MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2021 Vietnam National Olympiad
2021 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(7)
7
1
Hide problems
concyclic and collinear wanted, 5 circles related, starting with circumcircle
Let
A
B
C
ABC
A
BC
be an inscribed triangle in circle
(
O
)
(O)
(
O
)
. Let
D
D
D
be the intersection of the two tangent lines of
(
O
)
(O)
(
O
)
at
B
B
B
and
C
C
C
. The circle passing through
A
A
A
and tangent to
B
C
BC
BC
at
B
B
B
intersects the median passing
A
A
A
of the triangle
A
B
C
ABC
A
BC
at
G
G
G
. Lines
B
G
,
C
G
BG, CG
BG
,
CG
intersect
C
D
,
B
D
CD, BD
C
D
,
B
D
at
E
,
F
E, F
E
,
F
respectively. a) The line passing through the midpoint of
B
E
BE
BE
and
C
F
CF
CF
cuts
B
F
,
C
E
BF, CE
BF
,
CE
at
M
,
N
M, N
M
,
N
respectively. Prove that the points
A
,
D
,
M
,
N
A, D, M, N
A
,
D
,
M
,
N
belong to the same circle. b) Let
A
D
,
A
G
AD, AG
A
D
,
A
G
intersect the circumcircle of the triangles
D
B
C
,
G
B
C
DBC, GBC
D
BC
,
GBC
at
H
,
K
H, K
H
,
K
respectively. The perpendicular bisectors of
H
K
,
H
E
HK, HE
HK
,
H
E
, and
H
F
HF
H
F
cut
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
at
R
,
P
R, P
R
,
P
, and
Q
Q
Q
respectively. Prove that the points
R
,
P
R, P
R
,
P
, and
Q
Q
Q
are collinear.
6
1
Hide problems
a student divides all 30 marbles into 5 boxes, paints a few marbles
A student divides all
30
30
30
marbles into
5
5
5
boxes numbered
1
,
2
,
3
,
4
,
5
1, 2, 3, 4, 5
1
,
2
,
3
,
4
,
5
(after being divided, there may be a box with no marbles). a) How many ways are there to divide marbles into boxes (are two different ways if there is a box with a different number of marbles)? b) After dividing, the student paints those
30
30
30
marbles by a number of colors (each with the same color, one color can be painted for many marbles), so that there are no
2
2
2
marbles in the same box. have the same color and from any
2
2
2
boxes it is impossible to choose
8
8
8
marbles painted in
4
4
4
colors. Prove that for every division, the student must use no less than
10
10
10
colors to paint the marbles. c) Show a division so that with exactly
10
10
10
colors the student can paint the marbles that satisfy the conditions in question b).
4
1
Hide problems
a) s(n) = n/2 (n + 1- \varphi (n)) , b) s (n) = s (n + 2021) no solutions
For an integer
n
≥
2
n \geq 2
n
≥
2
, let
s
(
n
)
s (n)
s
(
n
)
be the sum of positive integers not exceeding
n
n
n
and not relatively prime to
n
n
n
. a) Prove that
s
(
n
)
=
n
2
(
n
+
1
−
φ
(
n
)
)
s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right)
s
(
n
)
=
2
n
(
n
+
1
−
φ
(
n
)
)
, where
φ
(
n
)
\varphi (n)
φ
(
n
)
is the number of integers positive cannot exceed
n
n
n
and are relatively prime to
n
n
n
. b) Prove that there is no integer
n
≥
2
n \geq 2
n
≥
2
such that
s
(
n
)
=
s
(
n
+
2021
)
s (n) = s (n + 2021)
s
(
n
)
=
s
(
n
+
2021
)
5
1
Hide problems
Polynomial and inequality
Let the polynomial
P
(
x
)
=
a
21
x
21
+
a
20
x
20
+
⋯
+
a
1
x
+
a
0
P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0
P
(
x
)
=
a
21
x
21
+
a
20
x
20
+
⋯
+
a
1
x
+
a
0
where
1011
≤
a
i
≤
2021
1011\leq a_i\leq 2021
1011
≤
a
i
≤
2021
for all
i
=
0
,
1
,
2
,
.
.
.
,
21.
i=0,1,2,...,21.
i
=
0
,
1
,
2
,
...
,
21.
Given that
P
(
x
)
P(x)
P
(
x
)
has an integer root and there exists an positive real number
c
c
c
such that
∣
a
k
+
2
−
a
k
∣
≤
c
|a_{k+2}-a_k|\leq c
∣
a
k
+
2
−
a
k
∣
≤
c
for all
k
=
0
,
1
,
.
.
.
,
19.
k=0,1,...,19.
k
=
0
,
1
,
...
,
19.
a) Prove that
P
(
x
)
P(x)
P
(
x
)
has an only integer root.b) Prove that
∑
k
=
0
10
(
a
2
k
+
1
−
a
2
k
)
2
≤
440
c
2
.
\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.
k
=
0
∑
10
(
a
2
k
+
1
−
a
2
k
)
2
≤
440
c
2
.
3
1
Hide problems
Merry Christmas
Let
△
A
B
C
\bigtriangleup ABC
△
A
BC
is not an isosceles triangle and is an acute triangle,
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
be the altitudes and
H
H
H
is the orthocenter .Let
I
I
I
is the circumcenter of
△
H
E
F
\bigtriangleup HEF
△
H
EF
and let
K
,
J
K,J
K
,
J
is the midpoint of
B
C
,
E
F
BC,EF
BC
,
EF
respectively.Let
H
J
HJ
H
J
intersects
(
I
)
(I)
(
I
)
again at
G
G
G
and
G
K
GK
G
K
intersects
(
I
)
(I)
(
I
)
at
L
≠
G
L\neq G
L
=
G
. a) Prove that
A
L
AL
A
L
is perpendicular to
E
F
EF
EF
. b) Let
A
L
AL
A
L
intersects
E
F
EF
EF
at
M
M
M
, the line
I
M
IM
I
M
intersects the circumcircle
△
I
E
F
\bigtriangleup IEF
△
I
EF
again at
N
N
N
,
D
N
DN
D
N
intersects
A
B
,
A
C
AB,AC
A
B
,
A
C
at
P
P
P
and
Q
Q
Q
respectively then prove that
P
E
,
Q
F
,
A
K
PE,QF,AK
PE
,
QF
,
A
K
are concurrent.
1
1
Hide problems
VMO 2021, P1
Let
(
x
n
)
(x_n)
(
x
n
)
define by
x
1
∈
(
0
;
1
2
)
x_1\in \left(0;\dfrac{1}{2}\right)
x
1
∈
(
0
;
2
1
)
and
x
n
+
1
=
3
x
n
2
−
2
n
x
n
3
x_{n+1}=3x_n^2-2nx_n^3
x
n
+
1
=
3
x
n
2
−
2
n
x
n
3
for all
n
≥
1
n\ge 1
n
≥
1
. a) Prove that
(
x
n
)
(x_n)
(
x
n
)
convergence to
0
0
0
.b) For each
n
≥
1
n\ge 1
n
≥
1
, let
y
n
=
x
1
+
2
x
2
+
⋯
+
n
x
n
y_n=x_1+2x_2+\cdots+n x_n
y
n
=
x
1
+
2
x
2
+
⋯
+
n
x
n
. Prove that
(
y
n
)
(y_n)
(
y
n
)
has a limit.
2
1
Hide problems
VMO 2021, P2
Find all function
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
such that
f
(
x
)
f
(
y
)
=
f
(
x
y
−
1
)
+
y
f
(
x
)
+
x
f
(
y
)
f(x)f(y)=f(xy-1)+yf(x)+xf(y)
f
(
x
)
f
(
y
)
=
f
(
x
y
−
1
)
+
y
f
(
x
)
+
x
f
(
y
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R