MathDB

p1. In the following

3

3

grid,

a, b, c

are numbers such that the sum of each row is listed at the right and the sum of each column is written below it: https://cdn.artofproblemsolving.com/attachments/d/9/4f6fd2bc959c25e49add58e6e09a7b7eed9346.png What is

n

?
p2. Suppose in your sock drawer of

14

socks there are 5 different colors and

3

different lengths present. One day, you decide you want to wear two socks that have both different colors and different lengths. Given only this information, what is the maximum number of choices you might have?
p3. The population of Arveymuddica is

2014

, which is divided into some number of equal groups. During an election, each person votes for one of two candidates, and the person who was voted for by

2/3

or more of the group wins. When neither candidate gets

2/3

of the vote, no one wins the group. The person who wins the most groups wins the election. What should the size of the groups be if we want to minimize the minimum total number of votes required to win an election?
p4. A farmer learns that he will die at the end of the year (day

365

, where today is day

0

) and that he has a number of sheep. He decides that his utility is given by ab where a is the money he makes by selling his sheep (which always have a fixed price) and

b

is the number of days he has left to enjoy the profit; i.e.,

365-k

where

k

is the day. If every day his sheep breed and multiply their numbers by

103/101

(yes, there are small, fractional sheep), on which day should he sell them all?
p5. Line segments

\overline{AB}

and

\overline{AC}

are tangent to a convex arc

BC

and

\angle BAC = \frac{\pi}{3}

. If

\overline{AB} = \overline{AC} = 3\sqrt3

, find the length of arc

BC

.
p6. Suppose that you start with the number

8

and always have two legal moves:

\bullet

Square the number

\bullet

Add one if the number is divisible by

8

or multiply by

4

otherwise How many sequences of

4

moves are there that return to a multiple of

8

?
p7. A robot is shuffling a

9

card deck. Being very well machined, it does every shuffle in exactly the same way: it splits the deck into two piles, one containing the

5

cards from the bottom of the deck and the other with the

4

cards from the top. It then interleaves the cards from the two piles, starting with a card from the bottom of the larger pile at the bottom of the new deck, and then alternating cards from the two piles while maintaining the relative order of each pile. The top card of the new deck will be the top card of the bottom pile. The robot repeats this shuffling procedure a total of n times, and notices that the cards are in the same order as they were when it started shuffling. What is the smallest possible value of

n

?
p8. A secant line incident to a circle at points

A

and

C

intersects the circle's diameter at point

B

with a

45^o

angle. If the length of

AB

1

and the length of

BC

7

, then what is the circle's radius?
p9. If a complex number

z

satisfies

z + 1/z = 1

, then what is

z^{96} + 1/z^{96}

?
p10. Let

a, b

be two acute angles where

\tan a = 5 \tan b

. Find the maximum possible value of

\sin (a - b)

.
p11. A pyramid, represented by

SABCD

has parallelogram

ABCD

as base (

A

is across from

C

) and vertex

S

. Let the midpoint of edge

SC

P

. Consider plane

AMPN

where

M

is on edge

SB

and

N

is on edge

SD

. Find the minimum value

r_1

and maximum value

r_2

\frac{V_1}{V_2}

where

V_1

is the volume of pyramid

SAMPN

and

V_2

is the volume of pyramid

SABCD

. Express your answer as an ordered pair

(r_1, r_2)

.
p12. A

5 \times 5

grid is missing one of its main diagonals. In how many ways can we place

5

pieces on the grid such that no two pieces share a row or column?
p13. There are

20

cities in a country, some of which have highways connecting them. Each highway goes from one city to another, both ways. There is no way to start in a city, drive along the highways of the country such that you travel through each city exactly once, and return to the same city you started in. What is the maximum number of roads this country could have?
p14. Find the area of the cyclic quadrilateral with side lengths given by the solutions to

x^4-10x^3+34x^2- 45x + 19 = 0.

p15. Suppose that we know

u_{0,m} = m^2 + m

and

u_{1,m} = m^2 + 3m

for all integers

m

, and that

u_{n-1,m} + u_{n+1,m} = u_{n,m-1} + u_{n,m+1}

Find

u_{30,-5}

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement