2016 ASDAN Math Tournament

Part of ASDAN Math Tournament

Subcontests

(27)

1

2016 Guts #27

Suppose that you are standing in the middle of a

100

meter long bridge. You take a random sequence of steps either

1

meter forward or

1

meter backwards each iteration. At each step, if you are currently at meter

n

\tfrac{n}{100}

1

meter forward, to meter

n+1

\tfrac{100-n}{100}

1

meter backward, to meter

n-1

. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter

0

100

C

be the actual answer and

A

be the answer you will submit. Your score will be given by

\max\{0,\lceil25-25|\log_6(\tfrac{A-C/2}{C/2})|^{0.8}\rceil\}

1

2016 Guts #26

The Euclidean Algorithm on inputs

a

b

is a way to find the greatest common divisor

\gcd(a,b)

. Suppose WLOG that

a>b

. On each step of the Euclidan Algorithm, we solve the equation

a=bq+r

q,r

0\leq r<b

, and repeat on

b

r

\gcd(a,b)=\gcd(b,r)

, and we repeat. If

r=0

, we are done. For example,

\gcd(100,15)=\gcd(15,10)=\gcd(10,5)=5

100=15\cdot6+10

15=10\cdot1+5

10=5\cdot2+0

. Thus, the Euclidean Algorithm here takes

3

steps. What is the largest number of steps that the Euclidean Algorithm can take on some integer inputs

a,b

0<a,b<10^{2016}

C

be the actual answer and

A

be the answer you submit. If

\tfrac{|A-C|}{C}>\tfrac{1}{2}

, then your score will be

0

. Otherwise, your score will be given by

\max\{0,\lceil25-2(\tfrac{|A-C|}{20})^{1/2.2}\rceil\}

1

2016 Guts #25

Find the best rational approximation

x

\sqrt[3]{2016}

|x-\sqrt[3]{2016}|

is as small as possible. You may either find an

x=\tfrac{a}{b}

a,b

are coprime integers or find a decimal approximation. Let

C

be the actual answer and

A

be the answer you submit. Your score will be given by

\lceil10+\tfrac{16.5}{0.1+e^{30|A-C|}}\rceil

\lceil x\rceil

denote the smallest integer which is

\geq x

1

2016 Guts #24

Alex, Bill, and Charlie want to play a game of DotA. They each come online at a uniformly random time between

8:00

8:05\text{ }\text{PM}

, and each person queues for

2

minutes. However, if any of them sees any other of them online while queuing, they merge parties and restart the queue, again waiting for

2

minutes starting from the merger time.
For example, suppose that Alex logs in at

8:00\text{ PM}

, Bill logs in at

8:01\text{ PM}

, and Charlie logs in at

8:02:30\text{ PM}

30

8:02\text{ PM}

8:01\text{ PM}

, Alex and Bill would merge parties and queue for

2

minutes starting at

8:01\text{ PM}

8:02:30\text{ PM}

, Charlie would merge with Alex and Bill’s party, since Alex and Bill have waited together for only

1.5

minutes.
What is the probability that they will play as a party of

3

1

2016 Guts #23

Find all quadruples of real numbers

(a,b,c,d)

that satisfy the system of equations: \begin{align*} a+4b+8c+4d&=53\\ 3a^2+4b^2+12c^2+2d^2&=159\\ 9a^3+4b^3+18c^3+d^3&=477. \end{align*}

1

2016 Guts #22

n\times n

Latin square is a

n\times n

grid that is filled with

n

1

n

2

\dots

n

n

's such that each column and row of the grid contains exactly one of each

1

2

\dots

n

. For example, the following is a valid

2\times2

\textstyle\begin{bmatrix}2&1\\1&2\end{bmatrix}

, but this is not:

\textstyle\begin{bmatrix}2&1\\2&1\end{bmatrix}

4\times4

Latin squares are there?

1

2016 Guts #21

Suppose that we have a

2\times5

grid, and we wish to write

0

1

's inside such that for any

2\times2

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

0

. The determinant of a

2\times2

\textstyle\begin{bmatrix}a&b\\c&d\end{bmatrix}

ad-bc

. For example, the following is a valid configuration: However, the following is not valid because the last

2\times2

sub-block has determinant

1

: How many such valid

2\times5

configurations are there?

1

2016 Guts #20

ABC

be a triangle such that

AB=9

BC=6

AC=10

2

D_1,D_2

BC

BC

is subdivided into

3

equal segments;

4

E_1,E_2,\dots,E_4

AC

AC

is subdivided into

5

equal segments; and

8

F_1,F_2,\dots,F_8

AB

AB

is subdivided into

9

equal segments. All possible cevians are drawn from

A

D_i

B

E_j

C

F_k

. At how many points in the interior of

\triangle ABC

2

cevians intersect?

1

2016 Guts #19

z\neq0

be a complex number satisfying

z^2=z+i|z|

|z|

denotes the length between the origin and

z

in the complex plane.) Find

z\cdot\overline{z}

\overline{z}=a-bi

is the complex conjugate of

z=a+bi

1

2016 Guts #18

Compute the number of nonnegative integer triples

(x,y,z)

4x+2y+z\leq36

1

2016 Guts #17

Consider triangle

ABC

AB=4

BC=11

CA=9

. The triangle is spun around a line that passes through

B

and the interior of the triangle (including the edges

BC

BA

). Of all possible lines with these constraints, what is the largest possible volume of the resulting solid?

1

2016 Guts #16

Let the notation

\underline{ABC}

denote the number compromised of the digits

A

B

C

0\leq A,B,C\leq9

\underline{ABC}=100A+10B+C

\underline{CCAAC}=10000C+1000C+100A+10A+C

(\underline{ABC})^2=\underline{CCAAC}

A

B

C

are distinct nonzero digits, find the

3

\underline{ABC}

2

2016 Guts #15

a

be the least positive integer with

20

positive divisors and

b

be the least positive integer with

16

positive divisors. What is

a+b

? (Note that for any integer

n

1

n

are considered divisors of

n

2016 Team #15

\omega_1

\omega_2

r_1<r_2

respectively and intersect at distinct points

X

Y

. The common external tangents intersect at point

Z

. The common tangent closer to

X

\omega_1

\omega_2

P

Q

respectively. Line

ZX

\omega_1

\omega_2

again at points

R

S

RP

SQ

intersect again at point

T

XT=8

XZ=15

XY=12

\tfrac{r_1}{r_2}

2

2016 Guts #14

In the diagram to the right, squares are drawn on the side of the triangle with side lengths

5

6

7

as shown below. The corners of adjacent squares are then connected. What is the area of the resulting hexagon?

2016 Team #14

x,y,z

are positive real numbers that satisfy \begin{align*} x+y+z&=xyz\\ \frac{x^2}{16(1+x^2)}=\frac{y^2}{25(1+y^2)}&=\frac{z^2}{36(1+z^2)}. \end{align*} Compute

\frac{x^2(1+x^2)^2}{x^2(1+z^2)^2}.

2

2016 Guts #13

\{a_n\}_{n=1}^\infty

is a sequence. The partial sums

\{s_n\}_{n=1}^\infty

s_n=\sum_{i=1}^na_i.

The Cesàro sums are then defined as

\{A_n\}_{n=1}^\infty

A_n=\frac{1}{n}\cdot\sum_{i=1}^ns_i.

a_n=(-1)^{n+1}

. What is the limit of the Cesàro sums of

\{a_n\}_{n=1}^\infty

n

goes to infinity?

2016 Team #13

Ash writes the positive integers from

1

2016

inclusive as a single positive integer

n=1234567891011\dots2016

. What is the result obtained by successively adding and subtracting the digits of

n

? (In other words, compute

1-2+3-4+5-6+7-8+9-1+0-1+\dots

2

2016 Guts #12

Find the number of real solutions

x

, in radians, to

\sin(x)=\frac{x}{1000}.

2016 Team #12

f(x)=\frac{2016^x}{2016^x+\sqrt{2016}}.

\sum_{k=0}^{2016}f\left(\frac{k}{2016}\right).

2

2016 Guts #11

Ebeneezer is painting the edges of a cube. He wants to paint the edges so that the colored edges form a loop that does not intersect itself. For example, the loop should not look like a “figure eight” shape. If two colorings are considered equivalent if there is a rotation of the cubes so that the colored edges are the same, what is the number of possible edge colorings?

2016 Team #11

ABC

be a triangle with

AB=2

BC=3

AC=4

. Consider all lines

XY

X

AC

Y

BC

\triangle XYC

has area equal to half that of

\triangle ABC

. What is the minimum possible length of

XY

6

6

6

6

6

6

6

10

10

10