MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia GMO TST
2018 Saudi Arabia GMO TST
2018 Saudi Arabia GMO TST
Part of
Saudi Arabia GMO TST
Subcontests
(4)
3
2
Hide problems
circumcenter of XYZ = circumcenter of A_1B_1C_1 iff incenter = orthocenter
Let
I
,
O
I, O
I
,
O
be the incenter, circumcenter of triangle
A
B
C
ABC
A
BC
and
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
be arbitrary points on the segments
A
I
,
B
I
,
C
I
AI, BI, CI
A
I
,
B
I
,
C
I
respectively. The perpendicular bisectors of
A
A
1
,
B
B
1
,
C
C
1
AA_1, BB_1, CC_1
A
A
1
,
B
B
1
,
C
C
1
intersect each other at
X
,
Y
X, Y
X
,
Y
and
Z
Z
Z
. Prove that the circumcenter of triangle
X
Y
Z
XYZ
X
Y
Z
coincides with
O
O
O
if and only if
I
I
I
is the orthocenter of triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
perpendicular wanted, arc midpoint, tangent lines to circles
Let
C
C
C
be a point lies outside the circle
(
O
)
(O)
(
O
)
and
C
S
,
C
T
CS, CT
CS
,
CT
are tangent lines of
(
O
)
(O)
(
O
)
. Take two points
A
,
B
A, B
A
,
B
on
(
O
)
(O)
(
O
)
with
M
M
M
is the midpoint of the minor arc
A
B
AB
A
B
such that
A
,
B
,
M
A, B, M
A
,
B
,
M
differ from
S
,
T
S, T
S
,
T
. Suppose that
M
S
,
M
T
MS, MT
MS
,
MT
cut line
A
B
AB
A
B
at
E
,
F
E, F
E
,
F
. Take
X
∈
O
S
X \in OS
X
∈
OS
and
Y
∈
O
T
Y \in OT
Y
∈
OT
such that
E
X
,
F
Y
EX, FY
EX
,
F
Y
are perpendicular to
A
B
AB
A
B
. Prove that
X
Y
X Y
X
Y
and
C
M
C M
CM
are perpendicular.
1
2
Hide problems
1 -1/2^{2^{n-1}}<sum 1/x+i<1/2^{2^{n}} if x_{n+1} = x_n^2 - x_n + 1, x_1 = 2
Let
{
x
n
}
\{x_n\}
{
x
n
}
be a sequence defined by
x
1
=
2
x_1 = 2
x
1
=
2
and
x
n
+
1
=
x
n
2
−
x
n
+
1
x_{n+1} = x_n^2 - x_n + 1
x
n
+
1
=
x
n
2
−
x
n
+
1
for
n
≥
1
n \ge 1
n
≥
1
. Prove that
1
−
1
2
2
n
−
1
<
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
<
1
−
1
2
2
n
1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}}
1
−
2
2
n
−
1
1
<
x
1
1
+
x
2
1
+
...
+
x
n
1
<
1
−
2
2
n
1
for all
n
n
n
gcd(a_1^n + a-1a_2 ...a_n, a_2^n +a_1a_2 ...a_n, ... , a_n^n +a_1a_2 ...a_n)
Let
n
n
n
be an odd positive integer with
n
>
1
n > 1
n
>
1
and let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,... , a_n
a
1
,
a
2
,
...
,
a
n
be positive integers such that gcd
(
a
1
,
a
2
,
.
.
.
,
a
n
)
=
1
(a_1, a_2,... , a_n) = 1
(
a
1
,
a
2
,
...
,
a
n
)
=
1
. Let
d
d
d
= gcd
(
a
1
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
,
a
2
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
,
.
.
.
,
a
n
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
)
(a_1^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, a_2^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, ... , a_n^n + a_1\cdot a_2 \cdot \cdot \cdot a_n)
(
a
1
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
,
a
2
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
,
...
,
a
n
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
)
. Show that the possible values of
d
d
d
are
d
=
1
,
d
=
2
d = 1, d = 2
d
=
1
,
d
=
2
4
2
Hide problems
two 2s and two 24s on a 13 x13 board , integers in adjacent cells differ by 1
In each of the cells of a
13
×
13
13 \times 13
13
×
13
board is written an integer such that the integers in adjacent cells differ by
1
1
1
. If there are two
2
2
2
s and two
24
24
24
s on this board, how many
13
13
13
s can there be?
max edges in a graph with 8 vertices that contains no cycle of length 4
In a graph with
8
8
8
vertices that contains no cycle of length
4
4
4
, at most how many edges can there be?
2
2
Hide problems
exists integer n such that p | n^3 - 3n + 1 where p a prime of form 9k + 1
Let
p
p
p
be a prime number of the form
9
k
+
1
9k + 1
9
k
+
1
. Show that there exists an integer n such that
p
∣
n
3
−
3
n
+
1
p | n^3 - 3n + 1
p
∣
n
3
−
3
n
+
1
.
similar integer, swapping 2 digits
Two positive integers
m
m
m
and
n
n
n
are called similar if one of them can be obtained from the other one by swapping two digits (note that a
0
0
0
-digit cannot be swapped with the leading digit). Find the greatest integer
N
N
N
such that N is divisible by
13
13
13
and any number similar to
N
N
N
is not divisible by
13
13
13
.