MathDB

1

2017 T10: Vieta Quadratic Interpolation

P(x) = x^3 - 6x - 2

has three real roots,

\alpha

,

\beta

, and

\gamma

. Depending on the assignment of the roots, there exist two different quadratics

Q

such that the graph of

y=Q(x)

pass through the points

(\alpha,\beta)

,

(\beta,\gamma)

, and

(\gamma,\alpha)

. What is the larger of the two values of

Q(1)

?

9

1

2017 T9: Many Tangent Circles

Circles

\omega_1

and

\omega_2

are externally tangent to each other. Circle

\Omega

is placed such that

\omega_1

is internally tangent to

\Omega

at

X

while

\omega_2

is internally tangent to

\Omega

at

Y

. Line

\ell

is tangent to

\omega_1

at

P

and

\omega_2

at

Q

and furthermore intersects

\Omega

at points

A

and

B

with

AP<AQ

. Suppose that

AP=2

,

PQ=4

, and

QB=3

. Compute the length of line segment

\overline{XY}

.

8

1

2017 T8: Strings Determined by Flipping Coins

Alice and Bob have a fair coin with sides labeled

C

and

M

, and they flip the coin repeatedly while recording the outcomes; for example, if they flip two

C

's then an

M

, they have

CCM

recorded. They play the following game: Alice chooses a four-character string

\mathcal A

, then Bob chooses two distinct three-character strings

\mathcal B_1

and

\mathcal B_2

such that neither is a substring of

\mathcal A

. Bob wins if

\mathcal A

shows up in the running record before either

\mathcal B_1

or

\mathcal B_2

do, and otherwise Alice wins. Given that Alice chooses

\mathcal A = CMMC

and Bob plays optimally, compute the probability that Bob wins.

7

1

2017 T7: Rational Approximation of sqrt(2)

Define

\{p_n\}_{n=0}^\infty\subset\mathbb N

and

\{q_n\}_{n=0}^\infty\subset\mathbb N

to be sequences of natural numbers as follows:
[*]

p_0=q_0=1

; [*]For all

n\in\mathbb N

,

q_n

is the smallest natural number such that there exists a natural number

p_n

with

\gcd(p_n,q_n)=1

satisfying

\dfrac{p_{n-1}}{q_{n-1}} < \dfrac{p_n}{q_n} < \sqrt 2.

Find

q_3

.

6

1

2017 T6: Cheating is Bad

George is taking a ten-question true-false exam, where the answer key has been selected uniformly at random; however, he doesn't know any of the answers! Luckily, a friend has helpfully hinted that no two consecutive questions have true as the correct answer. If George takes the exam and maximizes the expected number of questions he gets correct, how many of his answers are expected to be right?

5

1

2017 T5: A Computer Science Problem?

We have four registers,

R_1,R_2,R_3,R_4

, such that

R_i

initially contains the number

i

for

1\le i\le4

. We are allowed two operations:
[*] Simultaneously swap the contents of

R_1

and

R_3

as well as

R_2

and

R_4

.
[*] Simultaneously transfer the contents of

R_2

to

R_3

, the contents of

R_3

to

R_4

, and the contents of

R_4

to

R_2

. (For example if we do this once then

(R_1,R_2,R_3,R_4)=(1,4,2,3)

.)
Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?

4

1

2017 T4: tWinning

Say an odd positive integer

n > 1

is

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

if

p - 2 \mid n

for every prime

p \mid n

. Find the number of twinning integers less than 250.

3

1

2017 T3: Pat and Rick

Suppose Pat and Rick are playing a game in which they take turns writing numbers from

\{1, 2, \dots, 97\}

on a blackboard. In each round, Pat writes a number, then Rick writes a number; Rick wins if the sum of all the numbers written on the blackboard after

n

rounds is divisible by 100. Find the minimum positive value of

n

for which Rick has a winning strategy.

2

1

2017 T2: Quirky Manipulation

Suppose

x

,

y

, and

z

are nonzero complex numbers such that

(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3

. Compute

(x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1z\right).

1

2017 T1: Floors and Square Roots

Find the integer

n

such that

n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017.

Here, as usual,

\lfloor\cdot\rfloor

denotes the floor function.

2017 CMIMC Team

Subcontests

2017 T10: Vieta Quadratic Interpolation

2017 T9: Many Tangent Circles

2017 T8: Strings Determined by Flipping Coins

2017 T7: Rational Approximation of sqrt(2)

2017 T6: Cheating is Bad

2017 T5: A Computer Science Problem?

2017 T4: tWinning

2017 T3: Pat and Rick

2017 T2: Quirky Manipulation

2017 T1: Floors and Square Roots