MathDB

p1. What is

7\%

of one half of

11\%

20000

?
p2. Three circles centered at

A, B

, and

C

are tangent to each other. Given that

AB = 8

AC = 10

, and

BC = 12

, find the radius of circle

A

.
p3. How many positive integer values of

x

less than

2012

are there such that there exists an integer

y

for which

\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}

?
p4. The positive difference between

8

and twice

x

is equal to

11

more than

x

. What are all possible values of

x

?
p5. A region in the coordinate plane is bounded by the equations

x = 0

x = 6

y = 0

, and

y = 8

. A line through

(3, 4)

with slope

4

cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line?
p6. A polygon is composed of only angles of degrees

138

and

150

, with at least one angle of each degree. How many sides does the polygon have?
p7.

M, A, T, H

, and

L

are all not necessarily distinct digits, with

M \ne 0

and

L \ne 0

. Given that the sum

MATH +LMT

, where each letter represents a digit, equals

2012

, what is the average of all possible values of the three-digit integer

LMT

?
p8. A square with side length

\sqrt{10}

and two squares with side length

\sqrt{7}

share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square?
p9. Consider the sequence

2012, 12012, 20120, 20121, ...

. This sequence is the increasing sequence of all integers that contain “

2012

”. What is the

30

th term in this sequence?
p10. What is the coefficient of the

x^5

term in the simplified expansion of

(x +\sqrt{x} +\sqrt[3]{x})^{10}

?
PS. You had better use hide for answers.

Team Round

Problems(1)