Test State Interschool (PDF)

2017 FAMAT State Interschool

Welcome to the FAMAT Student Delegate 2017 State Interschool! Here are some frequently asked
questions that actually have never been asked. :)
What is this? This is a test written by the 2016-2017 FAMAT Student Delegate Officers for your
school to complete during the week leading up to the State Convention.
Test Breakdown: This test contains 8 sections, each containing a set of test items: questions, puzzles,
or problems. The maximum score for this test is 154 points.
#
1
2
3
4
5
6
7
8

Section
Math
Plots, Figures, and Visuals
Even More Visuals
Internet
Pop Culture
Fun With Words
Interactive
Jellybeans
Total

Points
32
21
19
9
26
23
12
12
154

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, which means something like asking a question on an internet
forum like Yahoo Answers is not be allowed.
How do we submit our answers? You must submit the answers using the answer document available
under the Downloads section of famat.org. Please download the answer sheet pdf, print single
sided, fill in your school (do not fill in the footer or any lines that say “For Grading Use Only”),
and handwrite your answers. Your school will turn in one answer sheet by Friday night of
the State Convention by 9 PM to the Convention Registration Desk (as stated on the State
Convention Letter). E-mailed submissions will not be accepted.
How do we dispute? Hello, FAMAT Student, we know that you love to dispute our tests. If you have
any questions or concerns about the test or if you have any issues using the answer document,
you may email 2017stateinterschool@famatdelegates.org during the testing period. We
will resolve your issues, if any clarifications must be made during the testing period, an errata will
be posted under Downloads and the FAMAT Student Organization Facebook Group.
A note on answer form: All answers must be exact unless otherwise stated. Stick to the generally
accepted answer forms (Rationalized denominators, simplified fractions and radicals, etc.). We
will accept equivalent answer forms within reason.

Page 1

1

2017 FAMAT State Interschool

Math
1. (2 points) The axis of a right circular cylinder with radius 1 unit is centered on the x-axis, and
extends infinitely in both directions. A second right circular cylinder with radius 1 unit has an
axis centered on the y-axis, and extends infinitely in both directions. Find the volume of the
intersection of the two cylinders.
2. (2 points) Let L be the locus of points consisting of the circumferences of two circles each with
radius 1 unit that are tangent to each other. Find the area of the largest square that can be
constructed such that each of its 4 vertices is in L.
3. (2 points) Solve for the four digit number W XY Z (Where W, X, Y, Z are digits) that has the
property W XY Z = W X · Y Z .
4. (2 points) Find the least common multiple of the sum of the numbers that equal the factors they
each have, the sum of the numbers less than 1000 that equal the sum of their factors (besides
themselves), the sum of the numbers between 1 and 60 (non-inclusive) whose factors are relatively
prime, and the sum of the numbers less than 60 that are less than the sum of their factors (besides
themselves).
5. (2 point) Let integers a, b, c satisfy:
a+b+c≡3
2

2

2

a +b +c ≡9

(mod 4)
(mod 16)

Find all positive integers k such that
a3 + b3 + c3 ≡ 27

(mod k)

is always true.
√
√
6. (2 point) Robert was taking a test, and got an answer of 10 2 + 9 3 for question
√ #999.
√
However the question then continued: “If the answer can be written as a b + c d,
where a, b, c, d are integers, find a + b + c + d.” Most people would’ve answered 24;
however, Robert realized there was a great chance to dispute. Find the sum of every possible
distinct value of a + b + c + d.
7. (2 point) For some relatively prime semiprimes l, u, the number lul has 2028 positive integer
divisors, including 1 and the number lul . (In other words, d(lul ) = 2028.) Find the sum of the
distinct values d(2lu) may obtain.
8. (2 points) Jhin, Jen, and Jyn play the following game: Jhin chooses a number randomly from the
interval [0, 2π] while Jen and Jyn randomly pick points on the circumference of a unit circle. Jhin
makes a straight slice connecting the two points, breaking the unit circle into two pieces. Consider
the area of the smaller of the two pieces. Jhin wins if his chosen number is larger than this area,
c1
1
and he loses otherwise. The probability Jhin wins can be written in the form
+
where
c2 c3 · π c4
c1 , c2 , c3 , c4 are positive integers and c1 , c2 are relatively prime. Find the sum c1 + 2c2 + 3c3 + 4c4 .

Page 2

2017 FAMAT State Interschool

9. (5 points) Define a word as any string of letters from the set a,b,c, ... ,x,y,z.
This days, it so hard find good word. All the word I hear are too big. I have pals. Help me find
they good word stat.
A person’s “good word stat” is the probability any given word that comes out of their mouth has
a length of at most 4 letters, over the space of all the (not necessarily distinct) words they’ve said
in infinite time. Write your answers rounded to the nearest integer percent.
(a) (1 point) Hi, my name is Viraj. I speak by randomly choosing (valid) words said within the
past 500 years or so, taking into account their frequencies. For example, I will say “the”
quite often.
(b) (1 point) Hi, my name is Adam. I speak by randomly choosing (valid) words said within the
past 500 years or so, without taking into account their frequencies. For example, I will say
“the” and “precalculus” with the same probability.
(c) (1 point) Hi, my name is Matt. When I speak, I have a 31 probability of ending the current
word I’m saying, and starting a new word with some random letter, and a 23 probability of
continuing the current word I’m saying by appending a new random letter.
(d) (2 points) Hi, my name is Mike. When I speak, I have a 21 probability of ending the current
word I’m saying, and starting a new word with some random letter, and a 12 probability of
continuing the current word I’m saying by appending some new random letters. Given that
I’m continuing the current word I’m saying, let k be a positive integer, and let L be the
current length of the current word. Then the probability of appending k new random letters
is inversely proportional to (kL)2 .
10. (3 points) Find the number of odd coefficients of the following expressions. For example, given
(x + 9)(x + 14)(x + 16) = x3 + 39x2 + 494x + 2016, it is easy to see there are exactly 2 odd
coefficients.
(a) (1 points)
(2x + 3x4 + 7x11 + 18)(3x14 + 2x71 + 8x314 + 299x8 )
(b) (1 point)
2017
Y

(x + i)

i=1

(c) (1 point)
2017
Y

(x2 + ix + i2 )

i=1

11. (4 points) After realizing you’ve solved every math problem ever published, you begin generating
your own problems. To do this, first you standardize notation. An expression is defined as any
string of symbols from the set 0123456789-+/*(). Then, you define a modifier as [a -> b],
where a and b are expressions. Applying a modifier to an expression replaces all instances of a
with b. For example, evaluating 3-5 after applying [- -> +] gives an answer of 8.

Page 3

2017 FAMAT State Interschool

(a) (0.5 points) Evaluate 2+2 after applying [2 -> 2+2] 4 times.
(b) (1 point) Evaluate 2*2 after applying [2 -> 2*2] 4 times.
(c) (1 point) Evaluate 1/1 after applying [/1 -> 1/1+1/10 4 times.
(d) (1.5 points) Evaluate 1+1 after applying [+ -> +1+1] 16 times.
√
12. (2 points) Let x be a real number. If tan(x) tan(150◦ − x) = − 5, what is the value of
sin(x) sin(150◦ − x)?
13. (2 points) Point N lies within triangle KIM such that ∠KIN = ∠M KN , ∠IKN = ∠IM N ,
and ∠IN M = 90◦ . What is the value of M N/KM ?

2

Plots, Figures, and Visuals

Part I: (0.5 points each, 5 total) Your FAMAT Student Delegate President loves visualizations, especially
when they are used for statistical analyses or for representing mathematical concepts. She often spends
hours scrolling through photos of plots on the Internet and wishing her plots could be just as beautiful.
It is 3 in the morning, and she has just spent the entire day looking at plots. She is confused. Help her
identify the following plots and figures:
[Hint: Numbers should not be relevant, identification can be made using distinctive features of different
types of plots]

(a)

(b)

(c)

(d)

Page 4

2017 FAMAT State Interschool

(e)

(f)

(g)

(h)

(i)

(j)

Part II: (16 points) After finally identifying all the plots that appeared earlier in this test (and learning
how to make them herself!), Kim decided to tone down her visualization web-surfing adventures by
focusing on the basics. She decided to look at colors instead, particularly HTML colors In the definition
used for this section, there are 140 defined HTML colors. The color names are the strings that do not
include any numbers or “#” symbols. An example is the color midnightblue. The following questions
deal with colors that Kim might have encountered on the second leg of her grand visualization odyssey.
1. (3 points) How many HTML color names have the substring medium in them? light? dark?
2. (1 point) What is the longest HTML color name (names do not have spaces)?

Page 5

2017 FAMAT State Interschool

3. (12 points; 0.5 per color) Name as many HTML colors you can find in the following figure, all are
one of 140 HTML colors, there are no repeats. Please write the colors in the corresponding line
in your answer sheet. The bottom right answer box corresponds with the bottom right color in
the array. You may write the colors with or without spaces.

3

Even More Visuals

Picture games from the 2016 Fall Interschool makes a comeback!
1. (6 points) What is something that the subjects of these four pictures have in common?

2. (4 points) These three groups of images, when interpreted, all share something in common. What
is it?

Page 6

2017 FAMAT State Interschool

3. (4 points) Identify what is pictured in the following:
(a) (Type of pasta)

(c) (Type of duck)

(d) (Specific position)
(b) (Type of knot)

Page 7

2017 FAMAT State Interschool

4. (2 points) What language is written below? (One point for identifying the language, one for
translating!)

5. (1 point) Which of the hands below would win in poker?

6. (1 point) Who painted this?

7. (1 points) In what country is this piece of public art located?

Page 8

4

2017 FAMAT State Interschool

An Adventure Through the Internet!
1. (5 points) What is the next sequence?
0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2...
1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2...
1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2...

2. (3 points) “Lazy Skunk” with a lisp = ?

3. (1 point) What would you tell Google to do if you wanted to rotate the screen 360◦ ?

5

An Adventure Through Pop Culture!
1. (9 points) After a long journey home from overseas, you overhear a conversation your father is
having with his best friend from high school. Figure out what they’re talking about:
“Ya know, they don’t make ’em like they used to. Back in the day, it was all
’bout the struggle of life. The common man, the common folk. ’Bout the
grime and dirt of the real world: the war and the battle. Now, seems like
every year everyone seems to be captivated by any old ball throwing sport;
looks like everyone just goes on and drops e’rything just to see it – their
brains are all mashed like potaters...Maybe they ought to go see sarg’ at
the medical tent, cuz they clearly have their eyes on backwards. But us old
fellers, we’ve left our mark on that list. The end of our stuff is on the same
caliber as them, I reckon. I saw it on a list–it said so itself. It made marks
on history that no ball-throwing half-time can do. I guess the good old days
are just old days now to these young folk.”
2. (1 point) Patrick is before Tom, who is before Matt. David is after Paul, who is after Jon. Who’s
first? (Hint: This question is making perfect sense. Are you keeping up?)