MathDB

2016 Algebra #3

Source:

12/24/2016

Let

A

denote the set of all integers

n

such that

1 \le n \le 10000

, and moreover the sum of the decimal digits of

n

is

2

. Find the sum of the squares of the elements of

A

.

2016 Combo #3

Source:

12/30/2016

Find the number of ordered pairs of integers

(a, b)

such that

a, b

are divisors of 720 but

ab

is not.

2016 Geo #3

Source:

12/30/2016

In the below picture,

T

is an equilateral triangle with a side length of

5

and

\omega

is a circle with a radius of

2

. The triangle and the circle have the same center. Let

X

be the area of the shaded region, and let

Y

be the area of the starred region. What is

X - Y

?

2016 Guts #3

Source:

12/24/2016

Let

PROBLEMZ

be a regular octagon inscribed in a circle of unit radius. Diagonals

MR

,

OZ

meet at

I

. Compute

LI

.

2016 Team #3

Source:

12/30/2016

Let

ABC

be an acute triangle with incenter

I

and circumcenter

O

. Assume that

\angle OIA = 90^{\circ}

. Given that

AI = 97

and

BC = 144

, compute the area of

\triangle ABC

.

2016 General #3: Rectangular Prisms

Source:

11/15/2016

Let

V

be a rectangular prism with integer side lengths. The largest face has area

240

and the smallest face has area

48

. A third face has area

x

, where

x

is not equal to

48

or

240

. What is the sum of all possible values of

x

?

HMMTgeometry3D geometryprism

2016 Theme #3: Triangle vertices as points

Source:

11/22/2016

The three points

A, B, C

form a triangle.

AB=4, BC=5, AC=6

. Let the angle bisector of

\angle A

intersect side

BC

at

D

. Let the foot of the perpendicular from

B

to the angle bisector of

\angle A

be

E

. Let the line through

E

parallel to

AC

meet

BC

at

F

. Compute

DF

.

HMMTgeometryangle bisector

3

Problems(7)