MathDB

2016 Algebra #8

Source:

8/6/2022

It is possible to express the sum

\sum_{n=1}^{24}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}

as

a\sqrt{2}+b\sqrt{3}

, for some integers

a

and

b

. Compute the ordered pair

(a,b)

.

2016Algebra Test

2016 Calculus #8

Source:

8/8/2022

Let

f

be a differentiable function such that

f'(0)=4

and

f(0)=3

. Compute

\lim_{x\rightarrow\infty}\left(\frac{f\left(\frac{1}{x}\right)}{f(0)}\right)^x.

2016Calculus Test

2016 Discrete #8

Source:

8/10/2022

Consider all fractions

\tfrac{a}{b}

where

1\leq b\leq100

and

0\leq a\leq b

. Of these fractions, let

\tfrac{m}{n}

be the smallest fraction such that

\tfrac{m}{n}>\tfrac{2}{7}

. What is

\tfrac{m}{n}

?

2016Discrete Math Test

2016 Geometry #8

Source:

8/11/2022

A circle with center

O

is drawn in the first quadrant of the 2D Cartesian plane (the quadrant with both positive

x

and

y

values) such that it lies tangent to the

x

and

y

-axes. A line is drawn with slope

m>1

and passing through the origin; the line intersects the circle at two points

A

and

B

, with

A

closer to the origin than

B

. Suppose that

ABO

is an equilateral triangle. Compute

m

.

2016Geometry Test

2016 Guts #8

Source:

8/14/2022

There are

n

integers

a

such that

0\leq a<91

and

a

is a solution to

x^3+8x^2-x+83\equiv0\pmod{91}

. What is

n

?

2016Guts Round

2016 Team #8

Source:

8/17/2022

Let

ABC

be a triangle with

AB=24

,

BC=30

, and

AC=36

. Point

M

lies on

BC

such that

BM=12

, and point

N

lies on

AC

such that

CN=20

. Let

X

be the intersection of

AM

and

BN

and let line

CX

intersect

AB

at point

L

. Compute

\frac{AX}{XM}+\frac{BX}{XN}+\frac{CX}{XL}.

2016team test

8

Problems(6)