MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2012 Vietnam National Olympiad
2012 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(4)
4
1
Hide problems
n boys and n girls in a row, students get candies
Let
n
n
n
be a natural number. There are
n
n
n
boys and
n
n
n
girls standing in a line, in any arbitrary order. A student
X
X
X
will be eligible for receiving
m
m
m
candies, if we can choose two students of opposite sex with
X
X
X
standing on either side of
X
X
X
in
m
m
m
ways. Show that the total number of candies does not exceed
1
3
n
(
n
2
−
1
)
.
\frac 13n(n^2-1).
3
1
n
(
n
2
−
1
)
.
2
2
Hide problems
Sequences a_n, b_n and P_k(x)=x^2+a_kx+b_k
Let
⟨
a
n
⟩
\langle a_n\rangle
⟨
a
n
⟩
and
⟨
b
n
⟩
\langle b_n\rangle
⟨
b
n
⟩
be two arithmetic sequences of numbers, and let
m
m
m
be an integer greater than
2.
2.
2.
Define
P
k
(
x
)
=
x
2
+
a
k
x
+
b
k
,
k
=
1
,
2
,
⋯
,
m
.
P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.
P
k
(
x
)
=
x
2
+
a
k
x
+
b
k
,
k
=
1
,
2
,
⋯
,
m
.
Prove that if the quadratic expressions
P
1
(
x
)
,
P
m
(
x
)
P_1(x), P_m(x)
P
1
(
x
)
,
P
m
(
x
)
do not have any real roots, then all the remaining polynomials also don't have real roots.
Odd natural numbers a,b; a|b^2+2 and b|a^2+2
Consider two odd natural numbers
a
a
a
and
b
b
b
where
a
a
a
is a divisor of
b
2
+
2
b^2+2
b
2
+
2
and
b
b
b
is a divisor of
a
2
+
2.
a^2+2.
a
2
+
2.
Prove that
a
a
a
and
b
b
b
are the terms of the series of natural numbers
⟨
v
n
⟩
\langle v_n\rangle
⟨
v
n
⟩
defined by
v
1
=
v
2
=
1
;
v
n
=
4
v
n
−
1
−
v
n
−
2
for
n
≥
3.
v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.
v
1
=
v
2
=
1
;
v
n
=
4
v
n
−
1
−
v
n
−
2
for
n
≥
3.
1
2
Hide problems
Sequence of reals x_1=3 and x_n={n+2}/{3n}(x_{n-1}+2).
Define a sequence
{
x
n
}
\{x_n\}
{
x
n
}
as:
{
x
1
=
3
x
n
=
n
+
2
3
n
(
x
n
−
1
+
2
)
for
n
≥
2.
\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.
⎩
⎨
⎧
x
1
=
3
x
n
=
3
n
n
+
2
(
x
n
−
1
+
2
)
for
n
≥
2.
Prove that this sequence has a finite limit as
n
→
+
∞
.
n\to+\infty.
n
→
+
∞.
Also determine the limit.
Group of 5 girls and 12 boys sitting in 17 seats
For a group of 5 girls, denoted as
G
1
,
G
2
,
G
3
,
G
4
,
G
5
G_1,G_2,G_3,G_4,G_5
G
1
,
G
2
,
G
3
,
G
4
,
G
5
and
12
12
12
boys. There are
17
17
17
chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met: (a) Each chair has a proper seat. (b) The order, from left to right, of the girls seating is
G
1
;
G
2
;
G
3
;
G
4
;
G
5
.
G_1; G_2; G_3; G_4; G_5.
G
1
;
G
2
;
G
3
;
G
4
;
G
5
.
(c) Between
G
1
G_1
G
1
and
G
2
G_2
G
2
there are at least three boys. (d) Between
G
4
G_4
G
4
and
G
5
G_5
G
5
there are at least one boy and most four boys. How many such arrangements are possible?
3
2
Hide problems
Cyclic quadrilateral ABCD and M=AB\cap CD, N=AD\cap BC
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with circumcentre
O
,
O,
O
,
and the pair of opposite sides not parallel with each other. Let
M
=
A
B
∩
C
D
M=AB\cap CD
M
=
A
B
∩
C
D
and
N
=
A
D
∩
B
C
.
N=AD\cap BC.
N
=
A
D
∩
BC
.
Denote, by
P
,
Q
,
S
,
T
;
P,Q,S,T;
P
,
Q
,
S
,
T
;
the intersection of the internal angle bisectors of
∠
M
A
N
\angle MAN
∠
M
A
N
and
∠
M
B
N
;
\angle MBN;
∠
MBN
;
∠
M
B
N
\angle MBN
∠
MBN
and
∠
M
C
N
;
\angle MCN;
∠
MCN
;
∠
M
D
N
\angle MDN
∠
M
D
N
and
∠
M
A
N
;
\angle MAN;
∠
M
A
N
;
∠
M
C
N
\angle MCN
∠
MCN
and
∠
M
D
N
.
\angle MDN.
∠
M
D
N
.
Suppose that the four points
P
,
Q
,
S
,
T
P,Q,S,T
P
,
Q
,
S
,
T
are distinct. (a) Show that the four points
P
,
Q
,
S
,
T
P,Q,S,T
P
,
Q
,
S
,
T
are concyclic. Find the centre of this circle, and denote it as
I
.
I.
I
.
(b) Let
E
=
A
C
∩
B
D
.
E=AC\cap BD.
E
=
A
C
∩
B
D
.
Prove that
E
,
O
,
I
E,O,I
E
,
O
,
I
are collinear.
VMO 2012 P7
Find all
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that: (a) For every real number
a
a
a
there exist real number
b
b
b
:
f
(
b
)
=
a
f(b)=a
f
(
b
)
=
a
(b) If
x
>
y
x>y
x
>
y
then
f
(
x
)
>
f
(
y
)
f(x)>f(y)
f
(
x
)
>
f
(
y
)
(c)
f
(
f
(
x
)
)
=
f
(
x
)
+
12
x
.
f(f(x))=f(x)+12x.
f
(
f
(
x
))
=
f
(
x
)
+
12
x
.