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Problems
Contests
National and Regional Contests
Vietnam Contests
VMEO = Vietnam Mathematical E-Olympiad
VMEO III 2006 Shortlist
VMEO III 2006 Shortlist
Part of
VMEO = Vietnam Mathematical E-Olympiad
Subcontests
(28)
N7
1
Hide problems
a-2006=\sum _{i=1}^{2006} 2^ia_i
Prove that there are only finitely positive integer
a
a
a
such that
a
−
2006
=
∑
i
=
1
2006
2
i
a
i
a-2006=\sum\limits_{i=1}^{2006} 2^ia_i
a
−
2006
=
i
=
1
∑
2006
2
i
a
i
with
{
a
i
}
\{a_i\}
{
a
i
}
as divisors (not necessary distinct) of
n
n
n
.
N2
1
Hide problems
n >= 2^{k-1} for k terms in geometric sequence of an arithmetic sequence
Let
a
1
,
a
2
,
.
.
.
a_1,a_2,...
a
1
,
a
2
,
...
be an arithmetic sequence with the common difference between terms is positive. Assume there are
k
k
k
terms of this sequence creates an geometric sequence with common ratio
d
d
d
. Prove that
n
≥
2
k
−
1
n\ge 2^{k-1}
n
≥
2
k
−
1
.
N3
1
Hide problems
4p^2-8p+1|x_i - (2p)^k odd prime p , x_{n+2}= 4x_{n+1}-x_n
Given odd prime
p
p
p
. Sequence
x
n
{x_n}
x
n
is defined by
x
n
+
2
=
4
x
n
+
1
−
x
n
x_{n+2}= 4x_{n+1}-x_n
x
n
+
2
=
4
x
n
+
1
−
x
n
. Choose
x
0
,
x
1
x_0,x_1
x
0
,
x
1
such that for every random positive integer
k
k
k
, there exists
i
∈
N
i\in \mathbb N
i
∈
N
such that
4
p
2
−
8
p
+
1
∣
x
i
−
(
2
p
)
k
4p^2-8p+1|x_i - (2p)^k
4
p
2
−
8
p
+
1∣
x
i
−
(
2
p
)
k
.
A10
1
Hide problems
0,a_1a_2... rational ? if a_{n+1}=[3a_n/2], a_1=2
Let
a
n
{a_n}
a
n
be a sequence defined by
a
1
=
2
a_1=2
a
1
=
2
,
a
n
+
1
=
[
3
a
n
2
]
a_{n+1}=\left[ \frac {3a_n}{2}\right]
a
n
+
1
=
[
2
3
a
n
]
∀
n
∈
N
\forall n \in \mathbb N
∀
n
∈
N
0.
a
1
a
2
.
.
.
0.a_1a_2...
0.
a
1
a
2
...
rational or irrational?
A9
1
Hide problems
P(...(P(x))...) has all roots from 1,2,..., mb
Is there any polynomial
P
(
x
)
P(x)
P
(
x
)
with degree
n
n
n
such that
P
(
.
.
.
(
P
(
x
)
)
.
.
.
)
⏟
m
t
i
m
e
s
P
\underbrace{P(...(P(x))...)}_{m\,\, times \,\, P}
m
t
im
es
P
P
(
...
(
P
(
x
))
...
)
has all roots from
1
,
2
,
.
.
.
,
m
n
1,2,..., mn
1
,
2
,
...
,
mn
?
A8
1
Hide problems
x_{n+1}=f(x_n) where f(x) = 3(|x|+|x-1|-|x+1|)
Let
f
(
x
)
=
3
(
∣
x
∣
+
∣
x
−
1
∣
−
∣
x
+
1
∣
)
f(x) = 3(|x|+|x-1|-|x+1|)
f
(
x
)
=
3
(
∣
x
∣
+
∣
x
−
1∣
−
∣
x
+
1∣
)
and let
x
n
+
1
=
f
(
x
n
)
x_{n+1}=f(x_n)
x
n
+
1
=
f
(
x
n
)
∀
n
≥
0
\forall n \ge 0
∀
n
≥
0
. How many real number
x
0
x_0
x
0
are there, that satisfy
x
0
=
x
2007
x_0=x_{2007}
x
0
=
x
2007
and
x
0
,
x
1
,
x
2
,
.
.
.
,
x
2006
x_0,x_1,x_2,...,x_{2006}
x
0
,
x
1
,
x
2
,
...
,
x
2006
are distinct?
G5
1
Hide problems
family of ratonal circles exists
Prove that there exists a family of rational circles with a distinct radius
{
(
O
n
)
}
\{(O_n)\}
{(
O
n
)}
(
n
=
1
,
2
,
3
,
.
.
.
)
(n = 1,2,3,...)
(
n
=
1
,
2
,
3
,
...
)
satisfying the property that for all natural indices
n
n
n
, circles
(
O
n
)
(O_n)
(
O
n
)
,
(
O
n
+
1
)
( O_{n+1})
(
O
n
+
1
)
,
(
O
n
+
2
)
(O_{n+2})
(
O
n
+
2
)
,
(
O
n
+
3
)
(O_{n+3})
(
O
n
+
3
)
are externally tangent like in the figure.https://cdn.artofproblemsolving.com/attachments/b/f/5655e677e7c4f203b63afe82c50088e9ef97f5.png
G4
1
Hide problems
OI passes through isogonal of I wrt triangle A'B'C', angle bisectors
Let
A
B
C
ABC
A
BC
be a triangle with circumscribed and inscribed circles
(
O
)
(O)
(
O
)
and
(
I
)
(I)
(
I
)
respectively.
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
are the bisectors of triangle
A
B
C
ABC
A
BC
. Prove that
O
I
OI
O
I
passes through the the isogonal conjugate of point
I
I
I
with respect to triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
.
G3
1
Hide problems
AB x BC + BC x CA + CA x AB>=OA^2 + OB^2 + OC^2 for tetrahedron, angles 60^o
The tetrahedron
O
A
B
C
OABC
O
A
BC
has all angles at vertex
O
O
O
equal to
6
0
o
60^o
6
0
o
. Prove that
A
B
⋅
B
C
+
B
C
⋅
C
A
+
C
A
⋅
A
B
≥
O
A
2
+
O
B
2
+
O
C
2
AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2
A
B
⋅
BC
+
BC
⋅
C
A
+
C
A
⋅
A
B
≥
O
A
2
+
O
B
2
+
O
C
2
N14
1
Hide problems
min integer, related to sum of digits
For any natural number
n
=
a
i
.
.
.
a
2
a
1
‾
n = \overline{a_i...a_2a_1}
n
=
a
i
...
a
2
a
1
, consider the number
T
(
n
)
=
10
∑
i
e
v
e
n
a
i
+
∑
i
o
d
d
a
i
.
T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i.
T
(
n
)
=
10
i
e
v
e
n
∑
a
i
+
i
o
dd
∑
a
i
.
Let's find the smallest positive integer
A
A
A
such that is sum of the natural numbers
n
1
,
n
2
,
.
.
.
,
n
148
n_1,n_2,...,n_{148}
n
1
,
n
2
,
...
,
n
148
as well as of
m
1
,
m
2
,
.
.
.
,
m
149
m_1,m_2,...,m_{149}
m
1
,
m
2
,
...
,
m
149
and matches the pattern:
A
=
n
1
+
n
2
+
.
.
.
+
n
148
=
m
1
+
m
2
+
.
.
.
+
m
149
A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}
A
=
n
1
+
n
2
+
...
+
n
148
=
m
1
+
m
2
+
...
+
m
149
T
(
n
1
)
=
T
(
n
2
)
=
.
.
.
=
T
(
n
148
)
T(n_1)=T(n_2)=...=T(n_{148})
T
(
n
1
)
=
T
(
n
2
)
=
...
=
T
(
n
148
)
T
(
m
1
)
=
T
(
m
2
)
=
.
.
.
=
T
(
m
148
)
T(m_1)=T(m_2)=...=T(m_{148})
T
(
m
1
)
=
T
(
m
2
)
=
...
=
T
(
m
148
)
N13
1
Hide problems
\phi (a) / a+ \phi (b)/ b <1 , euler phi function inequalities
Prove the following two inequalities: 1) If
n
>
49
n > 49
n
>
49
, then exist positive integers
a
,
b
>
1
a, b > 1
a
,
b
>
1
such that
a
+
b
=
n
a+b=n
a
+
b
=
n
and
ϕ
(
a
)
a
+
ϕ
(
b
)
b
<
1
\frac{\phi (a)}{a}+\frac{\phi (b)}{b}<1
a
ϕ
(
a
)
+
b
ϕ
(
b
)
<
1
2) If
n
>
4
n > 4
n
>
4
, then exist integer integers
a
,
b
>
1
a, b > 1
a
,
b
>
1
such that
a
+
b
=
n
a+b=n
a
+
b
=
n
and
ϕ
(
a
)
a
+
ϕ
(
b
)
b
>
1
\frac{\phi (a)}{a}+\frac{\phi (b)}{b}>1
a
ϕ
(
a
)
+
b
ϕ
(
b
)
>
1
N12
1
Hide problems
min integer in form (n^a-n^b)/ (n^c-n^d)
Given a positive integer
n
>
1
n > 1
n
>
1
. Find the smallest integer of the form
n
a
−
n
b
n
c
−
n
d
\frac{n^a-n^b}{n^c-n^d}
n
c
−
n
d
n
a
−
n
b
for all positive integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
.
N9
1
Hide problems
\sum_{1\le k \le m ,\,\, (k,m)=1} 1/k >= C \sum_{k=1}^{m} 1/k
Assume the
m
m
m
is a given integer greater than
1
1
1
. Find the largest number
C
C
C
such that for all
n
∈
N
n \in N
n
∈
N
we have
∑
1
≤
k
≤
m
,
(
k
,
m
)
=
1
1
k
≥
C
∑
k
=
1
m
1
k
\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}
1
≤
k
≤
m
,
(
k
,
m
)
=
1
∑
k
1
≥
C
k
=
1
∑
m
k
1
N4
1
Hide problems
f(n) is next positive integer whose all digits are divisible by n
Given the positive integer
n
n
n
, find the integer
f
(
n
)
f(n)
f
(
n
)
so that
f
(
n
)
f(n)
f
(
n
)
is the next positive integer that is always a number whose all digits are divisible by
n
n
n
.
N10
1
Hide problems
\phi (n) >= \pi (n) / 2
The notation
ϕ
(
n
)
\phi (n)
ϕ
(
n
)
is the number of positive integers smaller than
n
n
n
and coprime with
n
n
n
,
π
(
n
)
\pi (n)
π
(
n
)
is the number of primes that do not exceed
n
n
n
. Prove that for any natural number
n
>
1
n > 1
n
>
1
, we have
ϕ
(
n
)
≥
π
(
n
)
2
\phi (n) \ge \frac{\pi (n)}{2}
ϕ
(
n
)
≥
2
π
(
n
)
N11
1
Hide problems
composition of sets of form 2^{2^n}+1 and 6^{2^n}+1 is finite
Prove that the composition of the sets of one of the following two forms is finite: (a)
2
2
n
+
1
2^{2^n}+1
2
2
n
+
1
(b)
6
2
n
+
1
6^{2^n}+1
6
2
n
+
1
N1
1
Hide problems
problem related to largest k such that 2^k|n
f
(
n
)
f(n)
f
(
n
)
denotes the largest integer
k
k
k
such that that
2
k
∣
n
2^k|n
2
k
∣
n
.
2006
2006
2006
integers
a
i
a_i
a
i
are such that
a
1
<
a
2
<
.
.
.
<
a
2016
a_1<a_2<...<a_{2016}
a
1
<
a
2
<
...
<
a
2016
. Is it possible to find integers
k
k
k
where
1
≤
k
≤
2006
1 \le k\le 2006
1
≤
k
≤
2006
and
f
(
a
i
−
a
j
)
≠
k
f(a_i-a_j)\ne k
f
(
a
i
−
a
j
)
=
k
for every
1
≤
j
≤
i
≤
2006
1 \le j \le i \le 2006
1
≤
j
≤
i
≤
2006
?
N6
1
Hide problems
a+1|b^2+c^2 , b+1|c^2+a^2 , c+1|a^2+b^2
Find all sets of natural numbers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that
a
+
1
∣
b
2
+
c
2
,
b
+
1
∣
c
2
+
a
2
,
c
+
1
∣
a
2
+
b
2
.
a+1|b^2+c^2\,\, , b+1|c^2+a^2\,\,, c+1|a^2+b^2.
a
+
1∣
b
2
+
c
2
,
b
+
1∣
c
2
+
a
2
,
c
+
1∣
a
2
+
b
2
.
A6
1
Hide problems
f(x^2+f(y))=y^2+f(x) for all x,y \in N_m
The symbol
N
m
N_m
N
m
denotes the set of all integers not less than the given integer
m
m
m
. Find all functions
f
:
N
m
→
N
m
f: N_m \to N_m
f
:
N
m
→
N
m
such that
f
(
x
2
+
f
(
y
)
)
=
y
2
+
f
(
x
)
f(x^2+f(y))=y^2+f(x)
f
(
x
2
+
f
(
y
))
=
y
2
+
f
(
x
)
for all
x
,
y
∈
N
m
x,y \in N_m
x
,
y
∈
N
m
.
A5
1
Hide problems
1/3 (f(a+b-c)+f(b+c-a)+f(c+a-b))=f(\sqrt{(ab+bc+ca)/3) for a,b,c sidelengths
Find all continuous functions
f
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)
f : (0,+\infty) \to (0,+\infty)
f
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)
such that if
a
,
b
,
c
a, b, c
a
,
b
,
c
are the lengths of the sides of any triangle then it is satisfied that
f
(
a
+
b
−
c
)
+
f
(
b
+
c
−
a
)
+
f
(
c
+
a
−
b
)
3
=
f
(
a
b
+
b
c
+
c
a
3
)
\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)
3
f
(
a
+
b
−
c
)
+
f
(
b
+
c
−
a
)
+
f
(
c
+
a
−
b
)
=
f
(
3
ab
+
b
c
+
c
a
)
A3
1
Hide problems
(x+y+z)/(xy+yz+zx) <= 1+5/247} ( (x-y)^2+(y-z)^2+(z-x)^2 )
For positive real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
that satisfy
x
y
+
y
z
+
z
x
+
x
y
z
=
4
xy + yz + zx + xyz=4
x
y
+
yz
+
z
x
+
x
yz
=
4
, prove that
x
+
y
+
z
x
y
+
y
z
+
z
x
≤
1
+
5
247
⋅
(
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
)
\frac{x+y+z}{xy+yz+zx}\le 1+\frac{5}{247}\cdot \left( (x-y)^2+(y-z)^2+(z-x)^2\right)
x
y
+
yz
+
z
x
x
+
y
+
z
≤
1
+
247
5
⋅
(
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
)
A2
1
Hide problems
P(a^2+a+1) if P(x)=x^4+3x^2-9x+1
Given a polynomial
P
(
x
)
=
x
4
+
3
x
2
−
9
x
+
1
P(x)=x^4+3x^2-9x+1
P
(
x
)
=
x
4
+
3
x
2
−
9
x
+
1
. Calculate
P
(
α
2
+
α
+
1
)
P(\alpha^2+\alpha+1)
P
(
α
2
+
α
+
1
)
where
α
=
1
+
5
2
3
+
1
−
5
2
3
\alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}}
α
=
3
2
1
+
5
+
3
2
1
−
5
A1
1
Hide problems
f( x^2 + f(y) - y ) = (f(x))^2 - f(y)
Find all functions
f
:
R
→
R
f:R \to R
f
:
R
→
R
such that
f
(
x
2
+
f
(
y
)
−
y
)
=
(
f
(
x
)
)
2
−
f
(
y
)
f(x^2+f(y)-y) =(f(x))^2-f(y)
f
(
x
2
+
f
(
y
)
−
y
)
=
(
f
(
x
)
)
2
−
f
(
y
)
for all
x
,
y
∈
R
x,y \in R
x
,
y
∈
R
N8
1
Hide problems
exists m with (a_n, N) = 1 for all n = m, m + 1, ...,m + M
For every positive integer
n
n
n
, the symbol
a
n
/
b
n
a_n/b_n
a
n
/
b
n
is the simplest form of the fraction
1
+
1
/
2
+
.
.
.
+
1
/
n
1+1/2+...+1/n
1
+
1/2
+
...
+
1/
n
. Prove that for every pair of positive integers
(
M
,
N
)
(M, N)
(
M
,
N
)
we can always find a positive integer
m
m
m
where
(
a
n
,
N
)
=
1
(a_n, N) = 1
(
a
n
,
N
)
=
1
for all
n
=
m
,
m
+
1
,
.
.
.
,
m
+
M
n = m, m + 1, ...,m + M
n
=
m
,
m
+
1
,
...
,
m
+
M
.
G1
1
Hide problems
AC _|_ OQ wanted, 2 circles, tangent related
Given a circle
(
O
)
(O)
(
O
)
and a point
P
P
P
outside that circle.
M
M
M
is a point running on the circle
(
O
)
(O)
(
O
)
. The circle with center
I
I
I
and diameter
P
M
PM
PM
intersects circle
(
O
)
(O)
(
O
)
again at
N
N
N
. The tangent of
(
I
)
(I)
(
I
)
at
P
P
P
intersects
M
N
MN
MN
at
Q
Q
Q
. The line through
Q
Q
Q
perpendicular to
P
O
PO
PO
intersects
P
M
PM
PM
at
A
A
A
.
A
N
AN
A
N
intersects
(
O
)
(O)
(
O
)
further at
B
B
B
.
B
M
BM
BM
intersects
P
O
PO
PO
at
C
C
C
. Prove that
A
C
AC
A
C
is perpendicular to
O
Q
OQ
OQ
.
N5
1
Hide problems
diophantine x^4 + 5y^4 = z^4
Find all triples of integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
such that
x
4
+
5
y
4
=
z
4
x^4 + 5y^4 = z^4
x
4
+
5
y
4
=
z
4
.
G2
1
Hide problems
Hard geo
Given a triangle
A
B
C
ABC
A
BC
, incircle
(
I
)
(I)
(
I
)
touches
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Let
M
M
M
be a point inside
A
B
C
ABC
A
BC
. Prove that
M
M
M
lie on
(
I
)
(I)
(
I
)
if and only if one number among
A
E
⋅
S
B
M
C
,
B
F
⋅
S
C
M
A
,
C
D
⋅
S
A
M
B
\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}
A
E
⋅
S
BMC
,
BF
⋅
S
CM
A
,
C
D
⋅
S
A
MB
is sum of two remaining numbers (
S
A
B
C
S_{ABC}
S
A
BC
denotes the area of triangle
A
B
C
ABC
A
BC
)
A7
1
Hide problems
A problems related to floor function
Prove that for all
n
∈
Z
+
n\in\mathbb{Z}^+
n
∈
Z
+
, we have
∑
p
=
1
n
∑
q
=
1
p
⌊
−
1
+
8
q
+
(
2
p
−
1
)
2
2
⌋
=
−
n
(
n
+
1
)
(
n
+
2
)
3
\sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3}
p
=
1
∑
n
q
=
1
∑
p
⌊
−
2
1
+
8
q
+
(
2
p
−
1
)
2
⌋
=
−
3
n
(
n
+
1
)
(
n
+
2
)