MathDB

2013 General Problem 7

Source:

2/4/2013

A fly and an ant are on one corner of a unit cube. They wish to head to the opposite corner of the cube. The fly can fly through the interior of the cube, while the ant has to walk across the faces of the cube. How much shorter is the fly's path if both insects take the shortest path possible?

geometry3D geometryPythagorean Theorem

SMT 2013 Calculus #7

Source:

2/3/2013

The function

f(x)

has the property that, for some real positive constant

C

, the expression

\frac{f^{(n)}(x)}{n+x+C}

is independent of

n

for all nonnegative integers

n

, provided that

n+x+C\neq 0

. Given that

f'(0)=1

and

\int_{0}^{1}f(x) \, dx = C+(e-2)

, determine the value of

C

. Note:

f^{(n)}(x)

is the

n

-th derivative of

f(x)

, and

f^{(0)}(x)

is defined to be

f(x)

.

calculusfunctionintegrationderivative

2013 SMT Geometry #7

Source:

11/2/2014

ABCD

is a square such that

AB

lies on the line

y=x+4

and points

C

and

D

lie on the graph of parabola

y^2=x

. Compute the sum of all possible areas of

ABCD

.

geometryconicsparabola

SMT 2013 Algebra #7

Source:

2/3/2013

Find all real

x

that satisfy

\sqrt[3]{20x+\sqrt[3]{20x+13}}=13

.

algebrapolynomialquadratics

SMT 2013 Team #7

Source:

2/4/2013

Find all real values of

u

such that the curves

y=x^2+u

and

y=\sqrt{x-u}

intersect in exactly one point.

SMT 2013 Advanced Topics #7

Source:

12/31/2016

Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability

\tfrac12

the lowest note that is higher than the note he just played, and with probability

\tfrac12

the highest note that is lower than the note he just played. What is the probability that he plays the highest note on the piano before playing the lowest note?

2013Advanced Topics

7

Problems(6)