MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2017 Vietnam National Olympiad
2017 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(4)
4
1
Hide problems
Minimum k such that there is a k-balance label
Given an integer
n
>
1
n>1
n
>
1
and a
n
×
n
n\times n
n
×
n
grid
A
B
C
D
ABCD
A
BC
D
containing
n
2
n^2
n
2
unit squares, each unit square is colored by one of three colors: Black, white and gray. A coloring is called symmetry if each unit square has center on diagonal
A
C
AC
A
C
is colored by gray and every couple of unit squares which are symmetry by
A
C
AC
A
C
should be both colred by black or white. In each gray square, they label a number
0
0
0
, in a white square, they will label a positive integer and in a black square, a negative integer. A label will be called
k
k
k
-balance (with
k
∈
Z
+
k\in\mathbb{Z}^+
k
∈
Z
+
) if it satisfies the following requirements:i) Each pair of unit squares which are symmetry by
A
C
AC
A
C
are labelled with the same integer from the closed interval
[
−
k
,
k
]
[-k,k]
[
−
k
,
k
]
ii) If a row and a column intersectes at a square that is colored by black, then the set of positive integers on that row and the set of positive integers on that column are distinct.If a row and a column intersectes at a square that is colored by white, then the set of negative integers on that row and the set of negative integers on that column are distinct.a) For
n
=
5
n=5
n
=
5
, find the minimum value of
k
k
k
such that there is a
k
k
k
-balance label for the following grid[asy] size(4cm); pair o = (0,0); pair y = (0,5); pair z = (5,5); pair t = (5,0); dot("
A
A
A
", y, dir(180)); dot("
B
B
B
", z); dot("
C
C
C
", t); dot("
D
D
D
", o, dir(180)); fill((0,5)--(1,5)--(1,4)--(0,4)--cycle,gray); fill((1,4)--(2,4)--(2,3)--(1,3)--cycle,gray); fill((2,3)--(3,3)--(3,2)--(2,2)--cycle,gray); fill((3,2)--(4,2)--(4,1)--(3,1)--cycle,gray); fill((4,1)--(5,1)--(5,0)--(4,0)--cycle,gray); fill((0,3)--(1,3)--(1,1)--(0,1)--cycle,black); fill((2,5)--(4,5)--(4,4)--(2,4)--cycle,black); fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black); fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black); fill((4,3)--(5,3)--(5,2)--(4,2)--cycle,black); for (int i=0; i<=5; ++i) { draw((0,i)--(5,i)^^(i,0)--(i,5)); } [/asy] b) Let
n
=
2017
n=2017
n
=
2017
. Find the least value of
k
k
k
such that there is always a
k
k
k
-balance label for a symmetry coloring.
3
2
Hide problems
Not so hard geometry problem
Given an acute, non isoceles triangle
A
B
C
ABC
A
BC
and
(
O
)
(O)
(
O
)
be its circumcircle,
H
H
H
its orthocenter and
E
,
F
E, F
E
,
F
are the feet of the altitudes from
B
B
B
and
C
C
C
, respectively.
A
H
AH
A
H
intersects
(
O
)
(O)
(
O
)
at
D
D
D
(
D
≠
A
D\ne A
D
=
A
).a) Let
I
I
I
be the midpoint of
A
H
AH
A
H
,
E
I
EI
E
I
meets
B
D
BD
B
D
at
M
M
M
and
F
I
FI
F
I
meets
C
D
CD
C
D
at
N
N
N
. Prove that
M
N
⊥
O
H
MN\perp OH
MN
⊥
O
H
.b) The lines
D
E
DE
D
E
,
D
F
DF
D
F
intersect
(
O
)
(O)
(
O
)
at
P
,
Q
P,Q
P
,
Q
respectively (
P
≠
D
,
Q
≠
D
P\ne D,Q\ne D
P
=
D
,
Q
=
D
).
(
A
E
F
)
(AEF)
(
A
EF
)
meets
(
O
)
(O)
(
O
)
and
A
O
AO
A
O
at
R
,
S
R,S
R
,
S
respectively (
R
≠
A
,
S
≠
A
R\ne A, S\ne A
R
=
A
,
S
=
A
). Prove that
B
P
,
C
Q
,
R
S
BP,CQ,RS
BP
,
CQ
,
RS
are concurrent.
Projective geometry
Given an acute triangle
A
B
C
ABC
A
BC
and
(
O
)
(O)
(
O
)
be its circumcircle. Let
G
G
G
be the point on arc
B
C
BC
BC
that doesn't contain
O
O
O
of the circumcircle
(
I
)
(I)
(
I
)
of triangle
O
B
C
OBC
OBC
. The circumcircle of
A
B
G
ABG
A
BG
intersects
A
C
AC
A
C
at
E
E
E
and circumcircle of
A
C
G
ACG
A
CG
intersects
A
B
AB
A
B
at
F
F
F
(
E
≠
A
,
F
≠
A
E\ne A, F\ne A
E
=
A
,
F
=
A
).a) Let
K
K
K
be the intersection of
B
E
BE
BE
and
C
F
CF
CF
. Prove that
A
K
,
B
C
,
O
G
AK,BC,OG
A
K
,
BC
,
OG
are concurrent.b) Let
D
D
D
be a point on arc
B
O
C
BOC
BOC
(arc
B
C
BC
BC
containing
O
O
O
) of
(
I
)
(I)
(
I
)
.
G
B
GB
GB
meets
C
D
CD
C
D
at
M
M
M
,
G
C
GC
GC
meets
B
D
BD
B
D
at
N
N
N
. Assume that
M
N
MN
MN
intersects
(
O
)
(O)
(
O
)
at
P
P
P
nad
Q
Q
Q
. Prove that when
G
G
G
moves on the arc
B
C
BC
BC
that doesn't contain
O
O
O
of
(
I
)
(I)
(
I
)
, the circumcircle
(
G
P
Q
)
(GPQ)
(
GPQ
)
always passes through two fixed points.
1
2
Hide problems
Convergent sequence
Given
a
∈
R
a\in\mathbb{R}
a
∈
R
and a sequence
(
u
n
)
(u_n)
(
u
n
)
defined by
{
u
1
=
a
u
n
+
1
=
1
2
+
2
n
+
3
n
+
1
u
n
+
1
4
e
m
s
p
;
∀
n
∈
N
∗
\begin{cases} u_1=a\\ u_{n+1}=\frac{1}{2}+\sqrt{\frac{2n+3}{n+1}u_n+\frac{1}{4}} \forall n\in\mathbb{N}^* \end{cases}
{
u
1
=
a
u
n
+
1
=
2
1
+
n
+
1
2
n
+
3
u
n
+
4
1
e
m
s
p
;
∀
n
∈
N
∗
a) Prove that
(
u
n
)
(u_n)
(
u
n
)
is convergent sequence when
a
=
5
a=5
a
=
5
and find the limit of the sequence in that caseb) Find all
a
a
a
such that the sequence
(
u
n
)
(u_n)
(
u
n
)
is exist and is convergent.
Problem 5 of VMO 2017
Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
satisfying relation :
f
(
x
f
(
y
)
−
f
(
x
)
)
=
2
f
(
x
)
+
x
y
f(xf(y)-f(x))=2f(x)+xy
f
(
x
f
(
y
)
−
f
(
x
))
=
2
f
(
x
)
+
x
y
∀
x
,
y
∈
R
\forall x,y \in \mathbb{R}
∀
x
,
y
∈
R
2
2
Hide problems
Minimal polynomial
Is there an integer coefficients polynomial
P
(
x
)
P(x)
P
(
x
)
satisfying
{
P
(
1
+
2
3
)
=
1
+
2
3
P
(
1
+
5
)
=
2
+
3
5
\begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases}
{
P
(
1
+
3
2
)
=
1
+
3
2
P
(
1
+
5
)
=
2
+
3
5
Vietnam National Olympiad 2017 Problem 6
Prove that a)
∑
k
=
1
1008
k
C
2017
k
≡
0
\sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0
∑
k
=
1
1008
k
C
2017
k
≡
0
(mod
201
7
2
2017^2
201
7
2
)b)
∑
k
=
1
504
(
−
1
)
k
C
2017
k
≡
3
(
2
2016
−
1
)
\sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )
∑
k
=
1
504
(
−
1
)
k
C
2017
k
≡
3
(
2
2016
−
1
)
(mod
201
7
2
2017^2
201
7
2
)