MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2013 Vietnam National Olympiad
2013 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(4)
4
1
Hide problems
Count the number of number 2013
Write down some numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots, a_n
a
1
,
a
2
,
…
,
a
n
from left to right on a line. Step 1, we write
a
1
+
a
2
a_1+a_2
a
1
+
a
2
between
a
1
,
a
2
a_1,a_2
a
1
,
a
2
;
a
2
+
a
3
a_2+a_3
a
2
+
a
3
between
a
2
,
a
3
a_2,a_3
a
2
,
a
3
, …,
a
n
−
1
+
a
n
a_{n-1}+a_n
a
n
−
1
+
a
n
between
a
n
−
1
,
a
n
a_{n-1},a_n
a
n
−
1
,
a
n
, and then we have new sequence
b
=
(
a
1
,
a
1
+
a
2
,
a
2
,
a
2
+
a
3
,
a
3
,
…
,
a
n
−
1
,
a
n
−
1
+
a
n
,
a
n
)
b=(a_1, a_1+a_2,a_2,a_2+a_3,a_3, \ldots, a_{n-1}, a_{n-1}+a_n, a_n)
b
=
(
a
1
,
a
1
+
a
2
,
a
2
,
a
2
+
a
3
,
a
3
,
…
,
a
n
−
1
,
a
n
−
1
+
a
n
,
a
n
)
. Step 2, we do the same thing with sequence b to have the new sequence c again…. And so on. If we do 2013 steps, count the number of the number 2013 appear on the line if a)
n
=
2
n=2
n
=
2
,
a
1
=
1
,
a
2
=
1000
a_1=1, a_2=1000
a
1
=
1
,
a
2
=
1000
b)
n
=
1000
n=1000
n
=
1000
,
a
i
=
i
,
i
=
1
,
2
…
,
1000
a_i=i, i=1,2\ldots, 1000
a
i
=
i
,
i
=
1
,
2
…
,
1000
Sorry for my bad English [color=#008000]Moderator says: alternate phrasing here: https://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=516134
2
2
Hide problems
limit of squence
Define a sequence
{
a
n
}
\{a_n\}
{
a
n
}
as:
{
a
1
=
1
a
n
+
1
=
3
−
a
n
+
2
2
a
n
for
n
≥
1.
\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.
⎩
⎨
⎧
a
1
=
1
a
n
+
1
=
3
−
2
a
n
a
n
+
2
for
n
≥
1.
Prove that this sequence has a finite limit as
n
→
+
∞
n\to+\infty
n
→
+
∞
. Also determine the limit.
fixed circle
Let
A
B
C
ABC
A
BC
be a cute triangle.
(
O
)
(O)
(
O
)
is circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
.
D
D
D
is on arc
B
C
BC
BC
not containing
A
A
A
.Line
△
\triangle
△
moved through
H
H
H
(
H
H
H
is orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
cuts circumcircle of
△
A
B
H
\triangle ABH
△
A
B
H
,circumcircle
△
A
C
H
\triangle ACH
△
A
C
H
again at
M
,
N
M,N
M
,
N
respectively. a.Find
△
\triangle
△
satisfy
S
A
M
N
S_{AMN}
S
A
MN
max b.
d
1
,
d
2
d_{1},d_{2}
d
1
,
d
2
are the line through
M
M
M
perpendicular to
D
B
DB
D
B
,the line through
N
N
N
perpendicular to
D
C
DC
D
C
respectively.
d
1
d_{1}
d
1
cuts
d
2
d_{2}
d
2
at
P
P
P
.Prove that
P
P
P
move on a fixed circle.
1
2
Hide problems
system of equations
Solve with full solution:
{
(
sin
x
)
2
+
1
(
sin
x
)
2
+
(
cos
y
)
2
+
1
(
cos
y
)
2
=
20
y
x
+
y
(
sin
y
)
2
+
1
(
sin
y
)
2
+
(
cos
x
)
2
+
1
(
cos
x
)
2
=
20
x
x
+
y
\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y} \\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right.
⎩
⎨
⎧
(
sin
x
)
2
+
(
s
i
n
x
)
2
1
+
(
cos
y
)
2
+
(
c
o
s
y
)
2
1
=
x
+
y
20
y
(
sin
y
)
2
+
(
s
i
n
y
)
2
1
+
(
cos
x
)
2
+
(
c
o
s
x
)
2
1
=
x
+
y
20
x
function equation
Find all
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
that satisfies
f
(
0
)
=
0
,
f
(
1
)
=
2013
f(0)=0,f(1)=2013
f
(
0
)
=
0
,
f
(
1
)
=
2013
and
(
x
−
y
)
(
f
(
f
2
(
x
)
)
−
f
(
f
2
(
y
)
)
)
=
(
f
(
x
)
−
f
(
y
)
)
(
f
2
(
x
)
−
f
2
(
y
)
)
(x-y)(f(f^2(x))-f(f^2(y)))=(f(x)-f(y))(f^2(x)-f^2(y))
(
x
−
y
)
(
f
(
f
2
(
x
))
−
f
(
f
2
(
y
)))
=
(
f
(
x
)
−
f
(
y
))
(
f
2
(
x
)
−
f
2
(
y
))
Note:
f
2
(
x
)
=
(
f
(
x
)
)
2
f^2(x)=(f(x))^2
f
2
(
x
)
=
(
f
(
x
)
)
2
3
2
Hide problems
colinear and fixed point
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
C
ABC
A
BC
isn't a isosceles triangle.
(
I
)
(I)
(
I
)
is incircle of triangle 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. The line through
E
E
E
perpendicular to
B
I
BI
B
I
cuts
(
I
)
(I)
(
I
)
again at
K
K
K
. The line through
F
F
F
perpendicular to
C
I
CI
C
I
cuts
(
I
)
(I)
(
I
)
again at
L
L
L
.
J
J
J
is midpoint of
K
L
KL
K
L
. a) Prove that
D
,
I
,
J
D,I,J
D
,
I
,
J
collinear. b)
B
,
C
B,C
B
,
C
are fixed points,
A
A
A
is moved point such that
A
B
A
C
=
k
\frac{AB}{AC}=k
A
C
A
B
=
k
with
k
k
k
is constant.
I
E
,
I
F
IE,IF
I
E
,
I
F
cut
(
I
)
(I)
(
I
)
again at
M
,
N
M,N
M
,
N
respectively.
M
N
MN
MN
cuts
I
B
,
I
C
IB,IC
I
B
,
I
C
at
P
,
Q
P,Q
P
,
Q
respectively. Prove that bisector perpendicular of
P
Q
PQ
PQ
through a fixed point.
How many ordered 6-tuples satisfy the system
Find all ordered 6-tuples satisfy following system of modular equation:
a
b
+
a
′
b
′
≡
1
ab+a'b' \equiv 1
ab
+
a
′
b
′
≡
1
(mod 15)
b
c
+
b
′
c
′
≡
1
bc+b'c' \equiv 1
b
c
+
b
′
c
′
≡
1
(mod 15)
c
a
+
c
′
a
′
≡
1
ca+c'a' \equiv 1
c
a
+
c
′
a
′
≡
1
(mod 15) Given that
a
,
b
,
c
,
a
′
,
b
′
,
c
′
ϵ
(
0
;
1
;
2
;
.
.
.
;
14
)
a,b,c,a',b',c' \epsilon (0;1;2;...;14)
a
,
b
,
c
,
a
′
,
b
′
,
c
′
ϵ
(
0
;
1
;
2
;
...
;
14
)