2013fallinterschool final

2013 FAMAT Fall Interschool Test

What is this? This test contains forty questions/problems/puzzles. Some of these items have multiple

parts, and some of these items have only one part. The number of points each item is worth is given in

square brackets next to the item number, throughout the test. The maximum possible score on this test is

90 points.

Who can participate? Any student or sponsor associated with your chapter can help to solve the

problems. However, you may not consult alumni, parents, or any other humans outside of your chapter.

That being said, you may use the internet/calculators/books/etc., provided that you do not get help from

another human (e.g. asking a question on an internet forum would not be allowed).

How do we do this? You have until November 6, 2013 at 11:59pm EST to email your answers to

[email protected]

You must submit the answers using the answer document available under the Downloads section of

floridamao.org (presumably the same place you downloaded this test). You can open the document with

Microsoft Excel, Openoffice Calc, or Google Docs. If you have any questions about the test (e.g. think a

question is flawed) or if you are having issues using the answer document, you may email that address during

the testing period and a response will be given (if appropriate).

Note well! Throughout this test, the notation bxc denotes the floor function (also known as the greatest

integer function).

Good luck – and more importantly, have fun!

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[1] 1. The numbers 2013, 2014, and 2015 all have the same number of divisors. What is the smallest

natural number n such that n, n + 1, and n + 2 all have the same number of divisors?

[1] 2. Consider circle O with radius r . There is a square ABCD such that points A and B are on the

circumference of O and CD is tangent to O. If the sum of the areas of circle O and square ABCD is 2013,

determine brc.

[1] 3. There exists a function f (x) such that

f (sin4 θ + cos4 θ) = sin8 θ + cos8 θ

for all θ. Find f (2013).

[1] 4. Suppose a and b are real numbers such that

a2 − 3b2 =530

a

3a2 − b2 =954

b

What is a2 + b2?

[2] 5. For how many integers 1 ≤ n ≤ 100 is2n + n!

n + 1also an integer?

[1] 6.

∫ π4

0

16 cos 2x

9− cos 4xdx = arctan

m

n, where m and n are relatively prime positive integers. What is m + n?

[1] 7. Consider the set S containing all 24-tuples (x1, x2, · · · , x24) of integers such that 0 ≤ xk ≤ k for

1 ≤ k ≤ 24 and

x1 + 2x2 + 3x3 + · · ·+ 24x24 = 2450

Let (x1, x2, · · · , x24) be the average of all 24-tuples in S. What is x1 + x2 + · · ·+ x24?

[2] 8. For 4ABC,

AB + BC + AC = 61

cos∠A+ cos∠B + cos∠C =3

11

What is |AB · cos∠C + BC · cos∠A+ AC · cos∠B|?

[2] 9. A permutation {a1, a2, · · · , a8} of the numbers {2, 3, 4, · · · , 9} is called relative if gcd(ak , ak+1) = 1

for 1 ≤ k ≤ 7. How many relative permutations are there?

[1] 10. Two points A and B are chosen randomly from the inside of a circle Γ1. Circle Γ2 is the circle with

diameter AB. What is the probability that Γ2 is contained entirely within Γ1?

[2] 11. In the expression ∣∣ 1 2 3 · · · 2012 2013∣∣

each is randomly filled with either a plus or minus sign. For example, a possible value for the expression

is | − 1 + 2− 3 + 4 · · · − 2011 + 2012− 2013| = 1005. What is the expected value of the expression?

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[3] 12. Chords AB and CD of an ellipse intersect at focus F . If AF = 3, CF = 4, and BF = 5, what is DF?

[2] 13. Let Hn = 1 +1

2+

1

3+ · · ·+

1

n. Evaluate

limn→∞

n∑k=1

Hkk−Hn2

2.

[3] 14. A physicist designs a chamber shaped like an equilateral triangle and built with perfectly reflective

walls. He then cuts a small hole at each vertex of the chamber and aims a laser into the chamber through

one of these vertices such that the laser beam makes an angle of θ with the side of the chamber closest to

the beam (so that 0◦ < θ < 30◦). Each vertex of the chamber is also outfitted with a photodiode that can

detect when the laser beam is directed through it. If θm is the largest value of θ for which the laser beam

reflects 2013 times within the chamber before tripping a photodiode, find tan θm.

[3] 15. Let P (x) be the polynomial of degree 2012 such that P (k) = k! for k = 0, 1, 2, · · · , 2012. What is

the remainder when P (2013) is divided by 1000?

[3] 16. Francisco has a box with 2013 scraps of paper, each scrap labeled with a distinct number from 1 to

2013. He randomly chooses two scraps of paper from the box, suppose with the first scrap labeled m and

the second scrap labeled n. He then erases the label n on the second scrap, relabels the second scrap m (so

that both scraps are now labeled m), and places both scraps back in the box. What is the expected number

of times must Francisco repeat this process in order for all scraps in the box to be labeled with the same

number?

[2] 17. Let f : N→ N be the function such that∑d |n

f (d) = n2

for all natural numbers n. That is, the sum of f (d) over all divisors d of n equals n2. For example, setting

n = 6 yields f (1) + f (2) + f (3) + f (6) = 36. What is f (10!)?

[3] 18. Suppose f (x) and g(x) are nonconstant, infinitely differentiable functions satisfying the equations

f ′(x)

f (x)+g′(x)

g(x)= 1

f ′′(x)

f (x)+g′′(x)

g(x)=f ′′′(x)

f (x)+g′′′(x)

g(x)

for all x with f (x)g(x) 6= 0. Evaluatef ′′(2013)

f ′(2013)+g′′(2013)

g′(2013).

[≤ 1] 19. It’s time for a game theory classic! Provide a real number n from the interval (0, 2013] as your

answer. If n is the average of all answers provided, each school will receive 1−|2n − 3n|

4026points.

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[≤ 2] 20. Consider the number

N = 201320132013201320 + 201320132013201313 + 1

Provide a prime divisor d of N as your answer. The school(s) with the largest correct answer will be awarded

2 points. All other schools with correct answers will receive 2 ·(

1−|d − dmax|

N

)points, where dmax is the

largest correct answer given by a school.

[≤ 3] 21. Zoe has a valid 16 digit credit card number. The last eight digits of Zoe’s credit card number are

5903 4723. The first eight digits of her number consist of the numerals 2, 3, 5, 6, 7, 8, 9, 0 in some order.

Provide a list of N guesses for her credit card number as your answer, with N ≤ 8! = 40320. If Zoe’s credit

card number is not in a school’s list, this school will be awarded 0 points. Of the schools whose lists contain

Zoe’s credit card number, the school with the smallest value for N will be awarded 3 points. All other

schools whose lists contain the correct number will receive 3 ·(

1−|N − Nmin|

40320

)points, where Nmin is the

size of the smallest list containing Zoe’s number that is given by a school. (Note: no worries, Zoe. I didn’t

use your actual credit card number.)

[2] 22. In the following Skyscrapers puzzle, fill each cell of the 6 by 6 grid with a digit from 1 to 6 so that

no digits repeat in a given row or column. Each number in the grid also represents the height of a building,

and the clues on the outside of the grid indicate how many buildings can be seen when viewing the grid from

that direction. As in a real city, taller buildings block the view of smaller buildings behind them. For

example, if a row contained the numbers 615324, then only one building is visible from the left (6), but

three buildings are visible from the right (4, 5, 6). As your answer to this question, provide the numbers in

the row marked A, followed by the numbers in the row marked B.

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[2] 23. The answer is YOFVHSEBR in QDOZD.

V = ct2

` =u

j

v qz = −1

A =1

2r s sinT

FL = dHJ

Π = c(`2 − `1)

y =y

2q∫ ∞0

v−n2

un =

√q

2

o = cn + s

T = 2qh

Q = CHJ

[1 for each part] 24. Answer the following about math in movies:

(a) This graph can be seen in a film in which the main character finds its adjacency matrix. Name the

movie and calculate the determinant of the graph’s adjacency matrix.

(b) One film uses the example “ignore the blonde” to illustrate a particular concept in game theory. Name

the movie and the concept.

(c) This integral can be seen in a film in which the main character solves it using the tic-tac-toe method.

Name the movie and evaluate the indefinite integral.∫x2 sin x dx

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[2] 25. In a game of Pokemon Sapphire, you are attempting to catch the legendary Pokemon Kyogre.

Having battled Kyogre down to 4% of its initial HP, your Psyduck uses Hypnosis, causing Kyogre to fall

asleep. This sleep status ailment affects Kyogre for exactly three turns before he wakes up. Though your

Psyduck lacks the PP to use Hypnosis again, he has more than enough HP to withstand Kyogre’s attacks

indefinitely. Assuming that from the moment Kyogre first falls asleep, you throw a single standard Poke Ball

each turn, what is the expected number of standard Poke Balls necessary to capture Kyogre? Any answer

with an error of at most 0.5 Poke Balls will be accepted.

[3] 26. Researchers at SETI recently intercepted the following message sent to Earth by what appears to be

an advanced, semi-hostile alien civilization:

resistance is nearly futile

deduce our weapon override sequence from the facts below or perish

FF0000FFFF00000000

A52A2AFF0000A52A2A

EE82EEFF0000A52A2A

008000000000FF0000

FFFF00000000FFA500

FFA5004169E1FFFF00

FFA5004169E1008000

FFA500FFFFFF4169E1

FFFF00EE82EEEE82EE

[][][][][][][][][][][][][][][][][][]

808080EE82EEFFFFFF

Provide the override sequence as your answer and save the world.

[2] 27. While attempting to track down and steal the centuries-old treasure of Leonhard Euler,

internationally wanted thief Carmen Sandiego finds herself in the following location. Where in the world is

Carmen Sandiego?

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[3] 28.

“Time just keeps tickin’ clockwise/

this game of life pushin’ on/

tell me when my cells get a r-r-r-reset/

how many generations ’til the second dawn?”

- MC John Conway

[1 for each part] 29. All three have a number in common. What is it?

(a) Lewis Carroll, Douglas Adams, Jackie Robinson

(b) Jim Carrey, John Forbes Nash, Julius Caesar

(c) Jay-Z, Toto, Goldfinger

(d) Henry Fonda, Bruce Willis, Steve Martin

(e) Jack Skellington, Mega Man, Stanley Yelnats

(f) Mitt Romney, David Oxtoby, Asano Naganori

[1] 30. The island of Logicia is inhabited by three types of people: knights, who always tell the truth;

knaves, who always lie; and normals, who sometimes tell the truth and sometimes lie. In addition, there is an

honor-based system of ranking; knights are said to be of the highest rank, normals of middle rank, and

knaves of the lowest rank. During your travels, you meet three Logicians named Alex, Bradley, and Chico.

One of them is a knight, one is a knave, and one is normal. When you attempt to make polite conversation,

you receive the following cryptic responses:

Alex: Bradley is of a higher rank than Chico.

Bradley: Chico is of a higher rank than Alex.

Though neither of these responses answer your initial question of “how do you do today?,” you decide to

play along and ask Chico: “who has a higher rank, Alex or Bradley?” What does Chico answer?

[1] 31. There’s a little bit of space left on this page, so I figure we can fit in a quick puzzle. Which of the

following words is the odd one out?

repertoire, torturous, galahad, proprietor, alfalfa, typewriter, flagfall

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[3] 32. Wake up, soldier! No time for lazing about and dilly-dallying at Math Boot Camp. There are

conjectures to be made! Proofs to be written! Now drop down and solve me fifty; fifty partial differential

equations, that is. And while you’re at it, let’s check if you’ve been paying any attention at all. Who’s my

favorite mathematician?

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[1 for each part] 33. Each of the following excerpts is from an important mathematical work. Name each

work and its corresponding author(s).

(a) “A point is that which has no part. A line is breadthless length.”

(b) “From this proposition it will follow, when arithmetical addition is defined, that 1 + 1 = 2.”

(c) “But in our opinion, truths of this kind should be drawn from notions rather than notations.”

(d) “The alternation of motion is ever proportional to the motive force impressed; and is made in the

direction of the right line in which that force is impressed.”

[2] 34. In the following chess puzzle, it is white’s turn to move. Is it possible for white to force a

checkmate? If so, what is the minimum number of turns in which white can do so?

[3] 35. Over the past decade, the NSA has spent billions of dollars developing a new encryption technique,

a function known only by the codename f36. This function takes two integers as its input and outputs the

word encoded within those integers. As consulting cryptographer, it’s your job to verify the efficacy of the

encoding method, given a few values of f36:

f36(22, 35) = m

f36(887, 1295) = on

f36(15941, 46655) = cat

f36(1040453, 1679615) = math

f36(45856895, 60466175) = raven

f36(839608811942, 2821109907455) = applepie

f36(66050999355866, 101559956668415) = nevermore

f36(68108012021675783102, 170581728179578208255) = edgarallenpoe

Unfortunately for the NSA, you manage to crack the code in just an hour. What is the unique ordered pair

of integers (a, b) such that f36(a, b) = prettyweaknsa?

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[2 for part a, 1 for parts b and c] 36.

(a) Each digit of the following long division has been replaced either by a dot or the letter A. Every

occurrence of the letter A represents the same unique digit. Every dot simply indicates a digit (dots do

not necessarily represent the same digit). None of the dots are the A digit. What is the dividend of the

division problem?

(b) Let

f (n) =

∫ 10

xn − 1

ln xdx

Evaluate exp(f (3) + f (4) + f (5)), where exp(x) = ex .

(c) Who is the connection between these two problems?

[1] 37. Adam, Brian, Cathy, and Duchenka happen upon a river in the middle of the night. There’s a narrow

bridge across the river, but it only holds at most two people at a time. Since it’s pitch black outside, a torch

must be used for each crossing (and of course, the group only has one torch). Adam can cross the bridge

alone in 1 minute; Brian can cross alone in 2 minutes; Cathy in 5 minutes; and Duchenka in 10 minutes. If

two people cross the bridge together, they must cross at the pace of the slower person. What is the

minimum amount of time in which the whole group can cross the bridge?

[2] 38. In Russian, a male’s middle name is his patronymic; that is, his middle initial is the first letter of his

father’s first name. And, as expected, the first initial is the first letter of his first name. Consider the names

of all the males in a Russian family:

• A.N. Petrov

• B.M. Petrov

• G.K. Petrov

• K.M. Petrov

• K.T. Petrov

• M.M. Petrov

• M.N. Petrov

• N.K. Petrov

• N.M. Petrov

• N.T. Petrov

• T.M. Petrov

Given that every father has two sons, the patriarch of the family has four grandsons, and the patriarch’s

grandsons have two sons each, determine the patriarch.

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[1 for each part] 39. Answer the following questions about the university that I attend.

(a) What is the university that I attend?

(b) At my university, where does the course ’Honors Analysis in Rn’ meet each week? (building and room

number)

(c) Name one of the three Fields Medalists that are currently associated with my university.

(d) Let S be the set of all course numbers corresponding to mathematics classes offered by my university.

What is the largest element of S?

(e) One of the following facts about my university is false. Which one?

i. The saltwater school of economics has historically been defended by economists at my university.

ii. The author of Cat’s Cradle received his master’s degree from my university.

iii. The first self-sustaining nuclear reaction was conducted under the football field at my university.

iv. The behavioralist approach to political science originated at my university.

[2] 40. This is obviously necessary.

[1] 41. Thanks for participating in the 2013 Fall Interschool, and good luck during the

x = 32 · 23 · 7 + 503 · 2 FAMAT season! To collect your free point, simply provide x as your answer to this

question.

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2013fallinterschool final

Documents

smallestnatural number

test is90 points

item number

number of divisors

x1 x2 x24

areas of circle o

x24 of integers

circle withdiameter