MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia GMO TST
2015 Saudi Arabia GMO TST
2015 Saudi Arabia GMO TST
Part of
Saudi Arabia GMO TST
Subcontests
(4)
3
3
Hide problems
concurrent wanted, symmetric of projections of vertices on sides wrt centroid
Let
A
B
C
ABC
A
BC
be a triangle and
G
G
G
its centroid. Let
G
a
,
G
b
G_a, G_b
G
a
,
G
b
and
G
c
G_c
G
c
be the orthogonal projections of
G
G
G
on sides
B
C
,
C
A
BC, CA
BC
,
C
A
, respectively
A
B
AB
A
B
. If
S
a
,
S
b
S_a, S_b
S
a
,
S
b
and
S
c
S_c
S
c
are the symmetrical points of
G
a
,
G
b
G_a, G_b
G
a
,
G
b
, respectively
G
c
G_c
G
c
with respect to
G
G
G
, prove that
A
S
a
,
B
S
b
AS_a, BS_b
A
S
a
,
B
S
b
and
C
S
c
CS_c
C
S
c
are concurrent. Liana Topan
perpendicular wanted, orthocenter, circumcenter, midpoints related
Let
B
D
BD
B
D
and
C
E
CE
CE
be altitudes of an arbitrary scalene triangle
A
B
C
ABC
A
BC
with orthocenter
H
H
H
and circumcenter
O
O
O
. Let
M
M
M
and
N
N
N
be the midpoints of sides
A
B
AB
A
B
, respectively
A
C
AC
A
C
, and
P
P
P
the intersection point of lines
M
N
MN
MN
and
D
E
DE
D
E
. Prove that lines
A
P
AP
A
P
and
O
H
OH
O
H
are perpendicular.Liana Topan
right triangle wanted, perpendicular given , symmetric of vertex wrt midpoint
Let
A
B
C
ABC
A
BC
be a triangle, with
A
B
<
A
C
AB < AC
A
B
<
A
C
,
D
D
D
the foot of the altitude from
A
,
M
A, M
A
,
M
the midpoint of
B
C
BC
BC
, and
B
′
B'
B
′
the symmetric of
B
B
B
with respect to
D
D
D
. The perpendicular line to
B
C
BC
BC
at
B
′
B'
B
′
intersects
A
C
AC
A
C
at point
P
P
P
. Prove that if
B
P
BP
BP
and
A
M
AM
A
M
are perpendicular then triangle
A
B
C
ABC
A
BC
is right-angled.Liana Topan
1
3
Hide problems
2 ( b/(a + b) + bc /(b + c)+ ca/(c+ a) )+ 1 >= 6(ab + bc + ca)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
. Prove that
2
(
a
b
a
+
b
+
b
c
b
+
c
+
c
a
c
+
a
)
+
1
≥
6
(
a
b
+
b
c
+
c
a
)
2 \left( \frac{ab}{a + b} +\frac{bc}{b + c} +\frac{ca}{c+ a}\right)+ 1 \ge 6(ab + bc + ca)
2
(
a
+
b
ab
+
b
+
c
b
c
+
c
+
a
c
a
)
+
1
≥
6
(
ab
+
b
c
+
c
a
)
Trần Nam Dũng
x_nx_{n+2} - x^2_{n+1} = 4^{n-1}.
Let be given the sequence
(
x
n
)
(x_n)
(
x
n
)
defined by
x
1
=
1
x_1 = 1
x
1
=
1
and
x
n
+
1
=
3
x
n
+
⌊
x
n
5
⌋
x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor
x
n
+
1
=
3
x
n
+
⌊
x
n
5
⌋
for all
n
=
1
,
2
,
3
,
.
.
.
,
n = 1,2,3,...,
n
=
1
,
2
,
3
,
...
,
where
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the greatest integer that does not exceed
x
x
x
. Prove that for any positive integer
n
n
n
we have
x
n
x
n
+
2
−
x
n
+
1
2
=
4
n
−
1
x_nx_{n+2} - x^2_{n+1} = 4^{n-1}
x
n
x
n
+
2
−
x
n
+
1
2
=
4
n
−
1
Trần Nam Dũng
f(1) = 1, f(x + y) = f(x) + f(y),f(1/x)= f(x)/x^2
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
satisfying the following conditions (a)
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
, (b)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f(x + y) = f(x) + f(y)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
,
∀
(
x
,
y
)
∈
R
2
\forall (x,y) \in R^2
∀
(
x
,
y
)
∈
R
2
(c)
f
(
1
x
)
=
f
(
x
)
x
2
f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }
f
(
x
1
)
=
x
2
f
(
x
)
,
∀
x
∈
R
−
{
0
}
\forall x \in R -\{0\}
∀
x
∈
R
−
{
0
}
Trần Nam Dũng
4
3
Hide problems
p^2 divides $ k(k + 1)(k + 2) ... (k + p - 3) - 1
Let
p
p
p
be an odd prime number. Prove that there exists a unique integer
k
k
k
such that
0
≤
k
≤
p
2
0 \le k \le p^2
0
≤
k
≤
p
2
and
p
2
p^2
p
2
divides
k
(
k
+
1
)
(
k
+
2
)
.
.
.
(
k
+
p
−
3
)
−
1
k(k + 1)(k + 2) ... (k + p - 3) - 1
k
(
k
+
1
)
(
k
+
2
)
...
(
k
+
p
−
3
)
−
1
.Malik Talbi
s(n) \ge n , binomial remainder
For each positive integer
n
n
n
, define
s
(
n
)
=
∑
k
=
0
n
r
k
s(n) =\sum_{k=0}^n r_k
s
(
n
)
=
∑
k
=
0
n
r
k
, where
r
k
r_k
r
k
is the remainder when
(
n
k
)
n \choose k
(
k
n
)
is divided by
3
3
3
. Find all positive integers
n
n
n
such that
s
(
n
)
≥
n
s(n) \ge n
s
(
n
)
≥
n
.Malik Talbi
if p^3q^3 divides n^{pq} + 1 then either p^2 divides n + 1 or q^2 divides n + 1
Let
p
,
q
p, q
p
,
q
be two different odd prime numbers and
n
n
n
an integer such that
p
q
pq
pq
divides
n
p
q
+
1
n^{pq} + 1
n
pq
+
1
. Prove that if
p
3
q
3
p^3q^3
p
3
q
3
divides
n
p
q
+
1
n^{pq} + 1
n
pq
+
1
then either
p
2
p^2
p
2
divides
n
+
1
n + 1
n
+
1
or
q
2
q^2
q
2
divides
n
+
1
n + 1
n
+
1
.Malik Talbi
2
3
Hide problems
Salman has a number of stones, weights
In his bag, Salman has a number of stones. The weight of each stone is not greater than
0.5
0.5
0.5
kg and the total weight of the stones is not greater than
2.5
2.5
2.5
kg. Prove that Salman can divide his stones into
4
4
4
groups, each group has a total weight not greater than
1
1
1
kgTrần Nam Dũng
no of strictly increasing sequences of nonnegative integers with ...
Find the number of strictly increasing sequences of nonnegative integers with the first term
0
0
0
and the last term
15
15
15
, and among any two consecutive terms, exactly one of them is even.Lê Anh Vinh
2005 Combinatorics #7: Bishops on a Diagonal
What is the maximum number of bishops that can be placed on an
8
×
8
8 \times 8
8
×
8
chessboard such that at most three bishops lie on any diagonal?