MathDB

2017 Algebra #5

Source:

8/20/2022

Compute

\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.

2017Algebra Test

2017 Calculus #5

Source:

9/17/2022

Compute the maximum value attained by

f(x)=x^{1/x^2}

.

2017Calculus Test

2017 Discrete #5

Source:

10/12/2022

A

<span class='latex-italic'>shuffle</span>

is a permutation of the integers

1,2,3,4,5

. More formally, a shuffle is a function

f:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}

such that if

i\neq j

then

f(i)\neq f(j)

. For example,

12345\mapsto23154

denotes a shuffle

f

so that

f(1)=2

,

f(2)=3

,

f(3)=1

,

f(4)=5

, and

f(5)=4

. A shuffle can be repeated some number of times to obtain another shuffle. For example, if

f

is the shuffle

12345\mapsto23154

from above, then repeating

f

twice gives the shuffle

g(x)=f(f(x))

which is

12345\mapsto31245

. How many shuffles are there that, when repeated

6

times, give the shuffle

12345\mapsto12345

?

2017Discrete Math Test

2017 Geometry #5

Source:

11/19/2022

Regular hexagon

ABCDEF

has side length

2

. Line segment

BD

is drawn, and circle

O

is inscribed inside the pentagon

ABDEF

such that

O

is tangent to

AF

,

BD

, and

EF

. Compute the radius of

O

.

2017Geometry Test

2017 Guts #5

Source:

11/25/2022

Let

\alpha

and

\beta

be the two roots of

x^2+2017x+k

. What is the sum of the possible values of

k

so that the lines \begin{align*} y&=2\alpha x+2017^2\\ y&=3\alpha x+2017^3 \end{align*} are perpendicular?

2017Guts Round

5

Problems(5)