MathDB

p1. Let

f

be a function such that

f(x + y) = f(x) + f(y)

for all

x

and

y

. Assume

f(5) = 9

. Compute

f(2015)

.
p2. There are six cards, with the numbers

2, 2, 4, 4, 6, 6

on them. If you pick three cards at random, what is the probability that you can make a triangles whose side lengths are the chosen numbers?
p3. A train travels from Berkeley to San Francisco under a tunnel of length

10

kilometers, and then returns to Berkeley using a bridge of length

7

kilometers. If the train travels at

30

km/hr underwater and 60 km/hr above water, what is the train’s average speed in km/hr on the round trip?
p4. Given a string consisting of the characters A, C, G, U, its reverse complement is the string obtained by first reversing the string and then replacing A’s with U’s, C’s with G’s, G’s with C’s, and U’s with A’s. For example, the reverse complement of UAGCAC is GUGCUA. A string is a palindrome if it’s the same as its reverse. A string is called self-conjugate if it’s the same as its reverse complement. For example, UAGGAU is a palindrome and UAGCUA is self-conjugate. How many six letter strings with just the characters A, C, G (no U’s) are either palindromes or self-conjugate?
p5. A scooter has

2

wheels, a chair has

6

wheels, and a spaceship has

11

wheels. If there are

10

of these objects, with a total of

50

wheels, how many chairs are there?
p6. How many proper subsets of

\{1, 2, 3, 4, 5, 6\}

are there such that the sum of the elements in the subset equal twice a number in the subset?
p7. A circle and square share the same center and area. The circle has radius

1

and intersects the square on one side at points

A

and

B

. What is the length of

\overline{AB}

?
p8. Inside a circle, chords

AB

and

CD

intersect at

P

in right angles. Given that

AP = 6

BP = 12

and

CD = 15

, find the radius of the circle.
p9. Steven makes nonstandard checkerboards that have

29

squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
p10. John is organizing a race around a circular track and wants to put

3

water stations at

9

possible spots around the track. He doesn’t want any

2

water stations to be next to each other because that would be inefficient. How many ways are possible?
p11. In square

ABCD

, point

E

is chosen such that

CDE

is an equilateral triangle. Extend

CE

and

DE

F

and

G

AB

. Find the ratio of the area of

\vartriangle EFG

to the area of

\vartriangle CDE

.
p12. Let

S

be the number of integers from

2

8462

(inclusive) which does not contain the digit

1,3,5,7,9

. What is

S

?
p13. Let x, y be non zero solutions to

x^2 + xy + y^2 = 0

. Find

\frac{x^{2016} + (xy)^{1008} + y^{2016}}{(x + y)^{2016}}

.
p14. A chess contest is held among

10

players in a single round (each of two players will have a match). The winner of each game earns

2

points while loser earns none, and each of the two players will get

1

point for a draw. After the contest, none of the

10

players gets the same score, and the player of the second place gets a score that equals to

4/5

of the sum of the last

5

players. What is the score of the second-place player?
p15. Consider the sequence of positive integers generated by the following formula

a_1 = 3

a_{n+1} = a_n + a^2_n

for

n = 2, 3, ...

What is the tens digit of

a_{1007}

?
p16. Let

(x, y, z)

be integer solutions to the following system of equations

x^2z + y^2z + 4xy = 48

x^2 + y^2 + xyz = 24

Find

\sum x + y + z

where the sum runs over all possible

(x, y, z)

.
p17. Given that

x + y = a

and

xy = b

and

1 \le a, b \le 50

, what is the sum of all a such that

x^4 + y^4 - 2x^2y^2

is a prime squared?
p18. In

\vartriangle ABC

M

is the midpoint of

\overline{AB}

, point

N

is on side

\overline{BC}

. Line segments

\overline{AN}

and

\overline{CM}

intersect at

O

. If

AO = 12

CO = 6

, and

ON = 4

, what is the length of

OM

?
p19. Consider the following linear system of equations.

1 + a + b + c + d = 1

16 + 8a + 4b + 2c + d = 2

81 + 27a + 9b + 3c + d = 3

256 + 64a + 16b + 4c + d = 4

Find

a - b + c - d

.
p20. Consider flipping a fair coin

8

times. How many sequences of coin flips are there such that the string HHH never occurs?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

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