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CMIMC Problems
2017 CMIMC
2017 CMIMC Algebra
2017 CMIMC Algebra
Part of
2017 CMIMC
Subcontests
(10)
10
1
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2017 A10: A Surprising Polynomial FE
Let
c
c
c
denote the largest possible real number such that there exists a nonconstant polynomial
P
P
P
with
P
(
z
2
)
=
P
(
z
−
c
)
P
(
z
+
c
)
P(z^2)=P(z-c)P(z+c)
P
(
z
2
)
=
P
(
z
−
c
)
P
(
z
+
c
)
for all
z
z
z
. Compute the sum of all values of
P
(
1
3
)
P(\tfrac13)
P
(
3
1
)
over all nonconstant polynomials
P
P
P
satisfying the above constraint for this
c
c
c
.
9
1
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2017 A9: Sequence of Floors and Square Roots
Define a sequence
{
a
n
}
n
=
1
∞
\{a_{n}\}_{n=1}^{\infty}
{
a
n
}
n
=
1
∞
via
a
1
=
1
a_{1} = 1
a
1
=
1
and
a
n
+
1
=
a
n
+
⌊
a
n
⌋
a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor
a
n
+
1
=
a
n
+
⌊
a
n
⌋
for all
n
≥
1
n \geq 1
n
≥
1
. What is the smallest
N
N
N
such that
a
N
>
2017
a_{N} > 2017
a
N
>
2017
?
8
1
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2017 A8: Integer Linear Programming
Suppose
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots
…
,
a
10
a_{10}
a
10
are nonnegative integers such that
∑
k
=
1
10
a
k
=
15
and
∑
k
=
1
10
k
a
k
=
80.
\sum_{k=1}^{10}a_k=15\qquad\text{and}\qquad \sum_{k=1}^{10}ka_k = 80.
k
=
1
∑
10
a
k
=
15
and
k
=
1
∑
10
k
a
k
=
80.
Let
M
M
M
and
m
m
m
denote the maximum and minimum respectively of
∑
k
=
1
10
k
2
a
k
\sum_{k=1}^{10}k^2a_k
∑
k
=
1
10
k
2
a
k
. Compute
M
−
m
M-m
M
−
m
.
7
1
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2017 A7: Complicated System of Equations
Let
a
a
a
,
b
b
b
, and
c
c
c
be complex numbers satisfying the system of equations \begin{align*}\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}&=9,\\\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}&=32,\\\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}&=122.\end{align*} Find
a
b
c
abc
ab
c
.
6
1
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2017 A6: Maximization of Quintic
Suppose
P
P
P
is a quintic polynomial with real coefficients with
P
(
0
)
=
2
P(0)=2
P
(
0
)
=
2
and
P
(
1
)
=
3
P(1)=3
P
(
1
)
=
3
such that
∣
z
∣
=
1
|z|=1
∣
z
∣
=
1
whenever
z
z
z
is a complex number satisfying
P
(
z
)
=
0
P(z) = 0
P
(
z
)
=
0
. What is the smallest possible value of
P
(
2
)
P(2)
P
(
2
)
over all such polynomials
P
P
P
?
5
1
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2017 A5: Solution Set to Floor Equation
The set
S
S
S
of positive real numbers
x
x
x
such that
⌊
2
x
5
⌋
+
⌊
3
x
5
⌋
+
1
=
⌊
x
⌋
\left\lfloor\frac{2x}{5}\right\rfloor + \left\lfloor\frac{3x}{5}\right\rfloor + 1 = \left\lfloor x\right\rfloor
⌊
5
2
x
⌋
+
⌊
5
3
x
⌋
+
1
=
⌊
x
⌋
can be written as
S
=
⋃
j
=
1
∞
I
j
S = \bigcup_{j = 1}^{\infty} I_{j}
S
=
⋃
j
=
1
∞
I
j
, where the
I
i
I_{i}
I
i
are disjoint intervals of the form
[
a
i
,
b
i
)
=
{
x
∣
a
i
≤
x
<
b
i
}
[a_{i}, b_{i}) = \{x \, | \, a_i \leq x < b_i\}
[
a
i
,
b
i
)
=
{
x
∣
a
i
≤
x
<
b
i
}
and
b
i
≤
a
i
+
1
b_{i} \leq a_{i+1}
b
i
≤
a
i
+
1
for all
i
≥
1
i \geq 1
i
≥
1
. Find
∑
i
=
1
2017
(
b
i
−
a
i
)
\sum_{i=1}^{2017} (b_{i} - a_{i})
∑
i
=
1
2017
(
b
i
−
a
i
)
.
4
1
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2017 A4: Sum in Terms of e
It is well known that the mathematical constant
e
e
e
can be written in the form
e
=
1
0
!
+
1
1
!
+
1
2
!
+
⋯
e = \tfrac{1}{0!}+\tfrac{1}{1!}+\tfrac{1}{2!}+\cdots
e
=
0
!
1
+
1
!
1
+
2
!
1
+
⋯
. With this in mind, determine the value of
∑
j
=
3
∞
j
⌊
j
2
⌋
!
.
\sum_{j=3}^\infty\dfrac{j}{\lfloor\frac j2\rfloor!}.
j
=
3
∑
∞
⌊
2
j
⌋!
j
.
Express your answer in terms of
e
e
e
.
3
1
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2017 A3: Quadratic Polynomial Equation
Suppose
P
(
x
)
P(x)
P
(
x
)
is a quadratic polynomial with integer coefficients satisfying the identity
P
(
P
(
x
)
)
−
P
(
x
)
2
=
x
2
+
x
+
2016
P(P(x)) - P(x)^2 = x^2+x+2016
P
(
P
(
x
))
−
P
(
x
)
2
=
x
2
+
x
+
2016
for all real
x
x
x
. What is
P
(
1
)
P(1)
P
(
1
)
?
2
1
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2017 A2: Iterated Function
For nonzero real numbers
x
x
x
and
y
y
y
, define
x
∘
y
=
x
y
x
+
y
x\circ y = \tfrac{xy}{x+y}
x
∘
y
=
x
+
y
x
y
. Compute
2
1
∘
(
2
2
∘
(
2
3
∘
⋯
∘
(
2
2016
∘
2
2017
)
)
)
.
2^1\circ \left(2^2\circ \left(2^3\circ\cdots\circ\left(2^{2016}\circ 2^{2017}\right)\right)\right).
2
1
∘
(
2
2
∘
(
2
3
∘
⋯
∘
(
2
2016
∘
2
2017
)
)
)
.
1
1
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2017 A1: Animal Migration
The residents of the local zoo are either rabbits or foxes. The ratio of foxes to rabbits in the zoo is
2
:
3
2:3
2
:
3
. After
10
10
10
of the foxes move out of town and half the rabbits move to Rabbitretreat, the ratio of foxes to rabbits is
13
:
10
13:10
13
:
10
. How many animals are left in the zoo?