to maximize the number of handshakes. How many handshakes will there be?

15

2

computational geo, <ACB=30^o, ABP equilateral 2017 BMT Team 15

In triangle

ABC

, the angle at

C

is

30^o

, side

BC

has length

4

, and side

AC

has length

5

. Let

P

be the point such that triangle

ABP

is equilateral and non-overlapping with triangle

ABC

. Find the distance from

C

to

P

.

2017 BMT Individual 15

Alice and Bob live on the edges and vertices of the unit cube. Alice begins at point

(0, 0, 0)

and Bob begins at

(1, 1, 1)

. Every second, each of them chooses one of the three adjacent corners and walks at a constant rate of

1

unit per second along the edge until they reach the corner, after which they repeat the process. What is the expected amount of time in seconds before Alice and Bob meet?

13

2

common area of 4 equilaterals in a square 2017 BMT Team 13

4

equilateral triangles of side length

1

are drawn on the interior of a unit square, each one of which shares a side with one of the

4

sides of the unit square. What is the common area enclosed by all

4

equilateral triangles?

max area of triangle, tangent line 2017 BMT Individual 13

Two points are located

10

units apart, and a circle is drawn with radius

r

centered at one of the points. A tangent line to the circle is drawn from the other point. What value of

r

maximizes the area of the triangle formed by the two points and the point of tangency?

17

1

angle wanted, circumcenter, circle, 80-60-40 triangle 2017 BMT Individual 17

Triangle

ABC

is drawn such that

\angle A = 80^o

,

\angle B = 60^o

, and

\angle C = 40^o

. Let the circumcenter of

\vartriangle ABC

be

O

, and let

\omega

be the circle with diameter

AO

. Circle

\omega

intersects side

AC

at point

P

. Let M be the midpoint of side

BC

, and let the intersection of

\omega

and

PM

be

K

. Find the measure of

\angle MOK

.

16

1

max area of triangle of centroids of PBC, PAC,PAB 2017 BMT Individual 16

Let

ABC

be a triangle with

AB = 3

,

BC = 5

,

AC = 7

, and let

P

be a point in its interior. If

G_A

,

G_B

,

G_C

are the centroids of

\vartriangle PBC

,

\vartriangle PAC

,

\vartriangle PAB

, respectively, find the maximum possible area of

\vartriangle G_AG_BG_C

.

19

1

center of ellipse through midpoints of triangle 2017 BMT Individual 19

Let

T

be the triangle in the

xy

-plane with vertices

(0, 0)

,

(3, 0)

, and

\left(0, \frac32\right)

. Let

E

be the ellipse inscribed in

T

which meets each side of

T

at its midpoint. Find the distance from the center of

E

to

(0, 0)

5

5

5

5

5

5

5

5

5

2017 BMT Spring

Subcontests

2017 BMT Individual 20

2017 BMT Individual 18

2017 BMT Team 14

2017 BMT Individual 14

2017 BMT Team 12

2017 BMT Individual 12

2017 BMT Team 11

2017 BMT Individual 11

computational geo, <ACB=30^o, ABP equilateral 2017 BMT Team 15

2017 BMT Individual 15

common area of 4 equilaterals in a square 2017 BMT Team 13

max area of triangle, tangent line 2017 BMT Individual 13

angle wanted, circumcenter, circle, 80-60-40 triangle 2017 BMT Individual 17

max area of triangle of centroids of PBC, PAC,PAB 2017 BMT Individual 16

center of ellipse through midpoints of triangle 2017 BMT Individual 19

2017 BMT Spring

Subcontests

2017 BMT Individual 20

2017 BMT Individual 18

2017 BMT Team 14

2017 BMT Individual 14

2017 BMT Team 12

2017 BMT Individual 12

2017 BMT Team 11

2017 BMT Individual 11

computational geo, &lt;ACB=30^o, ABP equilateral 2017 BMT Team 15

2017 BMT Individual 15

common area of 4 equilaterals in a square 2017 BMT Team 13

max area of triangle, tangent line 2017 BMT Individual 13

angle wanted, circumcenter, circle, 80-60-40 triangle 2017 BMT Individual 17

max area of triangle of centroids of PBC, PAC,PAB 2017 BMT Individual 16

center of ellipse through midpoints of triangle 2017 BMT Individual 19

computational geo, <ACB=30^o, ABP equilateral 2017 BMT Team 15