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Cmimc 2017 A

This document contains a 10 question math test covering topics like ratios, polynomials, sequences, and complex numbers. It instructs students to write their answers on an answer sheet, providing their name, team, and test subject. Calculators are not permitted and answers must be reasonably simplified. Any errors should be reported in writing to a specific location by the end of lunch.

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Gulit Radian Wiyarno
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222 views2 pages

Cmimc 2017 A

This document contains a 10 question math test covering topics like ratios, polynomials, sequences, and complex numbers. It instructs students to write their answers on an answer sheet, providing their name, team, and test subject. Calculators are not permitted and answers must be reasonably simplified. Any errors should be reported in writing to a specific location by the end of lunch.

Uploaded by

Gulit Radian Wiyarno
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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1. Do not look at the test before the proctor starts the round.

2. This test consists of 10 short-answer problems to be solved in 60 minutes. Each question is


worth one point.
3. Write your name, team name, and team ID on your answer sheet. Circle the subject of the
test you are currently taking.
4. Write your answers in the corresponding boxes on the answer sheets.
5. No computational aids other than pencil/pen are permitted.
6. Answers must be reasonably simplified.
7. If you believe that the test contains an error, submit your protest in writing to Doherty 2302
by the end of lunch.
Algebra
1. The residents of the local zoo are either rabbits or foxes. The ratio of foxes to rabbits in the zoo is 2 : 3. After
10 of the foxes move out of town and half the rabbits move to Rabbitretreat, the ratio of foxes to rabbits is
13 : 10. How many animals are left in the zoo?
xy
2. For nonzero real numbers x and y, define x ◦ y = x+y . Compute

21 ◦ 22 ◦ 23 ◦ · · · ◦ 22016 ◦ 22017

.

3. Suppose P (x) is a quadratic polynomial with integer coefficients satisfying the identity
P (P (x)) − P (x)2 = x2 + x + 2016
for all real x. What is P (1)?
1 1 1
4. It is well known that the mathematical constant e can be written in the form e = 0! + 1! + 2! + · · · . With
this in mind, determine the value of
∞
X j
j
.
b c!
j=3 2

Express your answer in terms of e.


5. The set S of positive real numbers x such that
   
2x 3x
+ + 1 = bxc
5 5
S∞
can be written as S = j=1 Ij , where the Ii are disjoint intervals of the form [ai , bi ) = {x | ai ≤ x < bi } and
P2017
bi ≤ ai+1 for all i ≥ 1. Find i=1 (bi − ai ).
6. Suppose P is a quintic polynomial with real coefficients with P (0) = 2 and P (1) = 3 such that |z| = 1
whenever z is a complex number satisfying P (z) = 0. What is the smallest possible value of P (2) over all
such polynomials P ?
7. Let a, b, and c be complex numbers satisfying the system of equations
a b c
+ + = 9,
b+c c+a a+b
a2 b2 c2
+ + = 32,
b+c c+a a+b
a3 b3 c3
+ + = 122.
b+c c+a a+b
Find abc.
8. Suppose a1 , a2 , . . ., a10 are nonnegative integers such that
10
X 10
X
ak = 15 and kak = 80.
k=1 k=1
P10
Let M and m denote the maximum and minimum respectively of k=1 k 2 ak . Compute M − m.
√
9. Define a sequence {an }∞
n=1 via a1 = 1 and an+1 = an + b an c for all n ≥ 1. What is the smallest N such
that aN > 2017?
10. Let c denote the largest possible real number such that there exists a nonconstant polynomial P with
P (z 2 ) = P (z − c)P (z + c)
for all z. Compute the sum of all values of P ( 13 ) over all nonconstant polynomials P satisfying the above
constraint for this c.

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