MathDB

1

2013 BMT Individual 20

a_n

is defined such that

a_0 =\frac{1 + \sqrt3}{2}

and

a_{n+1} =\sqrt{a_n}

for

n \ge 0

. Evaluate

\prod_{k=0}^{\infty} 1 - a_k + a^2_k

19

1

2013 BMT Individual 19

Equilateral triangle

ABC

is inscribed in a circle. Chord

AD

meets

BC

at

E

. If

DE = 2013

, how many scenarios exist such that both

DB

and

DC

are integers (two scenarios are different if

AB

is different or

AD

is different)?

18

1

2013 BMT Individual 18

Paul and his pet octahedron like to play games together. For this game, the octahedron randomly draws an arrow on each of its faces pointing to one of its three edges. Paul then randomly chooses a face and progresses from face to adjacent face, as determined by the arrows on each face, and he wins if he reaches every face of the octahedron. What is the probability that Paul wins?

17

1

2013 BMT Individual 17

Let

N \ge 1

be a positive integer and

k

be an integer such that

1 \le k \le N

. Define the recurrence

x_n = \frac{x_{n-1} + x_{n-2} +... + x_{n-N}}{N}

for

n > N

and

x_k = 1

,

x_1 = x_2 = ... = x_{k-1} =x_{k+1} =.. = x_N = 0

. As

n

approaches infinity,

x_n

approaches some value. What is this value?

16

1

2013 BMT Individual 16

Find the sum of all possible

n

such that

n

is a positive integer and there exist

a, b, c

real numbers such that for every integer

m

, the quantity

\frac{2013m^3 + am^2 + bm + c}{n}

is an integer.

13

1

2013 BMT Individual 13

Let

f(n)

be a function from integers to integers. Suppose

f(11) = 1

, and

f(a)f(b) = f(a +b) + f(a - b)

, for all integers

a, b

. Find

f(2013)

.

12

1

2013 BMT Individual 12

Triangle

ABC

satisfies the property that

\angle A = a \log x

,

\angle B = a \log 2x

, and

\angle C = a \log 4x

radians, for some real numbers

a

and

x

. If the altitude to side

AB

has length

8

and the altitude to side

BC

has length

9

, find the area of

\vartriangle ABC

.

11

1

2013 BMT Individual 11

Let

t = (a, b, c)

, and let us define

f^1 (t) = (a + b, b + c, c + a)

and

f^k (t) = f^{k-1}(f^1(t))

for all

k > 1

. Furthermore, a permutation of

t

has the same elements, just in a different order (e.g.,

(b, c, a)

). If

f^{2013}(s)

is a permutation of

s

for some

s = (k, m, n)

, where

k, m

, and

n

are integers such that

|k|, |m|, |n|\le 10

, how many possible values of

s

are there?

15

1

computational geo <ABD=<BCD, <ABD=<ECD 2013 BMT Individual 15

Let

ABCD

be a convex quadrilateral with

\angle ABD = \angle BCD

,

AD = 1000

,

BD = 2000

,

BC = 2001

, and

DC = 1999

. Point

E

is chosen on segment

DB

such that

\angle ABD = \angle ECD

. Find

AE

.

14

1

computational geo with incircle 2013 BMT Individual 14

Triangle

ABC

has incircle

O

that is tangent to

AC

at

D

. Let

M

be the midpoint of

AC

.

E

lies on

BC

so that line

AE

is perpendicular to

BO

extended. If

AC = 2013

,

AB = 2014

,

DM = 249

, find

CE

.

P2

3

BMT 2013 Spring - Geometry P2

From a point

A

construct tangents to a circle centered at point

O

, intersecting the circle at

P

and

Q

respectively. Let

M

be the midpoint of

PQ

. If

K

and

L

are points on circle

O

such that

K, L

, and

A

are collinear, prove

\angle MKO = \angle MLO

.

BMT 2013 Spring - Discrete P2

Let

p

be an odd prime, and let

(p^p)!=mp^k

for some positive integers

m

and

k

. Find in terms of

p

the number of ordered pairs

(m,k)

satisfying

m+k\equiv0\pmod p

.

BMT 2014 Spring - Analysis P2

If

f(x)=x^n-7x^{n-1}+17x^{n-2}+a_{n-3}x^{n-3}+\ldots+a_0

is a real-valued function of degree

n>2

with all real roots, prove that no root has value greater than

4

and at least one root has value less than

0

or greater than

2

.

P1

3

BMT 2013 Spring - Geometry P1

Suppose a convex polygon has a perimeter of

1

. Prove that it can be covered with a circle of radius

1/4

.

BMT 2013 Spring - Discrete P1

Ahuiliztli is playing around with some coins (pennies, nickels, dimes, and quarters). She keeps grabbing

k

coins and calculating the value of her handful. After a while, she begins to notice that if

k

is even, she more often gets even sums, and if

k

is odd, she more often gets odd sums. Help her prove this true! Given

k

coins chosen uniformly and at random, prove that. the probability that the parity of

k

is the same as the parity of the

k

coins' value is greater than the probability that the parities are different.

BMT 2014 Spring - Analysis P1

Prove that for all positive integers

m

and

n

,

\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0

5

5

5

5

5

5

5

4

5

2013 BMT Spring

Subcontests

2013 BMT Individual 20

2013 BMT Individual 19

2013 BMT Individual 18

2013 BMT Individual 17

2013 BMT Individual 16

2013 BMT Individual 13

2013 BMT Individual 12

2013 BMT Individual 11

computational geo <ABD=<BCD, <ABD=<ECD 2013 BMT Individual 15

computational geo with incircle 2013 BMT Individual 14

BMT 2013 Spring - Geometry P2

BMT 2013 Spring - Discrete P2

BMT 2014 Spring - Analysis P2

BMT 2013 Spring - Geometry P1

BMT 2013 Spring - Discrete P1

BMT 2014 Spring - Analysis P1

2013 BMT Spring

Subcontests

2013 BMT Individual 20

2013 BMT Individual 19

2013 BMT Individual 18

2013 BMT Individual 17

2013 BMT Individual 16

2013 BMT Individual 13

2013 BMT Individual 12

2013 BMT Individual 11

computational geo &lt;ABD=&lt;BCD, &lt;ABD=&lt;ECD 2013 BMT Individual 15

computational geo with incircle 2013 BMT Individual 14

BMT 2013 Spring - Geometry P2

BMT 2013 Spring - Discrete P2

BMT 2014 Spring - Analysis P2

BMT 2013 Spring - Geometry P1

BMT 2013 Spring - Discrete P1

BMT 2014 Spring - Analysis P1

computational geo <ABD=<BCD, <ABD=<ECD 2013 BMT Individual 15