MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Viet Nam Math Olympiad For High School Students
2023 VN Math Olympiad For High School Students
2023 VN Math Olympiad For High School Students
Part of
Viet Nam Math Olympiad For High School Students
Subcontests
(11)
Problem 11
1
Hide problems
compute PR/PQ
Given a triangle
A
B
C
ABC
A
BC
inscribed in
(
O
)
(O)
(
O
)
with
2
2
2
symmedians
A
D
,
C
F
(
D
,
F
AD, CF(D,F
A
D
,
CF
(
D
,
F
are on the sides
B
C
,
A
B
,
BC, AB,
BC
,
A
B
,
respectively
)
.
).
)
.
The ray
D
F
DF
D
F
intersects
(
O
)
(O)
(
O
)
at
P
.
P.
P
.
The line passing through
P
P
P
and perpendicular to
O
A
OA
O
A
intersects
A
B
,
A
C
AB,AC
A
B
,
A
C
at
Q
,
R
,
Q,R,
Q
,
R
,
respectively
.
.
.
Compute the ratio
P
R
P
Q
.
\dfrac{PR}{PQ}.
PQ
PR
.
Problem 10
2
Hide problems
angle condition with Lemoine point
Given a triangle
A
B
C
ABC
A
BC
with Lemoine point
L
.
L.
L
.
Choose points
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
on the segments
L
A
,
L
B
,
L
C
,
LA,LB,LC,
L
A
,
L
B
,
L
C
,
respectively such that:
∠
X
B
A
=
∠
Y
A
B
,
∠
X
C
A
=
∠
Z
A
C
.
\angle XBA=\angle YAB,\angle XCA=\angle ZAC.
∠
XB
A
=
∠
Y
A
B
,
∠
XC
A
=
∠
Z
A
C
.
Prove that:
∠
Z
B
C
=
∠
Y
C
B
.
\angle ZBC=\angle YCB.
∠
ZBC
=
∠
Y
CB
.
(x(x+1)(x+2)(x+3))^{2^{2023}}+1is irreducible in Q[x]
Prove that: the polynomial
(
x
(
x
+
1
)
(
x
+
2
)
(
x
+
3
)
)
2
2023
+
1
(x(x+1)(x+2)(x+3))^{2^{2023}}+1
(
x
(
x
+
1
)
(
x
+
2
)
(
x
+
3
)
)
2
2023
+
1
is irreducible in
Q
[
x
]
.
\mathbb{Q}[x].
Q
[
x
]
.
Problem 8
2
Hide problems
inequality with Lemoine point
Given a triangle
A
B
C
ABC
A
BC
with symmedians
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
concurrent at Lemoine point
L
(
D
,
E
,
F
L(D,E,F
L
(
D
,
E
,
F
are on the sides
B
C
,
C
A
,
A
B
,
BC,CA,AB,
BC
,
C
A
,
A
B
,
respectively
)
.
).
)
.
Prove that:
L
A
+
L
B
+
L
C
≥
2
(
L
D
+
L
E
+
L
F
)
.
LA+LB+LC\ge 2(LD+LE+LF).
L
A
+
L
B
+
L
C
≥
2
(
L
D
+
L
E
+
L
F
)
.
(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1 is irreducible
Prove that: for all positive integers
n
≥
2
,
n\ge 2,
n
≥
2
,
the polynomial
(
x
2
−
1
)
2
(
x
2
−
1
)
2
.
.
.
(
x
2
−
2023
)
2
+
1
(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1
(
x
2
−
1
)
2
(
x
2
−
1
)
2
...
(
x
2
−
2023
)
2
+
1
is irreducible in
Q
[
x
]
.
\mathbb{Q}[x].
Q
[
x
]
.
Problem 7
2
Hide problems
prove that AB=AC in each case
Given a triangle
A
B
C
ABC
A
BC
with symmedians
B
E
,
C
F
(
E
,
F
BE,CF(E,F
BE
,
CF
(
E
,
F
are on the sides
C
A
,
A
B
,
CA,AB,
C
A
,
A
B
,
respectively
)
)
)
intersecting at Lemoine point
L
.
L.
L
.
Prove that:
A
B
=
A
C
AB=AC
A
B
=
A
C
in each case: a)
L
B
=
L
C
.
LB=LC.
L
B
=
L
C
.
b)
B
E
=
C
F
.
BE=CF.
BE
=
CF
.
P(x^2) is irreducible in Q[x]
Given a polynomial with integer coefficents
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
a
0
,
n
≥
1
P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0,n\ge 1
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
,
n
≥
1
satisfying these conditions: i)
∣
a
0
∣
|a_0|
∣
a
0
∣
is not a perfect square. ii)
P
(
x
)
P(x)
P
(
x
)
is irreducible in
Q
[
x
]
.
\mathbb{Q}[x].
Q
[
x
]
.
Prove that:
P
(
x
2
)
P(x^2)
P
(
x
2
)
is irreducible in
Q
[
x
]
.
\mathbb{Q}[x].
Q
[
x
]
.
Problem 6
2
Hide problems
the Lemoine point is the midpoint of the altitude
a) Given a triangle
A
B
C
ABC
A
BC
with
∠
B
A
C
=
9
0
∘
\angle BAC=90^{\circ}
∠
B
A
C
=
9
0
∘
and the altitude
A
H
(
H
AH(H
A
H
(
H
is on the side
B
C
)
.
BC).
BC
)
.
Prove that: the Lemoine point of the triangle
A
B
C
ABC
A
BC
is the midpoint of
A
H
.
AH.
A
H
.
b) If a triangle has its Lemoine point is the midpoint of
1
1
1
in
3
3
3
symmedian segments, does that triangle need to be a right triangle? Explain why.
irreducible in Q[x]
Prove that these polynomials are irreducible in
Q
[
x
]
:
\mathbb{Q}[x]:
Q
[
x
]
:
a)
x
p
p
!
+
x
p
−
1
(
p
−
1
)
!
+
.
.
.
+
x
2
2
+
x
+
1
,
\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,
p
!
x
p
+
(
p
−
1
)!
x
p
−
1
+
...
+
2
x
2
+
x
+
1
,
with
p
p
p
is a prime number. b)
x
2
n
+
1
,
x^{2^n}+1,
x
2
n
+
1
,
with
n
n
n
is a positive integer.
Problem 5
2
Hide problems
vector equality, Lemoine point
Given a triangle
A
B
C
ABC
A
BC
with Lemoine point
L
.
L.
L
.
Let
a
=
B
C
,
b
=
C
A
,
c
=
A
B
.
a=BC, b=CA,c=AB.
a
=
BC
,
b
=
C
A
,
c
=
A
B
.
Prove that:
a
2
L
A
→
+
b
2
L
B
→
+
c
2
L
C
→
=
0
→
.
{a^2}\overrightarrow {LA} + {b^2}\overrightarrow {LB} + {c^2}\overrightarrow {LC} = \overrightarrow 0 .
a
2
L
A
+
b
2
L
B
+
c
2
L
C
=
0
.
Perron's criterion
Given a polynomial
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
a
0
∈
Z
[
x
]
P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0\in \mathbb{Z}[x]
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
∈
Z
[
x
]
with degree
n
≥
2
n\ge 2
n
≥
2
and
a
o
≠
0.
a_o\ne 0.
a
o
=
0.
Prove that if
∣
a
n
−
1
∣
>
1
+
∣
a
n
−
2
∣
+
.
.
.
+
∣
a
1
∣
+
∣
a
0
∣
|a_{n-1}|>1+|a_{n-2}|+...+|a_1|+|a_0|
∣
a
n
−
1
∣
>
1
+
∣
a
n
−
2
∣
+
...
+
∣
a
1
∣
+
∣
a
0
∣
, then
P
(
x
)
P(x)
P
(
x
)
is irreducible in
Z
[
x
]
.
\mathbb{Z}[x].
Z
[
x
]
.
Problem 4
2
Hide problems
the length of symmedian and angle bisector
Determine whether or not the length of symmedian is not greater than the length of the angle bisector drawn from the same vertex?
irreducible in Z[x] iff irreducible in Q[x]
Prove that: a polynomial is irreducible in
Z
[
x
]
\mathbb{Z}[x]
Z
[
x
]
if and only if it is irreducible in
Q
[
x
]
.
\mathbb{Q}[x].
Q
[
x
]
.
Problem 3
2
Hide problems
<APB+ <MPC=180
Given a triangle
A
B
C
ABC
A
BC
isosceles at
A
.
A.
A
.
A point
P
P
P
lying inside the triangle such that
∠
P
B
C
=
∠
P
C
A
\angle PBC=\angle PCA
∠
PBC
=
∠
PC
A
and let
M
M
M
be the midpoint of
B
C
.
BC.
BC
.
Prove that:
∠
A
P
B
+
∠
M
P
C
=
18
0
∘
.
\angle APB+ \angle MPC =180^{\circ}.
∠
A
PB
+
∠
MPC
=
18
0
∘
.
Eisenstein's criterion
Given a polynomial with integer coefficents with degree
n
>
0
:
n>0:
n
>
0
:
P
(
x
)
=
a
n
x
n
+
.
.
.
+
a
1
x
+
a
0
.
P(x)=a_nx^n+...+a_1x+a_0.
P
(
x
)
=
a
n
x
n
+
...
+
a
1
x
+
a
0
.
Assume that there exists a prime number
p
p
p
satisfying these conditions: i)
p
∣
a
i
p|a_i
p
∣
a
i
for all
0
≤
i
<
n
,
0\le i<n,
0
≤
i
<
n
,
ii)
p
∤
a
n
,
p\nmid a_n,
p
∤
a
n
,
iii)
p
2
∤
a
0
.
p^2\nmid a_0.
p
2
∤
a
0
.
Prove that
P
(
x
)
P(x)
P
(
x
)
is irreducible in
Z
[
x
]
.
\mathbb{Z}[x].
Z
[
x
]
.
Problem 2
2
Hide problems
3 symmedians are concurrent
Prove that:
3
3
3
symmedians of a triangle are concurrent at a point; the concurrent point is called the Lemoine point of the given triangle.
original polynomial
a) Given a prime number
p
p
p
and
2
2
2
polynomials
P
(
x
)
=
a
n
x
n
+
.
.
.
+
a
1
x
+
a
0
;
Q
(
x
)
=
b
m
x
m
+
.
.
.
+
b
1
x
+
b
0
.
P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0.
P
(
x
)
=
a
n
x
n
+
...
+
a
1
x
+
a
0
;
Q
(
x
)
=
b
m
x
m
+
...
+
b
1
x
+
b
0
.
We know that the product
P
(
x
)
Q
(
x
)
P(x)Q(x)
P
(
x
)
Q
(
x
)
is a polynomial whose coefficents are all divisible by
p
.
p.
p
.
Prove that: at least
1
1
1
in
2
2
2
polynomials
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
has all coefficents are all divisible by
p
.
p.
p
.
b) Prove that the product of
2
2
2
original polynomials is a original polynomial.
Problem 1
2
Hide problems
symmedian ratio
Given a triangle
A
B
C
ABC
A
BC
with
A
D
AD
A
D
is the
A
−
A-
A
−
symmedian
(
D
(D
(
D
is on the side
B
C
)
.
BC).
BC
)
.
Prove that:
D
B
D
C
=
A
B
2
A
C
2
.
\dfrac{DB}{DC}=\dfrac{AB^2}{AC^2}.
D
C
D
B
=
A
C
2
A
B
2
.
(x-1)(x-2)(x-3)-1 is irreducible in Z[x]
Prove that the polynomial
P
(
x
)
=
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
−
1
P(x)=(x-1)(x-2)(x-3)-1
P
(
x
)
=
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
−
1
is irreducible in
Z
[
x
]
.
\mathbb{Z}[x].
Z
[
x
]
.
Problem 9
2
Hide problems
nice problem about 2 Lemoine points, concurrent
Given a quadrilateral
A
B
C
D
ABCD
A
BC
D
inscribed in
(
O
)
(O)
(
O
)
. Let
L
,
J
L, J
L
,
J
be the Lemoine point of
△
A
B
C
\triangle ABC
△
A
BC
and
△
A
C
D
\triangle ACD
△
A
C
D
.Prove that:
A
C
,
B
D
,
L
J
AC, BD, LJ
A
C
,
B
D
,
L
J
are concurrent.
(x(x+1))^{2^{2023}}+1is irreducible in Q[x]
Prove that: the polynomial
(
x
(
x
+
1
)
)
2
2023
+
1
(x(x+1))^{2^{2023}}+1
(
x
(
x
+
1
)
)
2
2023
+
1
is irreducible in
Q
[
x
]
.
\mathbb{Q}[x].
Q
[
x
]
.