2014 ASDAN Math Tournament

Part of ASDAN Math Tournament

Subcontests

(25)

1

2014 General #25

300

couples (one man, one woman) are invited to a party. Everyone at the party either always tells the truth or always lies. Exactly

2/3

of the men say their partner always tells the truth and the remaining

1/3

say their partner always lies. Exactly

2/3

of the women say their partner is the same type as themselves and the remaining

1/3

say their partner is different. Find

a

, the maximum possible number of people who tell the truth, and

b

, the minimum possible number of people who tell the truth. Express your answer as

(a,b)

1

2014 General #24

It's pouring down rain, and the amount of rain hitting point

(x,y)

f(x,y)=|x^3+2x^2y-5xy^2-6y^3|.

If you start at the origin

(0,0)

, find all the possibilities for

m

y=mx

is a straight line along which you could walk without any rain falling on you.

1

2014 General #23

ABC

have side lengths

AB=11

BC=7

AC=12

D

AC

E

AB

\angle CDE=90^\circ

and the area of triangle

CDE

is maximized. Find the area of triangle

CDE

1

2014 General #22

2

dollars. Bananas cost

3

dollars. Oranges cost

5

dollars. Compute the number of distinct baskets of fruit such that there are

100

pieces of fruit and the basket costs

300

dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit.

1

2014 General #21

A bitstring of length

\ell

is a sequence of

\ell

0

1

's in a row. How many bitstrings of length

2014

2012

0

1

1

2014 General #20

ABCD

is a parallelogram, and circle

S

2

) is inscribed insider

ABCD

S

is tangent to all four line segments

AB

BC

CD

DA

. One of the internal angles of the parallelogram is

60^\circ

. What is the maximum possible area of

ABCD

1

2014 General #19

f(x)+2f(4-x)=x+8

f(16)

1

2014 General #18

A two-digit positive integer is

<span class='latex-italic'>primeable</span>

if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is

<span class='latex-italic'>unripe</span>

. Compute the total number of unripe integers.

1

2014 General #17

Given that the line

y=mx+k

intersects the parabola

y=ax^2+bx+c

at two points, compute the product of the two

x

-coordinates of these points in terms of

a

b

c

k

m

1

2014 General #16

Compute the number of geometric sequences of length

3

where each number is a positive integer no larger than

10

2

2014 General #15

Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?

2014 Team #15

A point is "bouncing" inside a unit equilateral triangle with vertices

(0,0)

(1,0)

(1/2,\sqrt{3}/2)

. The point moves in straight lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point starts at the origin and begins motion in the direction of

(1,1)

. After the ball has traveled a cumulative distance of

30\sqrt{2}

, compute its distance from the origin.

2

2014 General #14

Patricia has a rectangular painting that she wishes to frame. The frame must also be rectangular and will extend

3\text{ cm}

outward from each of the four sides of the painting. When the painting is framed, the area of the frame not covered by the painting is

108\text{ cm}^2

. What is the perimeter of the painting alone (without the frame)?

2014 Team #14

Consider a round table on which

2014

people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated

2

seats away from the person at the head of the table.

2

2014 General #13

S_1

is inscribed inside circle

C_1

, which is inscribed inside square

S_2

, which is inscribed inside circle

C_2

, which is inscribed inside square

S_3

, which is inscribed inside circle

C_3

, which is inscribed inside square

S_4

a

be the side length of

S_4

b

be the side length of

S_1

\tfrac{a}{b}

2014 Team #13

\alpha,\beta,\gamma

be the three real roots of the polynomial

x^3-x^2-2x+1=0

. Find all possible values of

\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}

2

2014 General #12

Consider a rectangular tiled room with dimensions

m\times n

, where the tiles are

1\times1

in size. Compute all ordered pairs

(m,n)

m\leq n

such that the number of tiles on the perimeter is equal to the number of tiles in the interior (i.e. not on the perimeter).

2014 Team #12

Find the last two digits of

\tbinom{200}{100}

. Express the answer as an integer between

0

99

. (e.g. if the last two digits are

05

5

2

2014 General #11

Mr. Ambulando is at the intersection of

5^{\text{th}}

\text{A St}

, and needs to walk to the intersection of

1^{\text{st}}

\text{F St}

. There's an accident at the intersection of

4^{\text{th}}

\text{B St}

, which he'd like to avoid.

Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take?

2014 Team #11

In the following system of equations

|x+y|+|y|=|x-1|+|y-1|=2,

find the sum of all possible

x

5

5

5

5

5

5

5

8

8

8