MathDB
Problems
Contests
Undergraduate contests
IberoAmerican Olympiad For University Students
2006 IberoAmerican Olympiad For University Students
2006 IberoAmerican Olympiad For University Students
Part of
IberoAmerican Olympiad For University Students
Subcontests
(7)
7
1
Hide problems
Homomorphisms s.t. f(f(x))=f(x) - OIMU 2006 Problem 7
Consider the multiplicative group
A
=
{
z
∈
C
∣
z
200
6
k
=
1
,
0
<
k
∈
Z
}
A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}
A
=
{
z
∈
C
∣
z
200
6
k
=
1
,
0
<
k
∈
Z
}
of all the roots of unity of degree
200
6
k
2006^k
200
6
k
for all positive integers
k
k
k
.Find the number of homomorphisms
f
:
A
→
A
f:A\to A
f
:
A
→
A
that satisfy
f
(
f
(
x
)
)
=
f
(
x
)
f(f(x))=f(x)
f
(
f
(
x
))
=
f
(
x
)
for all elements
x
∈
A
x\in A
x
∈
A
.
6
1
Hide problems
Convolution identity - OIMU 2006 Problem 6
Let
x
0
(
t
)
=
1
x_0(t)=1
x
0
(
t
)
=
1
,
x
k
+
1
(
t
)
=
(
1
+
t
k
+
1
)
x
k
(
t
)
x_{k+1}(t)=(1+t^{k+1})x_k(t)
x
k
+
1
(
t
)
=
(
1
+
t
k
+
1
)
x
k
(
t
)
for all
k
≥
0
k\geq 0
k
≥
0
;
y
n
,
0
(
t
)
=
1
y_{n,0}(t)=1
y
n
,
0
(
t
)
=
1
,
y
n
,
k
(
t
)
=
t
n
−
k
+
1
−
1
t
k
−
1
y
n
,
k
−
1
(
t
)
y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t)
y
n
,
k
(
t
)
=
t
k
−
1
t
n
−
k
+
1
−
1
y
n
,
k
−
1
(
t
)
for all
n
≥
0
n\geq 0
n
≥
0
,
1
≤
k
≤
n
1\leq k \leq n
1
≤
k
≤
n
.Prove that
∑
j
=
0
n
−
1
(
−
1
)
j
x
n
−
j
−
1
(
t
)
y
n
,
j
(
t
)
=
1
−
(
−
1
)
n
2
\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2}
∑
j
=
0
n
−
1
(
−
1
)
j
x
n
−
j
−
1
(
t
)
y
n
,
j
(
t
)
=
2
1
−
(
−
1
)
n
for all
n
≥
1
n\geq 1
n
≥
1
.
5
1
Hide problems
Regular n-gon, \prod(5-ai^2)=F_n^2 - OIMU 2006 Problem 5
A regular
n
n
n
-gon is inscribed in a circle of radius
1
1
1
. Let
a
1
,
⋯
,
a
n
−
1
a_1,\cdots,a_{n-1}
a
1
,
⋯
,
a
n
−
1
be the distances of one of the vertices of the polygon to all the other vertices. Prove that
(
5
−
a
1
2
)
⋯
(
5
−
a
n
−
1
2
)
=
F
n
2
(5-a_1^2)\cdots(5-a_{n-1}^2)=F_n^2
(
5
−
a
1
2
)
⋯
(
5
−
a
n
−
1
2
)
=
F
n
2
where
F
n
F_n
F
n
is the
n
t
h
n^{th}
n
t
h
term of the Fibonacci sequence
1
,
1
,
2
,
⋯
1,1,2,\cdots
1
,
1
,
2
,
⋯
4
1
Hide problems
Alternating integral sum is zero - OIMU 2006 Problem 4
Prove that for any interval
[
a
,
b
]
[a,b]
[
a
,
b
]
of real numbers and any positive integer
n
n
n
there exists a positive integer
k
k
k
and a partition of the given interval
a
=
x
(
0
)
<
x
(
1
)
<
x
(
2
)
<
⋯
<
x
(
k
−
1
)
<
x
(
k
)
=
b
a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b
a
=
x
(
0
)
<
x
(
1
)
<
x
(
2
)
<
⋯
<
x
(
k
−
1
)
<
x
(
k
)
=
b
such that
∫
x
(
0
)
x
(
1
)
f
(
x
)
d
x
+
∫
x
(
2
)
x
(
3
)
f
(
x
)
d
x
+
⋯
=
∫
x
(
1
)
x
(
2
)
f
(
x
)
d
x
+
∫
x
(
3
)
x
(
4
)
f
(
x
)
d
x
+
⋯
\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots
∫
x
(
0
)
x
(
1
)
f
(
x
)
d
x
+
∫
x
(
2
)
x
(
3
)
f
(
x
)
d
x
+
⋯
=
∫
x
(
1
)
x
(
2
)
f
(
x
)
d
x
+
∫
x
(
3
)
x
(
4
)
f
(
x
)
d
x
+
⋯
for all polynomials
f
f
f
with real coefficients and degree less than
n
n
n
.
3
1
Hide problems
Polynomial recursive sequence - OIMU 2006 Problem 3
Let
p
1
(
x
)
=
p
(
x
)
=
4
x
3
−
3
x
p_1(x)=p(x)=4x^3-3x
p
1
(
x
)
=
p
(
x
)
=
4
x
3
−
3
x
and
p
n
+
1
(
x
)
=
p
(
p
n
(
x
)
)
p_{n+1}(x)=p(p_n(x))
p
n
+
1
(
x
)
=
p
(
p
n
(
x
))
for each positive integer
n
n
n
. Also, let
A
(
n
)
A(n)
A
(
n
)
be the set of all the real roots of the equation
p
n
(
x
)
=
x
p_n(x)=x
p
n
(
x
)
=
x
.Prove that
A
(
n
)
⊆
A
(
2
n
)
A(n)\subseteq A(2n)
A
(
n
)
⊆
A
(
2
n
)
and that the product of the elements of
A
(
n
)
A(n)
A
(
n
)
is the average of the elements of
A
(
2
n
)
A(2n)
A
(
2
n
)
.
2
1
Hide problems
Trigonometric equation always has root - OIMU 2006 Problem 2
Prove that for any positive integer
n
n
n
and any real numbers
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
we have that the equation
a
1
sin
(
x
)
+
a
2
sin
(
2
x
)
+
⋯
+
a
n
sin
(
n
x
)
=
b
1
cos
(
x
)
+
b
2
cos
(
2
x
)
+
⋯
+
b
n
cos
(
n
x
)
a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)
a
1
sin
(
x
)
+
a
2
sin
(
2
x
)
+
⋯
+
a
n
sin
(
n
x
)
=
b
1
cos
(
x
)
+
b
2
cos
(
2
x
)
+
⋯
+
b
n
cos
(
n
x
)
has at least one real root.
1
1
Hide problems
Distance between sets of fractions - OIMU 2006 Problem 1
Let
m
,
n
m,n
m
,
n
be positive integers greater than
1
1
1
. We define the sets
P
m
=
{
1
m
,
2
m
,
⋯
,
m
−
1
m
}
P_m=\left\{\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}{m}\right\}
P
m
=
{
m
1
,
m
2
,
⋯
,
m
m
−
1
}
and
P
n
=
{
1
n
,
2
n
,
⋯
,
n
−
1
n
}
P_n=\left\{\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}{n}\right\}
P
n
=
{
n
1
,
n
2
,
⋯
,
n
n
−
1
}
.Find the distance between
P
m
P_m
P
m
and
P
n
P_n
P
n
, that is defined as
min
{
∣
a
−
b
∣
:
a
∈
P
m
,
b
∈
P
n
}
\min\{|a-b|:a\in P_m,b\in P_n\}
min
{
∣
a
−
b
∣
:
a
∈
P
m
,
b
∈
P
n
}