MathDB

p1. Ten math students take a test, and the average score on the test is

28

. If five students had an average of

15

, what was the average of the other five students' scores?
p2. If

a\otimes b = a^2 + b^2 + 2ab

, find

(-5\otimes 7) \otimes 4

.
p3. Below is a

3 \times 4

grid. Fill each square with either

1

2

3

. No two squares that share an edge can have the same number. After filling the grid, what is the

4

-digit number formed by the bottom row? https://cdn.artofproblemsolving.com/attachments/9/6/7ef25fc1220d1342be66abc9485c4667db11c3.png
p4. What is the angle in degrees between the hour hand and the minute hand when the time is

6:30

?
p5. In a small town, there are some cars, tricycles, and spaceships. (Cars have

4

wheels, tricycles have

3

wheels, and spaceships have

6

wheels.) Among the vehicles, there are

24

total wheels. There are more cars than tricycles and more tricycles than spaceships. How many cars are there in the town?
p6. You toss five coins one after another. What is the probability that you never get two consecutive heads or two consecutive tails?
p7. In the below diagram,

\angle ABC

and

\angle BCD

are right angles. If

\overline{AB} = 9

\overline{BD} = 13

, and

\overline{CD} = 5

, calculate

\overline{AC}

. https://cdn.artofproblemsolving.com/attachments/7/c/8869144e3ea528116e2d93e14a7896e5c62229.png
p8. Out of

100

customers at a market,

80

purchased oranges,

60

purchased apples, and

70

purchased bananas. What is the least possible number of customers who bought all three items?
p9. Francis, Ted and Fred planned to eat cookies after dinner. But one of them sneaked o earlier and ate the cookies all by himself. The three say the following: Francis: Fred ate the cookies. Fred: Ted did not eat the cookies. Ted: Francis is lying. If exactly one of them is telling the truth, who ate all the cookies?
p11. Let

ABC

be a triangle with a right angle at

A

. Suppose

\overline{AB} = 6

and

\overline{AC} = 8

. If

AD

is the perpendicular from

A

BC

, what is the length of

AD

?
p12. How many three digit even numbers are there with an even number of even digits?
p13. Three boys, Bob, Charles and Derek, and three girls, Alice, Elizabeth and Felicia are all standing in one line. Bob and Derek are each adjacent to precisely one girl, while Felicia is next to two boys. If Alice stands before Charles, who stands before Elizabeth, determine the number of possible ways they can stand in a line.
p14. A man

5

foot,

10

inches tall casts a

14

foot shadow.

20

feet behind the man, a flagpole casts ashadow that has a

9

foot overlap with the man's shadow. How tall (in inches) is the flagpole?
p15. Alvin has a large bag of balls. He starts throwing away balls as follows: At each step, if he has

n

balls and 3 divides

n

, then he throws away a third of the balls. If

3

does not divide

n

but

2

divides

n

, then he throws away half of them. If neither

3

nor

2

divides

n

, he stops throwing away the balls. If he began with

1458

balls, after how many steps does he stop throwing away balls?
p16. Oski has

50

coins that total to a value of

82

cents. You randomly steal one coin and find out that you took a quarter. As to not enrage Oski, you quickly put the coin back into the collection. However, you are both bored and curious and decide to randomly take another coin. What is the probability that this next coin is a penny? (Every coin is either a penny, nickel, dime or quarter).
p17. Let

ABC

be a triangle. Let

M

be the midpoint of

BC

. Suppose

\overline{MA} = \overline{MB} = \overline{MC} = 2

and

\angle ACB = 30^o

. Find the area of the triangle.
p18. A spirited integer is a positive number representable in the form

20^n + 13k

for some positive integer

n

and any integer

k

. Determine how many spirited integers are less than

2013

.
p19. Circles of radii

20

and

13

are externally tangent at

T

. The common external tangent touches the circles at

A

, and

B

, respectively where

A \ne B

. The common internal tangent of the circles at

T

intersects segment

AB

X

. Find the length of

AX

.
p20. A finite set of distinct, nonnegative integers

\{a_1, ... , a_k\}

is called admissible if the integer function

f(n) = (n + a_1) ... (n + a_k)

has no common divisor over all terms; that is,

gcd \left(f(1), f(2),... f(n)\right) = 1

for any integer

n

. How many admissible sets only have members of value less than

10

\{4\}

and

\{0, 2, 6\}

are such sets, but

\{4, 9\}

and

\{1, 3, 5\}

are not.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

p1. If Bob takes

6

hours to build

4

houses, how many hours will he take to build

12

houses?
p2. Compute the value of

\frac12+ \frac16+ \frac{1}{12} + \frac{1}{20}

.
p3. Given a line

2x + 5y = 170

, find the sum of its

x

- and

y

-intercepts.
p4. In some future year, BmMT will be held on Saturday, November

19

th. In that year, what day of the week will April Fool’s Day (April

1

st) be?
p5. We distribute

78

penguins among

10

people in such a way that no person has the same number of penguins and each person has at least one penguin. If Mr. Popper (one of the

10

people) wants to take as many penguins as possible, what is the largest number of penguins that Mr. Popper can take?
p6. A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry?
p7. Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections are distinct?
p8. The pages of a book are consecutively numbered from

1

through

480

. How many times does the digit

8

appear in this numbering?
p9. A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length

2

, what is the area of the flower?
p10. There are two non-consecutive positive integers

a, b

such that

a^2 - b^2 = 291

. Find

a

and

b

.
p11. Let

ABC

be an equilateral triangle. Let

P, Q, R

be the midpoints of the sides

BC

CA

and

AB

respectively. Suppose the area of triangle

PQR

1

. Among the

6

points

A, B, C, P, Q, R

, how many distinct triangles with area

1

have vertices from that set of

6

points?
p12. A positive integer is said to be binary-emulating if its base three representation consists of only

0

s and

1

s. Determine the sum of the first

15

binary-emulating numbers.
p13. Professor

X

can choose to assign homework problems from a set of problems labeled

1

30

, inclusive. No two problems in his assignment can share a common divisor greater than

1

. What is the maximum number of problems that Professor

X

can assign?
p14. Trapezoid

ABCD

has legs (non-parallel sides)

BC

and

DA

of length

5

and

6

respectively, and there exists a point

X

CD

such that

\angle XBC = \angle XAD = \angle AXB = 90^o

. Find the area of trapezoid

ABCD

.
p15. Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner? (Two games are said to be identical if either player wins/loses rounds in the same order in both games.)
p16. Let

ABC

be a triangle and M be the midpoint of

BC

. If

AB = AM = 5

and

BC = 12

, what is the area of triangle

ABC

?
p17. A positive integer

n

is called good if it can be written as

5x+ 8y = n

for positive integers

x, y

. Given that

42

43

44

45

and

46

are good, what is the largest n that is not good?
p18. Below is a

7 \times 7

square with each of its unit squares labeled

1

49

in order. We call a square contained in the figure good if the sum of the numbers inside it is odd. For example, the entire square is good because it has an odd sum of

1225

. Determine the number of good squares in the figure.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 https://cdn.artofproblemsolving.com/attachments/9/2/1039c3319ae1eab7102433694acc20fb995ebb.png
p19. A circle of integer radius

r

has a chord

PQ

of length

8

. There is a point

X

on chord

PQ

such that

\overline{PX} = 2

and

\overline{XQ} = 6

. Call a chord

AB

euphonic if it contains

X

and both

\overline{AX}

and

\overline{XB}

are integers. What is the minimal possible integer

r

such that there exist

6

euphonic chords for

X

?
p20. On planet Silly-Math, two individuals may play a game where they write the number

324000

on a whiteboard and take turns dividing the number by prime powers – numbers of the form

p^k

for some prime

p

and positive integer

k

. Divisions are only legal if the resulting number is an integer. The last player to make a move wins. Determine what number the first player should select to divide

324000

by in order to ensure a win.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2013 BmMT

Subcontests

2013 BmMT Individual Round - Berkley mini Math Tournament

2013 BmMT Team Round - Berkley mini Math Tournament Fall