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National and Regional Contests
Vietnam Contests
Viet Nam Math Olympiad For High School Students
2022 VN Math Olympiad For High School Students
2022 VN Math Olympiad For High School Students
Part of
Viet Nam Math Olympiad For High School Students
Subcontests
(8)
Problem 8
2
Hide problems
AP is perpendicular to QR iff AB=AC or 2BC^2=AB^2+AC^2
Given the triangle
A
B
C
ABC
A
BC
with
T
T
T
is its Fermat–Torricelli point. Let
(
N
a
)
(N_a)
(
N
a
)
be the circumcircle of
△
T
B
C
\triangle TBC
△
TBC
. Choose a point
X
X
X
on
(
N
a
)
(N_a)
(
N
a
)
such that
T
X
TX
TX
is perpendicular to
B
C
BC
BC
. The segment
B
C
BC
BC
intersects
(
T
N
a
X
)
(TN_aX)
(
T
N
a
X
)
at
D
D
D
. Similar definition of points
Y
,
Z
,
E
,
F
Y, Z, E, F
Y
,
Z
,
E
,
F
. The reflection lines of the Euler line of
△
A
B
C
\triangle ABC
△
A
BC
wrt
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
intersect
X
D
,
Y
E
,
Z
F
XD, YE, ZF
X
D
,
Y
E
,
ZF
at
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
, respectively.Prove that:
A
P
AP
A
P
is perpendicular to
Q
R
QR
QR
if and only if
A
B
=
A
C
AB = AC
A
B
=
A
C
or
2
B
C
2
=
A
B
2
+
A
C
2
2BC^2 = AB^2 + AC^2
2
B
C
2
=
A
B
2
+
A
C
2
.
k(m) is even
Given Fibonacci sequence
(
F
n
)
,
(F_n),
(
F
n
)
,
and a positive integer
m
m
m
, denote
k
(
m
)
k(m)
k
(
m
)
by the smallest positive integer satisfying
F
n
+
k
(
m
)
≡
F
n
(
m
o
d
m
)
,
F_{n+k(m)}\equiv F_n(\bmod m),
F
n
+
k
(
m
)
≡
F
n
(
mod
m
)
,
for all natural numbers
n
n
n
.Prove that:
k
(
m
)
k(m)
k
(
m
)
is even for all
m
>
2.
m>2.
m
>
2.
Problem 7
2
Hide problems
Fermat point, orthocenter
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
,
∠
B
,
∠
C
<
12
0
∘
\angle A,\angle B,\angle C <120^{\circ}
∠
A
,
∠
B
,
∠
C
<
12
0
∘
,
T
T
T
is its Fermat-Torricelli point. Let
H
a
,
H
b
,
H
c
H_a, H_b, H_c
H
a
,
H
b
,
H
c
be the orthocenter of triangles
T
B
C
,
T
C
A
,
T
A
B
TBC, TCA, TAB
TBC
,
TC
A
,
T
A
B
, respectively. a) Prove that:
T
T
T
is the centroid of the
△
H
a
H
b
H
c
\triangle H_aH_bH_c
△
H
a
H
b
H
c
. b) Denote
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively by the intersections of
H
c
H
b
H_cH_b
H
c
H
b
and the segment
B
C
BC
BC
,
H
c
H
a
H_cH_a
H
c
H
a
and the segment
C
A
CA
C
A
,
H
a
H
b
H_aH_b
H
a
H
b
and the segment
A
B
AB
A
B
. Prove that: the triangle
D
E
F
DEF
D
EF
is equilateral. c) Prove that: the lines passing through
D
,
E
,
F
D, E, F
D
,
E
,
F
and are respectively perpendicular to
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
are concurrent at a point. Let that point be
S
S
S
. d) Prove that:
T
S
TS
TS
is parallel to the Euler line of the triangle
A
B
C
ABC
A
BC
.
Fibonacci sequence, congruent, k(2^s)
Given Fibonacci sequence
(
F
n
)
,
(F_n),
(
F
n
)
,
and a positive integer
m
m
m
, denote
k
(
m
)
k(m)
k
(
m
)
by the smallest positive integer satisfying
F
n
+
k
(
m
)
≡
F
n
(
m
o
d
m
)
,
F_{n+k(m)}\equiv F_n(\bmod m),
F
n
+
k
(
m
)
≡
F
n
(
mod
m
)
,
for all natural numbers
n
n
n
,
s
s
s
is a positive integer. Prove that: a)
F
3.2
s
−
1
≡
0
(
m
o
d
2
s
)
{F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s})
F
3.2
s
−
1
≡
0
(
mod
2
s
)
and
F
3.2
s
−
1
+
1
≡
1
(
m
o
d
2
s
)
.
{F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}).
F
3.2
s
−
1
+
1
≡
1
(
mod
2
s
)
.
b)
k
(
2
s
)
=
3.
2
s
−
1
.
k({2^s}) = {3.2^{s - 1}}.
k
(
2
s
)
=
3.
2
s
−
1
.
Problem 6
2
Hide problems
the same distances
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
,
∠
B
,
∠
C
<
12
0
∘
\angle A,\angle B,\angle C <120^{\circ}
∠
A
,
∠
B
,
∠
C
<
12
0
∘
,
T
T
T
is its Fermat-Torricelli point. Let
G
G
G
be the centroid of
△
A
B
C
\triangle ABC
△
A
BC
.Prove that: the distances from
G
G
G
to the perpendicular bisectors of
T
A
,
T
B
,
T
C
TA, TB, TC
T
A
,
TB
,
TC
are the same.
Fibonacci sequence, k(p), congruent
Given Fibonacci sequence
(
F
n
)
,
(F_n),
(
F
n
)
,
and a positive integer
m
m
m
, denote
k
(
m
)
k(m)
k
(
m
)
by the smallest positive integer satisfying
F
n
+
k
(
m
)
≡
F
n
(
m
o
d
m
)
,
F_{n+k(m)}\equiv F_n(\bmod m),
F
n
+
k
(
m
)
≡
F
n
(
mod
m
)
,
for all natural numbers
n
n
n
,
p
p
p
is an odd prime such that
p
≡
±
1
(
m
o
d
5
)
p \equiv \pm 1(\bmod 5)
p
≡
±
1
(
mod
5
)
. Prove that: a)
F
p
+
1
≡
0
(
m
o
d
p
)
.
{F_{p + 1}} \equiv 0(\bmod p).
F
p
+
1
≡
0
(
mod
p
)
.
b)
k
(
p
)
∣
2
p
+
2.
k(p)|2p+2.
k
(
p
)
∣2
p
+
2.
c)
k
(
p
)
k(p)
k
(
p
)
is divisible by
4.
4.
4.
Problem 5
2
Hide problems
Steiner's problem
Given a convex quadrilateral
M
N
P
Q
MNPQ
MNPQ
. Assume that there exists 2 points
U
,
V
U, V
U
,
V
inside
M
N
P
Q
MNPQ
MNPQ
satifying:
∠
M
U
N
=
∠
M
U
V
=
∠
N
U
V
=
∠
Q
V
U
=
∠
P
V
U
=
∠
P
V
Q
\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQ
∠
M
U
N
=
∠
M
U
V
=
∠
N
U
V
=
∠
Q
V
U
=
∠
P
V
U
=
∠
P
V
Q
Consider another 2 points
X
,
Y
X, Y
X
,
Y
in the plane. Prove that the sum
X
M
+
X
N
+
X
Y
+
Y
P
+
Y
Q
XM + XN + XY + YP + YQ
XM
+
XN
+
X
Y
+
Y
P
+
Y
Q
get its minimum value iff
X
≡
U
,
Y
≡
V
X\equiv U, Y\equiv V
X
≡
U
,
Y
≡
V
.
Fibonacci sequence, congruent mod p
Given Fibonacci sequence
(
F
n
)
,
(F_n),
(
F
n
)
,
and a positive integer
m
m
m
, denote
k
(
m
)
k(m)
k
(
m
)
by the smallest positive integer satisfying
F
n
+
k
(
m
)
≡
F
n
(
m
o
d
m
)
,
F_{n+k(m)}\equiv F_n(\bmod m),
F
n
+
k
(
m
)
≡
F
n
(
mod
m
)
,
for all natural numbers
n
n
n
,
p
p
p
is an odd prime such that
p
≡
±
1
(
m
o
d
5
)
p \equiv \pm 1(\bmod 5)
p
≡
±
1
(
mod
5
)
. Prove that: a)
5
p
−
1
2
≡
1
(
m
o
d
p
)
.
{5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p).
5
2
p
−
1
≡
1
(
mod
p
)
.
b)
F
p
−
1
≡
0
(
m
o
d
p
)
.
{F_{p - 1}} \equiv 0(\bmod p).
F
p
−
1
≡
0
(
mod
p
)
.
c)
k
(
p
)
∣
p
−
1.
k(p)|p-1.
k
(
p
)
∣
p
−
1.
Problem 4
2
Hide problems
vector and lengths (in)equality
Assume that
△
A
B
C
\triangle ABC
△
A
BC
is acute. Let
a
=
B
C
,
b
=
C
A
,
c
=
A
B
a=BC, b=CA, c=AB
a
=
BC
,
b
=
C
A
,
c
=
A
B
. a) Denote
H
H
H
by the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that:
a
.
H
A
→
H
A
+
b
.
H
B
→
H
B
+
c
.
H
C
→
H
C
=
0
→
.
a.\frac{{\overrightarrow {HA} }}{{HA}} + b.\frac{{\overrightarrow {HB} }}{{HB}} + c.\frac{{\overrightarrow {HC} }}{{HC}} = \overrightarrow 0 .
a
.
H
A
H
A
+
b
.
H
B
H
B
+
c
.
H
C
H
C
=
0
.
b) Consider a point
P
P
P
lying on the plane. Prove that the sum:
a
P
a
+
b
P
B
+
c
P
C
aPa+bPB+cPC
a
P
a
+
b
PB
+
c
PC
get its minimum value iff
P
≡
H
P\equiv H
P
≡
H
.
determine k(2),k(4),k(5),k(10)
Given Fibonacci sequence
(
F
n
)
,
(F_n),
(
F
n
)
,
and a positive integer
m
m
m
, denote
k
(
m
)
k(m)
k
(
m
)
by the smallest positive integer satisfying
F
n
+
k
(
m
)
≡
F
n
(
m
o
d
m
)
,
F_{n+k(m)}\equiv F_n(\bmod m),
F
n
+
k
(
m
)
≡
F
n
(
mod
m
)
,
for all natural numbers
n
n
n
. a) Prove that: For all
m
1
,
m
2
∈
Z
+
m_1,m_2\in \mathbb{Z^+}
m
1
,
m
2
∈
Z
+
, we have:
k
(
[
m
1
,
m
2
]
)
=
[
k
(
m
1
)
,
k
(
m
2
)
]
.
k([m_1,m_2])=[k(m_1),k(m_2)].
k
([
m
1
,
m
2
])
=
[
k
(
m
1
)
,
k
(
m
2
)]
.
(Here
[
a
,
b
]
[a,b]
[
a
,
b
]
is the least common multiple of
a
,
b
.
a,b.
a
,
b
.
)b) Determine
k
(
2
)
,
k
(
4
)
,
k
(
5
)
,
k
(
10
)
.
k(2),k(4),k(5),k(10).
k
(
2
)
,
k
(
4
)
,
k
(
5
)
,
k
(
10
)
.
Problem 3
2
Hide problems
fermat point, vector (in)equalities
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
,
∠
B
,
∠
C
<
12
0
∘
\angle A,\angle B,\angle C <120^{\circ}
∠
A
,
∠
B
,
∠
C
<
12
0
∘
,
T
T
T
is its Fermat-Torricelli point. Consider a point
P
P
P
lying on the same plane with
△
A
B
C
\triangle ABC
△
A
BC
. Prove that: a)
T
A
→
T
A
+
T
B
→
T
B
+
T
C
→
T
C
=
0
→
.
\dfrac{\overrightarrow {TA}}{TA}+\dfrac{\overrightarrow {TB}}{TB}+\dfrac{\overrightarrow {TC}}{TC}=\overrightarrow {0}.
T
A
T
A
+
TB
TB
+
TC
TC
=
0
.
b)
P
A
+
P
B
+
P
C
≥
P
A
→
.
T
A
→
T
A
+
P
B
→
.
T
B
→
T
B
+
P
C
→
.
T
C
→
T
C
.
PA + PB + PC \ge \frac{{\overrightarrow {PA} \overrightarrow {.TA} }}{{TA}} + \frac{{\overrightarrow {PB} .\overrightarrow {TB} }}{{TB}} + \frac{{\overrightarrow {PC} \overrightarrow {.TC} }}{{TC}}.
P
A
+
PB
+
PC
≥
T
A
P
A
.
T
A
+
TB
PB
.
TB
+
TC
PC
.
TC
.
c)
P
A
+
P
B
+
P
C
≥
T
A
+
T
B
+
T
C
PA + PB + PC \ge TA + TB + TC
P
A
+
PB
+
PC
≥
T
A
+
TB
+
TC
and the equality occurs iff
P
≡
T
P\equiv T
P
≡
T
.
infinitely numbers that are divisible by N
Given a positive integer
N
N
N
.Prove that: there are infinitely elements of the Fibonacci sequence that are divisible by
N
N
N
.
Problem 2
2
Hide problems
k and k(m)
Given Fibonacci sequence
(
F
n
)
,
(F_n),
(
F
n
)
,
and a positive integer
m
m
m
. a) Prove that: there exists integers
0
≤
i
<
j
≤
m
2
0\le i<j\le m^2
0
≤
i
<
j
≤
m
2
such that
F
i
≡
F
j
(
m
o
d
m
)
F_i\equiv F_j (\bmod m)
F
i
≡
F
j
(
mod
m
)
and
F
i
+
1
≡
F
j
+
1
(
m
o
d
m
)
F_{i+1}\equiv F_{j+1}(\bmod m)
F
i
+
1
≡
F
j
+
1
(
mod
m
)
.b) Prove that: there exists a positive integer
k
k
k
such that
F
n
+
k
≡
F
n
(
m
o
d
m
)
,
F_{n+k}\equiv F_n(\bmod m),
F
n
+
k
≡
F
n
(
mod
m
)
,
for all natural numbers
n
n
n
.*Denote
k
(
m
)
k(m)
k
(
m
)
by the smallest positive integer satisfying
F
n
+
k
(
m
)
≡
F
n
(
m
o
d
m
)
,
F_{n+k(m)}\equiv F_n(\bmod m),
F
n
+
k
(
m
)
≡
F
n
(
mod
m
)
,
for all natural numbers
n
n
n
*. c) Prove that:
k
(
m
)
k(m)
k
(
m
)
is the smallest positive integer such that
F
k
(
m
)
≡
0
(
m
o
d
m
)
F_{k(m)}\equiv 0(\bmod m)
F
k
(
m
)
≡
0
(
mod
m
)
and
F
k
(
m
)
+
1
≡
1
(
m
o
d
m
)
F_{k(m)+1}\equiv 1(\bmod m)
F
k
(
m
)
+
1
≡
1
(
mod
m
)
.d) Given a positive integer
k
k
k
. Prove that:
F
n
+
k
≡
F
n
(
m
o
d
m
)
F_{n+k}\equiv F_n(\bmod m)
F
n
+
k
≡
F
n
(
mod
m
)
for all natural numbers
n
n
n
iff
k
⋮
k
(
m
)
k\vdots k(m)
k
⋮
k
(
m
)
.
another way to construct Fermat point
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
,
∠
B
,
∠
C
<
12
0
∘
\angle A,\angle B,\angle C <120^{\circ}
∠
A
,
∠
B
,
∠
C
<
12
0
∘
,
T
T
T
is its Fermat-Torricelli point. Construct 3 equilateral triangles
B
C
D
,
C
A
E
,
A
B
F
BCD, CAE, ABF
BC
D
,
C
A
E
,
A
BF
outside
△
A
B
C
\triangle ABC
△
A
BC
Prove that:
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
are concurrent at
T
T
T
.
Problem 1
2
Hide problems
only 1 Fermat-Torricelli point inside a triangle
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
,
∠
B
,
∠
C
<
12
0
∘
\angle A,\angle B,\angle C <120^{\circ}
∠
A
,
∠
B
,
∠
C
<
12
0
∘
.Prove that: there is exactly one point
T
T
T
inside
△
A
B
C
\triangle ABC
△
A
BC
such that
∠
B
T
C
=
∠
C
T
A
=
∠
A
T
B
=
12
0
∘
\angle BTC=\angle CTA=\angle ATB=120^{\circ}
∠
BTC
=
∠
CT
A
=
∠
A
TB
=
12
0
∘
. (
T
T
T
is called Fermat-Torricelli point of
△
A
B
C
\triangle ABC
△
A
BC
)
basic Fibonacci sequence properties
Given Fibonacci sequence
(
F
n
)
(F_n)
(
F
n
)
a) Prove that: for all
u
,
v
∈
N
,
u
≥
1
u,v\in \mathbb{N}, u\ge 1
u
,
v
∈
N
,
u
≥
1
, we have:
F
u
+
v
=
F
u
−
1
F
v
+
F
u
F
v
+
1
.
F_{u+v}=F_{u-1}F_{v}+F_{u}F_{v+1}.
F
u
+
v
=
F
u
−
1
F
v
+
F
u
F
v
+
1
.
b) Prove that: for all
n
∈
N
,
n
≥
1
n\in \mathbb{N}, n\ge 1
n
∈
N
,
n
≥
1
, we have:
F
2
n
=
F
n
(
F
n
−
1
+
F
n
+
1
)
,
F_{2n}=F_n(F_{n-1}+F_{n+1}),
F
2
n
=
F
n
(
F
n
−
1
+
F
n
+
1
)
,
F
2
n
+
1
=
F
n
2
+
F
n
+
1
2
.
F_{2n+1}=F_n^2+F_{n+1}^2.
F
2
n
+
1
=
F
n
2
+
F
n
+
1
2
.