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Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2017 Saudi Arabia Pre-TST + Training Tests
2017 Saudi Arabia Pre-TST + Training Tests
Part of
Saudi Arabia Pre-TST + Training Tests
Subcontests
(9)
8
1
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2017 points on the plane, n segments
There are
2017
2017
2017
points on the plane, no three of them are collinear. Some pairs of the points are connected by
n
n
n
segments. Find the smallest value of
n
n
n
so that there always exists two disjoint segments in any case.
7
1
Hide problems
y^3 = 8x^6 + 2x^3 y -y^2
Find all pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
y
3
=
8
x
6
+
2
x
3
y
−
y
2
y^3 = 8x^6 + 2x^3 y -y^2
y
3
=
8
x
6
+
2
x
3
y
−
y
2
.
6
1
Hide problems
convex polygon is divided into some triangles, set of vertices and edges
A convex polygon is divided into some triangles. Let
V
V
V
and
E
E
E
be respectively the set of vertices and the set of egdes of all triangles (each vertex in
V
V
V
may be some vertex of the polygon or some point inside the polygon). The polygon is said to be good if the following conditions hold: i. There are no
3
3
3
vertices in
V
V
V
which are collinear. ii. Each vertex in
V
V
V
belongs to an even number of edges in
E
E
E
. Find all good polygon.
4
1
Hide problems
n roots of polynomial form an arithmetic sequence
Does there exist an integer
n
≥
3
n \ge 3
n
≥
3
and an arithmetic sequence
a
0
,
a
1
,
.
.
.
,
a
n
a_0, a_1, ... , a_n
a
0
,
a
1
,
...
,
a
n
such that the polynomial
a
n
x
n
+
.
.
.
+
a
1
x
+
a
0
a_nx^n +... + a_1x + a_0
a
n
x
n
+
...
+
a
1
x
+
a
0
has
n
n
n
roots which also form an arithmetic sequence?
2
1
Hide problems
4950 ants, n groups, and bosses of ants
There are
4950
4950
4950
ants. Assume that, for any three ants
A
,
B
A, B
A
,
B
and
C
C
C
, if the ant
A
A
A
is the boss of the ant
B
B
B
, and the ant
B
B
B
is the boss of the ant
C
C
C
then the ant
A
A
A
is also the boss of the ant
C
C
C
. We want to divide the ants into
n
n
n
groups so that in any group, either any two ants have the boss relationship or any two ants do not have the boss relationship. Find the smallest of
n
n
n
we can always do in any case.
1
1
Hide problems
n^k + mn^l + 1 divides n^{k+l }- 1
Let
m
,
n
,
k
m, n, k
m
,
n
,
k
and
l
l
l
be positive integers with
n
≠
1
n \ne 1
n
=
1
such that
n
k
+
m
n
l
+
1
n^k + mn^l + 1
n
k
+
m
n
l
+
1
divides
n
k
+
l
−
1
n^{k+l }- 1
n
k
+
l
−
1
. Prove that either
m
=
1
m = 1
m
=
1
and
l
=
2
k
l = 2k
l
=
2
k
, or
l
∣
k
l | k
l
∣
k
and
m
=
n
k
−
l
−
1
n
l
−
1
m =\frac{n^{k-l} - 1}{n^l - 1}
m
=
n
l
−
1
n
k
−
l
−
1
.
9
1
Hide problems
right angle wanted, projection of circumcenter on symmedian related
Let
A
B
C
ABC
A
BC
be a triangle inscribed in circle
(
O
)
(O)
(
O
)
, with its altitudes
B
H
b
,
C
H
c
BH_b, CH_c
B
H
b
,
C
H
c
intersect at orthocenter
H
H
H
(
H
b
∈
A
C
H_b \in AC
H
b
∈
A
C
,
H
c
∈
A
B
H_c \in AB
H
c
∈
A
B
).
H
b
H
c
H_bH_c
H
b
H
c
meets
B
C
BC
BC
at
P
P
P
. Let
N
N
N
be the midpoint of
A
H
,
L
AH, L
A
H
,
L
be the orthogonal projection of
O
O
O
on the symmedian with respect to angle
A
A
A
of triangle
A
B
C
ABC
A
BC
. Prove that
∠
N
L
P
=
9
0
o
\angle NLP = 90^o
∠
N
L
P
=
9
0
o
.
5
1
Hide problems
tangent circumcircles wanted
Let
A
B
C
ABC
A
BC
be an acute triangle inscribed in circle
(
O
)
(O)
(
O
)
, with orthocenter
H
H
H
. Median
A
M
AM
A
M
of triangle
A
B
C
ABC
A
BC
intersects circle
(
O
)
(O)
(
O
)
at
A
A
A
and
N
N
N
.
A
H
AH
A
H
intersects
(
O
)
(O)
(
O
)
at
A
A
A
and
K
K
K
. Three lines
K
N
,
B
C
KN, BC
K
N
,
BC
and line through
H
H
H
and perpendicular to
A
N
AN
A
N
intersect each other and form triangle
X
Y
Z
X Y Z
X
Y
Z
. Prove that the circumcircle of triangle
X
Y
Z
X Y Z
X
Y
Z
is tangent to
(
O
)
(O)
(
O
)
.
3
1
Hide problems
tangents circumcircles wanted and given (iff condition)
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Ray
A
D
AD
A
D
meets ray
B
C
BC
BC
at
P
P
P
. Let
O
,
O
′
O,O'
O
,
O
′
be the circumcenters of triangles
P
C
D
,
P
A
B
PCD, PAB
PC
D
,
P
A
B
, respectively,
H
,
H
′
H,H'
H
,
H
′
be the orthocenters of triangles
P
C
D
,
P
A
B
PCD, PAB
PC
D
,
P
A
B
, respectively. Prove that circumcircle of triangle
D
O
C
DOC
D
OC
is tangent to circumcircle of triangle
A
O
′
B
AO'B
A
O
′
B
if and only if circumcircle of triangle
D
H
C
DHC
DH
C
is tangent to circumcircle of triangle
A
H
′
B
AH'B
A
H
′
B
.