MathDB

BMT 2014 Spring - Geometry 9

Source:

12/29/2021

Let

ABC

be a triangle. Construct points

B'

and

C'

such that

ACB'

and

ABC'

are equilateral triangles that have no overlap with

\vartriangle ABC

. Let

BB'

and

CC'

intersect at X. If

AX = 3

,

BC = 4

, and

CX = 5

, find the area of quadrilateral

BCB'C'

. .

geometry

2014 BMT Team 9 limit

Source:

1/6/2022

Two different functions

f, g

of

x

are selected from the set of real-valued functions

\left \{sin x, e^{-x}, x \ln x, \arctan x, \sqrt{x^2 + x} -\sqrt{x^2 + x} -x, \frac{1}{x} \right \}

to create a product function

f(x)g(x)

. For how many such products is

\lim_{x\to infty} f(x)g(x)

finite?

calculus

BMT 2014 Spring - Analysis 9

Source:

1/6/2022

Find

\alpha

such that

\lim_{x\to0^+}x^\alpha I(x)=a\enspace\text{given}\enspace I(x)=\int^\infty_0\sqrt{1+t}\cdot e^{-xt}dt

where

a

is a nonzero real number.

calculuslimitsintegration

BMT 2014 Spring - Individual 9

Source:

1/22/2022

Suppose

a_1, a_2, ...

and

b_1, b_2,...

are sequences satisfying

a_n + b_n = 7

,

a_n = 2b_{n-1} - a_{n-1}

, and

b_n = 2a_{n-1} - b_{n-1}

, for all

n

. If

a_1 = 2

, find

(a_{2014})^2 - (b_{2014})^2

. .

algebra

BMT 2014 Spring - Discrete 9

Source:

1/6/2022

Leo and Paul are at the Berkeley BART station and are racing to San Francisco. Leo is planning to take the line that takes him directly to SF, and because he has terrible BART luck, his train will arrive in some integer number of minutes, with probability

\frac i{210}

for

1\le i\le20

at any given minute. Paul will take a second line, whose trains always arrive before Leo’s train, with uniform probability. However, Paul must also make a transfer to a 3rd line, whose trains arrive with uniform probability between

0

and

10

minutes after Paul reaches the transfer station. What is the probability that Leo gets to SF before Paul does?

probabilitycombinatorics

9

Problems(5)