MathDB

2017 Team #2: Double Positive Detector

Source:

2/19/2017

Does there exist a two-variable polynomial

P(x, y)

with real number coefficients such that

P(x, y)

is positive exactly when

x

and

y

are both positive?

algebrapolynomial

2017 Combinatorics #2: Ways to insert +'s

Source:

2/19/2017

How many ways are there to insert

+

's between the digits of

111111111111111

(fifteen

1

's) so that the result will be a multiple of

30

?

2017 Algebra/NT #2: Geometric Series

Source:

2/19/2017

Find the value of

\sum_{1\le a<b<c} \frac{1}{2^a3^b5^c}

(i.e. the sum of

\frac{1}{2^a3^b5^c}

over all triples of positive integers

(a, b, c)

satisfying

a<b<c

)

geometric series

2017 Geometry #2: Maximum Possible Area

Source:

2/19/2017

Let

ABC

be a triangle with

AB=13

,

BC=14

, and

CA=15

. Let

\ell

be a line passing through two sides of triangle

ABC

. Line

\ell

cuts triangle

ABC

into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?

geometryperimeter

2017 Theme #2

Source:

5/8/2018

Horizontal parallel segments

AB=10

and

CD=15

are the bases of trapezoid

ABCD

. Circle

\gamma

of radius

6

has center within the trapezoid and is tangent to sides

AB

,

BC

, and

DA

. If side

CD

cuts out an arc of

\gamma

measuring

120^{\circ}

, find the area of

ABCD

.

geometry

2017 General #2

Source:

5/8/2018

Determine the sum of all distinct real values of

x

such that

||| \cdots ||x|+x| \cdots |+x|+x|=1

where there are

2017

x

s in the equation.

algebra

2017 Guts #2: Perimeter of triangle in circle

Source:

2/21/2017

Let

A

,

B

,

C

,

D

,

E

,

F

be

6

points on a circle in that order. Let

X

be the intersection of

AD

and

BE

,

Y

is the intersection of

AD

and

CF

, and

Z

is the intersection of

CF

and

BE

.

X

lies on segments

BZ

and

AY

and

Y

lies on segment

CZ

. Given that

AX = 3

,

BX = 2

,

CY = 4

,

DY = 10

,

EZ = 16

, and

FZ = 12

, find the perimeter of triangle

XYZ

.

geometryperimeter

2

Problems(7)