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Undergraduate contests
IMC
2008 IMC
2008 IMC
Part of
IMC
Subcontests
(6)
6
2
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IMC 2008 Day 1 P6 - Permutation Distances
For a permutation
σ
∈
S
n
\sigma\in S_n
σ
∈
S
n
with
(
1
,
2
,
…
,
n
)
↦
(
i
1
,
i
2
,
…
,
i
n
)
(1,2,\dots,n)\mapsto(i_1,i_2,\dots,i_n)
(
1
,
2
,
…
,
n
)
↦
(
i
1
,
i
2
,
…
,
i
n
)
, define D(\sigma) \equal{} \sum_{k \equal{} 1}^n |i_k \minus{} k| Let Q(n,d) \equal{} \left|\left\{\sigma\in S_n : D(\sigma) \equal{} d\right\}\right| Show that when
d
≥
2
n
d \geq 2n
d
≥
2
n
,
Q
(
n
,
d
)
Q(n,d)
Q
(
n
,
d
)
is an even number.
IMC 2008 Day 2 P6 - Hilbert space
Let
H
\mathcal{H}
H
be an infinite-dimensional Hilbert space, let
d
>
0
d>0
d
>
0
, and suppose that
S
S
S
is a set of points (not necessarily countable) in
H
\mathcal{H}
H
such that the distance between any two distinct points in
S
S
S
is equal to
d
d
d
. Show that there is a point
y
∈
H
y\in\mathcal{H}
y
∈
H
such that \left\{\frac{\sqrt{2}}{d}(x\minus{}y): \ x\in S\right\} is an orthonormal system of vectors in
H
\mathcal{H}
H
.
5
2
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IMC 2008 Day 1 P5 - Automorphisms of Subgroups
Does there exist a finite group
G
G
G
with a normal subgroup
H
H
H
such that
∣
Aut
H
∣
>
∣
Aut
G
∣
|\text{Aut } H| > |\text{Aut } G|
∣
Aut
H
∣
>
∣
Aut
G
∣
? Disprove or provide an example. Here the notation
∣
Aut
X
∣
|\text{Aut } X|
∣
Aut
X
∣
for some group
X
X
X
denotes the number of isomorphisms from
X
X
X
to itself.
IMC 2008 Day 2 P5 - Determinant
Let
n
n
n
be a positive integer, and consider the matrix A \equal{} (a_{ij})_{1\leq i,j\leq n} where a_{ij} \equal{} 1 if i\plus{}j is prime and a_{ij} \equal{} 0 otherwise. Prove that |\det A| \equal{} k^2 for some integer
k
k
k
.
4
2
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IMC 2008 Day 1 P4 - Better Triples
We say a triple of real numbers
(
a
1
,
a
2
,
a
3
)
(a_1,a_2,a_3)
(
a
1
,
a
2
,
a
3
)
is better than another triple
(
b
1
,
b
2
,
b
3
)
(b_1,b_2,b_3)
(
b
1
,
b
2
,
b
3
)
when exactly two out of the three following inequalities hold:
a
1
>
b
1
a_1 > b_1
a
1
>
b
1
,
a
2
>
b
2
a_2 > b_2
a
2
>
b
2
,
a
3
>
b
3
a_3 > b_3
a
3
>
b
3
. We call a triple of real numbers special when they are nonnegative and their sum is
1
1
1
. For which natural numbers
n
n
n
does there exist a collection
S
S
S
of special triples, with |S| \equal{} n, such that any special triple is bettered by at least one element of
S
S
S
?
IMC 2008 Day 2 P4 - Polynomial with degree > 5
Let
Z
[
x
]
\mathbb{Z}[x]
Z
[
x
]
be the ring of polynomials with integer coefficients, and let
f
(
x
)
,
g
(
x
)
∈
Z
[
x
]
f(x), g(x) \in\mathbb{Z}[x]
f
(
x
)
,
g
(
x
)
∈
Z
[
x
]
be nonconstant polynomials such that
g
(
x
)
g(x)
g
(
x
)
divides
f
(
x
)
f(x)
f
(
x
)
in
Z
[
x
]
\mathbb{Z}[x]
Z
[
x
]
. Prove that if the polynomial f(x)\minus{}2008 has at least 81 distinct integer roots, then the degree of
g
(
x
)
g(x)
g
(
x
)
is greater than 5.
3
2
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IMC 2008 Day 1 P3
Let
p
p
p
be a polynomial with integer coefficients and let
a
1
<
a
2
<
⋯
<
a
k
a_1<a_2<\cdots <a_k
a
1
<
a
2
<
⋯
<
a
k
be integers. Given that
p
(
a
i
)
≠
0
∀
i
=
1
,
2
,
⋯
,
k
p(a_i)\ne 0\forall\; i=1,2,\cdots, k
p
(
a
i
)
=
0∀
i
=
1
,
2
,
⋯
,
k
.(a) Prove
∃
a
∈
Z
\exists\; a\in \mathbb{Z}
∃
a
∈
Z
such that
p
(
a
i
)
∣
p
(
a
)
∀
i
=
1
,
2
,
…
,
k
p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k
p
(
a
i
)
∣
p
(
a
)
∀
i
=
1
,
2
,
…
,
k
(b) Does there exist
a
∈
Z
a\in \mathbb{Z}
a
∈
Z
such that
∏
i
=
1
k
p
(
a
i
)
∣
p
(
a
)
\prod_{i=1}^{k}p(a_i)\mid p(a)
i
=
1
∏
k
p
(
a
i
)
∣
p
(
a
)
IMC 2008 Day 2 P3 - Power of 2 divides binomial sum
Let
n
n
n
be a positive integer. Prove that 2^{n\minus{}1} divides \sum_{0\leq k < n/2} \binom{n}{2k\plus{}1}5^k.
2
2
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IMC 2008 Day 1 P2
Denote by
V
\mathbb{V}
V
the real vector space of all real polynomials in one variable, and let
γ
:
V
→
R
\gamma :\mathbb{V}\to \mathbb{R}
γ
:
V
→
R
be a linear map. Suppose that for all
f
,
g
∈
V
f,g\in \mathbb{V}
f
,
g
∈
V
with
γ
(
f
g
)
=
0
\gamma(fg)=0
γ
(
f
g
)
=
0
we have
γ
(
f
)
=
0
\gamma(f)=0
γ
(
f
)
=
0
or
γ
(
g
)
=
0
\gamma(g)=0
γ
(
g
)
=
0
. Prove that there exist
c
,
x
0
∈
R
c,x_0\in \mathbb{R}
c
,
x
0
∈
R
such that \gamma(f)=cf(x_0) \forall f\in \mathbb{V}
IMC 2008 Day 2 P2 - Ellipses with common focus
Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.
1
2
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IMC 2008 Day 1 P1
Find all continuous functions
f
:
R
→
R
f: \mathbb{R}\to \mathbb{R}
f
:
R
→
R
such that f(x)-f(y)\in \mathbb{Q} \text{ for all } x-y\in\mathbb{Q}
IMC 2008 Day 2 P1 - Polynomial divisor
Let
n
,
k
n, k
n
,
k
be positive integers and suppose that the polynomial x^{2k}\minus{}x^k\plus{}1 divides x^{2n}\plus{}x^n\plus{}1. Prove that x^{2k}\plus{}x^k\plus{}1 divides x^{2n}\plus{}x^n\plus{}1.