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Contests
National and Regional Contests
Vietnam Contests
VMEO = Vietnam Mathematical E-Olympiad
VMEO IV 2015
VMEO IV 2015
Part of
VMEO = Vietnam Mathematical E-Olympiad
Subcontests
(12)
12.1
2
Hide problems
(a - b)/(a + b) is rational if a^3 + b^3 + c^3 - 3abc is rational
Given a set
S
⊂
R
+
S \subset R^+
S
⊂
R
+
,
S
≠
∅
S \ne \emptyset
S
=
∅
such that for all
a
,
b
,
c
∈
S
a, b, c \in S
a
,
b
,
c
∈
S
(not necessarily distinct) then
a
3
+
b
3
+
c
3
−
3
a
b
c
a^3 + b^3 + c^3 - 3abc
a
3
+
b
3
+
c
3
−
3
ab
c
is rational number. Prove that for all
a
,
b
∈
S
a, b \in S
a
,
b
∈
S
then
a
−
b
a
+
b
\frac{a - b}{a + b}
a
+
b
a
−
b
is also rational.
a^2+b^2+c^2-ab-bc-ca>= k |(a^3-b^3)/(a+b)+(b^3-c^3)/(b+c)+({c^3-a^3)/(c+a)
Find the largest constant
k
k
k
such that the inequality
a
2
+
b
2
+
c
2
−
a
b
−
b
c
−
c
a
≥
k
∣
a
3
−
b
3
a
+
b
+
b
3
−
c
3
b
+
c
+
c
3
−
a
3
c
+
a
∣
a^2+b^2+c^2-ab-bc-ca \ge k \left|\frac{a^3-b^3}{a+b}+\frac{b^3-c^3}{b+c}+\frac{c^3-a^3}{c+a}\right|
a
2
+
b
2
+
c
2
−
ab
−
b
c
−
c
a
≥
k
a
+
b
a
3
−
b
3
+
b
+
c
b
3
−
c
3
+
c
+
a
c
3
−
a
3
holds for any for non negative real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
with
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
>
0
(a+b)(b+c)(c+a)>0
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
>
0
.
12.2
2
Hide problems
ST // BC , circumcircle, point on angle bisector
Given a triangle
A
B
C
ABC
A
BC
inscribed in circle
(
O
)
(O)
(
O
)
and let
P
P
P
be a point on the interior angle bisector of
B
A
C
BAC
B
A
C
.
P
B
PB
PB
,
P
C
PC
PC
cut
C
A
CA
C
A
,
A
B
AB
A
B
at
E
,
F
E,F
E
,
F
respectively. Let
E
F
EF
EF
meet
(
O
)
(O)
(
O
)
at
M
,
N
M,N
M
,
N
. The line that is perpendicular to
P
M
PM
PM
,
P
N
PN
PN
at
M
,
N
M,N
M
,
N
respectively intersect
(
O
)
(O)
(
O
)
at
S
,
T
S, T
S
,
T
different from
M
,
N
M,N
M
,
N
. Prove that
S
T
∥
B
C
ST \parallel BC
ST
∥
BC
.
n has at least k distinct prime divisors, n | 2^{\sigma(n)}-1
Given a positive integer
k
k
k
. Prove that there are infinitely many positive integers
n
n
n
satisfy the following conditions at the same time: a)
n
n
n
has at least
k
k
k
distinct prime divisors b) All prime divisors other than
3
3
3
of
n
n
n
have the form
4
t
+
1
4t+1
4
t
+
1
, with
t
t
t
some positive integer. c)
n
∣
2
σ
(
n
)
−
1
n | 2^{\sigma(n)}-1
n
∣
2
σ
(
n
)
−
1
Here
σ
(
n
)
\sigma(n)
σ
(
n
)
demotes the sum of the positive integer divisors of
n
n
n
.
11.4
1
Hide problems
2015 students standing in rows, not all tell the truth
Students in a school are arranged in an order that when you count from left to right, there will be
n
n
n
students in the first row,
n
−
1
n-1
n
−
1
students in the second row,
n
−
2
n - 2
n
−
2
students in the third row,... until there is one student in the
n
n
n
th row. All the students face to the first row. For example, here is an arrangement for
n
=
5
n = 5
n
=
5
, where each
∗
*
∗
represents one student:
∗
*
∗
∗
∗
* *
∗
∗
∗
∗
∗
* * *
∗
∗
∗
∗
∗
∗
∗
* * * *
∗
∗
∗
∗
∗
∗
∗
∗
∗
* * * * *
∗
∗
∗
∗
∗
( first row)Each student will pick one of two following statement (except the student standing at the beginning of the row): i) The guy before me is telling the truth, while the guy standing next to him on the left is lying. ii) The guy before me is lying, while the guy standing next to him on the left is telling the truth.For
n
=
2015
n = 2015
n
=
2015
, find the maximum number of students telling the truth. (A student is lying if what he said is not true. Otherwise, he is telling the truth.)
11.2
2
Hide problems
isosceles wanted, isosceles 80-80-20 and 50-50-80 , equilateral
Given an isosceles triangle
B
A
C
BAC
B
A
C
with vertex angle
∠
B
A
C
=
2
0
o
\angle BAC =20^o
∠
B
A
C
=
2
0
o
. Construct an equilateral triangle
B
D
C
BDC
B
D
C
such that
D
,
A
D,A
D
,
A
are on the same side wrt
B
C
BC
BC
. Construct an isosceles triangle
D
E
B
DEB
D
EB
with vertex angle
∠
E
D
B
=
8
0
o
\angle EDB = 80^o
∠
E
D
B
=
8
0
o
and
C
,
E
C,E
C
,
E
are on the different sides wrt
D
B
DB
D
B
. Prove that the triangle
A
E
C
AEC
A
EC
is isosceles at
E
E
E
.
UK has a fixed point when isogonal points P ,Q move , many projections
Let
A
B
C
ABC
A
BC
be a triangle with two isogonal points
P
P
P
and
Q
Q
Q
. Let
D
,
E
D, E
D
,
E
be the projection of
P
P
P
on
A
B
AB
A
B
,
A
C
AC
A
C
.
G
G
G
is the projection of
Q
Q
Q
on
B
C
BC
BC
.
U
U
U
is the projection of
G
G
G
on
D
E
DE
D
E
,
L
L
L
is the projection of
P
P
P
on
A
Q
AQ
A
Q
,
K
K
K
is the symmetric of
L
L
L
wrt
U
G
UG
U
G
. Prove that
U
K
UK
U
K
passes through a fixed point when
P
P
P
and
Q
Q
Q
vary.
10.3
2
Hide problems
diophantine a^2 + b^2 + c^2 =3(ab + bc + ca)
Find all triples of integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
satisfying
a
2
+
b
2
+
c
2
=
3
(
a
b
+
b
c
+
c
a
)
.
a^2 + b^2 + c^2 =3(ab + bc + ca).
a
2
+
b
2
+
c
2
=
3
(
ab
+
b
c
+
c
a
)
.
n^m | 5^{n^k} + 1
Given a positive integer
k
k
k
. Find the condition of positive integer
m
m
m
over
k
k
k
such that there exists only one positive integer
n
n
n
satisfying
n
m
∣
5
n
k
+
1
,
n^m | 5^{n^k} + 1,
n
m
∣
5
n
k
+
1
,
10.1
2
Hide problems
rationals wanted, a\alpha^2 + b\alpha+ c = \sqrt{d} if \alpha^3 = \alpha + 1
Given a real number
α
\alpha
α
satisfying
α
3
=
α
+
1
\alpha^3 = \alpha + 1
α
3
=
α
+
1
. Determine all
4
4
4
-tuples of rational numbers
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
satisfying:
a
α
2
+
b
α
+
c
=
d
.
a\alpha^2 + b\alpha+ c = \sqrt{d}.
a
α
2
+
b
α
+
c
=
d
.
limit of S_n where a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2
Where
n
n
n
is a positive integer, the sequence
a
n
a_n
a
n
is determined by the formula
a
n
+
1
=
1
a
1
+
a
2
+
.
.
.
+
a
n
−
2
,
a
1
=
1.
a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2, \,a_1 = 1.
a
n
+
1
=
a
1
+
a
2
+
...
+
a
n
1
−
2
,
a
1
=
1.
Find the limit of the sequence
S
n
S_n
S
n
defined by
S
n
=
a
1
+
a
2
+
.
.
.
+
a
n
S_n=a_1 + a_2 +... + a_n
S
n
=
a
1
+
a
2
+
...
+
a
n
.
10.4
2
Hide problems
numbers 1 to n^2 in a nxn board, prison break 4
In the movie ”Prison break
4
4
4
”. Michael Scofield has to break into The Company. There, he encountered a kind of code to protect Scylla from being taken away. This code require picking out every number in a
2015
×
2015
2015\times 2015
2015
×
2015
grid satisfying: i) The number right above of this number is
≡
1
m
o
d
2
\equiv 1 \mod 2
≡
1
mod
2
ii) The number right on the right of this number is
≡
2
m
o
d
3
\equiv 2 \mod 3
≡
2
mod
3
iii) The number right below of this number is
≡
3
m
o
d
4
\equiv 3 \mod 4
≡
3
mod
4
iv) The number right on the right of this number is
≡
4
m
o
d
5
\equiv 4 \mod 5
≡
4
mod
5
. How many number does Schofield have to choose? Also, in a
n
×
n
n\times n
n
×
n
grid, the numbers from
1
1
1
to
n
2
n^2
n
2
are arranged in the following way : On the first row, the numbers are written in an ascending order
1
,
2
,
3
,
4
,
.
.
.
,
n
1, 2, 3, 4, ..., n
1
,
2
,
3
,
4
,
...
,
n
, each cell has one number. On the second row, the number are written in descending order
2
n
,
2
n
−
1
,
2
n
−
2
,
.
.
.
,
n
+
1
2n, 2n -1, 2n- 2, ..., n + 1
2
n
,
2
n
−
1
,
2
n
−
2
,
...
,
n
+
1
. On the third row, it is ascending order again
2
n
+
1
,
2
n
+
2
,
.
.
.
,
3
n
2n + 1, 2n + 2, ..., 3n
2
n
+
1
,
2
n
+
2
,
...
,
3
n
. The numbers are written like that until
n
n
n
th row. For example, this is how a
3
3
3
×
\times
×
3
3
3
board looks like:https://cdn.artofproblemsolving.com/attachments/8/7/0a5c8aba6543fd94fd24ae4b9a30ef8a32d3bd.png
VMEO IV October P4
Let
n
∈
Z
+
n\in\mathbb{Z}^+
n
∈
Z
+
. Arrange
n
n
n
students
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
on a circle such that the distances between them are.equal. They each receives a number of candies such that the total amount of candies is
m
≥
n
m\geq n
m
≥
n
. A configuration is called balance if for an arbitrary student
A
i
A_i
A
i
, there will always be a regular polygon taking
A
i
A_i
A
i
as one of its vertices, and every student standing at the vertices of this polygon has an equal number of candies.a) Given
n
n
n
, find the least
m
m
m
such that we can create a balance configuration.b) In a move, a student can give a candy to the student standing next to him (no matter left or right) on one condition that the receiver has less candies than the giver. Prove that if
n
n
n
is the product of at most
2
2
2
prime numbers and
m
m
m
satisfies the condition in a), then no matter how we distribute the candies at the beginning, one can always create a balance configuration after a finite number of moves.
11.1
2
Hide problems
Integral points
On Cartesian plane, given a line defined by
y
=
x
+
1
2
y=x+\frac{1}{\sqrt{2}}
y
=
x
+
2
1
. a) Prove that every circle has center
I
∈
d
I\in d
I
∈
d
and radius is
1
8
\frac{1}{8}
8
1
has no integral point inside. b) Find the greatest
k
>
0
k>0
k
>
0
such that the distance of every integral points to
d
d
d
is greater or equal than
k
k
k
.
a^2 + b^2 + c^2 <= (k^2 + 1)(ab + bc + ca) if a/b+b/c+ c/a= (k + 1)^2 + 2/(k+1)
Let
k
≥
0
k \ge 0
k
≥
0
and
a
,
b
,
c
a, b, c
a
,
b
,
c
be three positive real numbers such that
a
b
+
b
c
+
c
a
=
(
k
+
1
)
2
+
2
k
+
1
.
\frac{a}{b}+\frac{b}{c}+ \frac{c}{a}= (k + 1)^2 + \frac{2}{k+ 1}.
b
a
+
c
b
+
a
c
=
(
k
+
1
)
2
+
k
+
1
2
.
Prove that
a
2
+
b
2
+
c
2
≤
(
k
2
+
1
)
(
a
b
+
b
c
+
c
a
)
.
a^2 + b^2 + c^2 \le (k^2 + 1)(ab + bc + ca).
a
2
+
b
2
+
c
2
≤
(
k
2
+
1
)
(
ab
+
b
c
+
c
a
)
.
12.3
2
Hide problems
VMEO IV P3/12 - Junior Division
Find all integes
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
that form an arithmetic progression satisfying
d
−
c
+
1
d-c+1
d
−
c
+
1
is prime number and
a
+
b
2
+
c
3
=
d
2
b
a+b^2+c^3=d^2b
a
+
b
2
+
c
3
=
d
2
b
PA=PL wanted, circumcircle, symmetrics, projection,
Triangle
A
B
C
ABC
A
BC
is inscribed in circle
(
O
)
(O)
(
O
)
.
P
P
P
is a point on arc
B
C
BC
BC
that does not contain
A
A
A
such that
A
P
AP
A
P
is the symmedian of triangle
A
B
C
ABC
A
BC
.
E
,
F
E ,F
E
,
F
are symmetric of
P
P
P
wrt
C
A
,
A
B
CA, AB
C
A
,
A
B
respectively .
K
K
K
is symmetric of
A
A
A
wrt
E
F
EF
EF
.
L
L
L
is the projection of
K
K
K
on the line passing through
A
A
A
and parallel to
B
C
BC
BC
. Prove that
P
A
=
P
L
PA=PL
P
A
=
P
L
.
12.4
2
Hide problems
VMEO IV P4/12 - Junior Division
Six mathematician sit around a round table. Each of them has a number and they do the following transformation: Each time, two mathematician sitting next to each other is chosen, they will add
1
1
1
to their own number. Is it possible to make all the six numbers equal if the initial numbers are a) 6,5,4,3,2,1 b) 7,5,3,2,1,4
S_n=\sum_{k=1}^{2^n}T(K), where T(k) = number of 11 in binary repres. of k
We call the tribi of a positive integer
k
k
k
(denoted
T
(
k
)
T(k)
T
(
k
)
) the number of all pairs
11
11
11
in the binary representation of
k
k
k
. e.g
T
(
1
)
=
T
(
2
)
=
0
,
T
(
3
)
=
1
,
T
(
4
)
=
T
(
5
)
=
0
,
T
(
6
)
=
1
,
T
(
7
)
=
2.
T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.
T
(
1
)
=
T
(
2
)
=
0
,
T
(
3
)
=
1
,
T
(
4
)
=
T
(
5
)
=
0
,
T
(
6
)
=
1
,
T
(
7
)
=
2.
Calculate
S
n
=
∑
k
=
1
2
n
T
(
K
)
S_n=\sum_{k=1}^{2^n}T(K)
S
n
=
∑
k
=
1
2
n
T
(
K
)
.
11.3
2
Hide problems
numbers less than 2015 divisible by [ \sqrt[3]{n} ]
How many natural number
n
n
n
less than
2015
2015
2015
that is divisible by
⌊
n
3
⌋
\lfloor\sqrt[3]{n}\rfloor
⌊
3
n
⌋
?
VMEO IV p3/11
Find all positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying
(
a
,
b
)
=
(
b
,
c
)
=
(
c
,
a
)
=
1
(a,b)=(b,c)=(c,a)=1
(
a
,
b
)
=
(
b
,
c
)
=
(
c
,
a
)
=
1
and
{
a
2
+
b
∣
b
2
+
c
b
2
+
c
∣
c
2
+
a
\begin{cases} a^2+b\mid b^2+c\\ b^2+c\mid c^2+a \end{cases}
{
a
2
+
b
∣
b
2
+
c
b
2
+
c
∣
c
2
+
a
and none of prime divisors of
a
2
+
b
a^2+b
a
2
+
b
are congruent to
1
1
1
modulo
7
7
7
10.2
2
Hide problems
DE = DF if <BEH = <C and <CFH =<B
Given a triangle
A
B
C
ABC
A
BC
with obtuse
∠
A
\angle A
∠
A
and attitude
A
H
AH
A
H
with
H
∈
B
C
H \in BC
H
∈
BC
. Let
E
,
F
E,F
E
,
F
on
C
A
CA
C
A
,
A
B
AB
A
B
satisfying
∠
B
E
H
=
∠
C
\angle BEH = \angle C
∠
BE
H
=
∠
C
and
∠
C
F
H
=
∠
B
\angle CFH = \angle B
∠
CF
H
=
∠
B
. Let
B
E
BE
BE
cut
C
F
CF
CF
at
D
D
D
. Prove that
D
E
=
D
F
DE = DF
D
E
=
D
F
.
Isogonal conjugate problem
Given triangle
A
B
C
ABC
A
BC
and
P
,
Q
P,Q
P
,
Q
are two isogonal conjugate points in
△
A
B
C
\triangle ABC
△
A
BC
.
A
P
,
A
Q
AP,AQ
A
P
,
A
Q
intersects
(
Q
B
C
)
(QBC)
(
QBC
)
and
(
P
B
C
)
(PBC)
(
PBC
)
at
M
,
N
M,N
M
,
N
, respectively (
M
,
N
M,N
M
,
N
be inside triangle
A
B
C
ABC
A
BC
) 1. Prove that
M
,
N
,
P
,
Q
M,N,P,Q
M
,
N
,
P
,
Q
locate on a circle - named
(
I
)
(I)
(
I
)
2.
M
N
∩
P
Q
MN\cap PQ
MN
∩
PQ
at
J
J
J
. Prove that
I
J
IJ
I
J
passed through a fixed line when
P
,
Q
P,Q
P
,
Q
changed