MathDB

2009 Algebra #10 - Points on Cubic that Form a Rectangle

Source:

1/2/2012

Let

f(x)=2x^3-2x

. For what positive values of

a

do there exist distinct

b,c,d

such that

(a,f(a)),(b,f(b)),(c,f(c)),(d,f(d))

is a rectangle?

geometryrectangle

2009 Calculus #10: Definite Integral Closed Form

Source:

6/23/2012

Let

a

and

b

be real numbers satisfying

a>b>0

. Evaluate

\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.

Express your answer in terms of

a

and

b

.

calculusintegrationtrigonometryfunctionderivativegeometric series

2009 Combinatorics #10 - Rearrangement of First n Integers

Source:

1/7/2012

Given a rearrangement of the numbers from

1

to

n

, each pair of consecutive elements

a

and

b

of the sequence can be either increasing (if

a < b

) or decreasing (if

b < a

). How many rearrangements of the numbers from

1

to

n

have exactly two increasing pairs of consecutive elements? Express your answer in terms of

n

.

2009 Geometry #4: Incircles and Kites

Source:

6/23/2012

A kite is a quadrilateral whose diagonals are perpendicular. Let kite

ABCD

be such that

\angle B = \angle D = 90^\circ

. Let

M

and

N

be the points of tangency of the incircle of

ABCD

to

AB

and

BC

respectively. Let

\omega

be the circle centered at

C

and tangent to

AB

and

AD

. Construct another kite

AB^\prime C^\prime D^\prime

that is similar to

ABCD

and whose incircle is

\omega

. Let

N^\prime

be the point of tangency of

B^\prime C^\prime

to

\omega

. If

MN^\prime \parallel AC

, then what is the ratio of

AB:BC

?

geometryratio

2009 Geometry #10: Extensions and Circles

Source:

6/23/2012

Points

A

and

B

lie on circle

\omega

. Point

P

lies on the extension of segment

AB

past

B

. Line

\ell

passes through

P

and is tangent to

\omega

. The tangents to

\omega

at points

A

and

B

intersect

\ell

at points

D

and

C

respectively. Given that

AB=7

,

BC=2

, and

AD=3

, compute

BP

.

geometrytrigonometryinequalitiestrig identitiesLaw of Cosinestriangle inequalitypower of a point

10

Problems(5)