MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia GMO TST
2013 Saudi Arabia GMO TST
2013 Saudi Arabia GMO TST
Part of
Saudi Arabia GMO TST
Subcontests
(4)
4
3
Hide problems
perpenducilar wanted, circle with diamater an altitude related
In acute triangle
A
B
C
ABC
A
BC
, points
D
D
D
and
E
E
E
are the feet of the perpendiculars from
A
A
A
to
B
C
BC
BC
and
B
B
B
to
C
A
CA
C
A
, respectively. Segment
A
D
AD
A
D
is a diameter of circle
ω
\omega
ω
. Circle
ω
\omega
ω
intersects sides
A
C
AC
A
C
and
A
B
AB
A
B
at
F
F
F
and
G
G
G
(other than
A
A
A
), respectively. Segment
B
E
BE
BE
intersects segments
G
D
GD
G
D
and
G
F
GF
GF
at
X
X
X
and
Y
Y
Y
respectively. Ray
D
Y
DY
D
Y
intersects side
A
B
AB
A
B
at
Z
Z
Z
. Prove that lines
X
Z
XZ
XZ
and
B
C
BC
BC
are perpendicular
F_n + 2 = F_{n+1} + 1 = F_{n+2} mod m, Fibonacci
Let
F
0
=
0
,
F
1
=
1
F_0 = 0, F_1 = 1
F
0
=
0
,
F
1
=
1
and
F
n
+
1
=
F
n
+
F
n
−
1
F_{n+1} = F_n + F_{n-1}
F
n
+
1
=
F
n
+
F
n
−
1
, for all positive integer
n
n
n
, be the Fibonacci sequence. Prove that for any positive integer
m
m
m
there exist infinitely many positive integers
n
n
n
such that
F
n
+
2
≡
F
n
+
1
+
1
≡
F
n
+
2
F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2}
F
n
+
2
≡
F
n
+
1
+
1
≡
F
n
+
2
mod
m
m
m
.
a^2 + b^2 divides both a^3 + 1 and b^3 + 1
Find all pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
a
2
+
b
2
a^2 + b^2
a
2
+
b
2
divides both
a
3
+
1
a^3 + 1
a
3
+
1
and
b
3
+
1
b^3 + 1
b
3
+
1
.
3
3
Hide problems
there are exactly 29 non-similar regular n-pointed stars
Define a regular
n
n
n
-pointed star to be a union of
n
n
n
lines segments
P
1
P
2
,
P
2
P
3
,
.
.
.
,
P
n
P
1
P_1P_2, P_2P_3, ..., P_nP_1
P
1
P
2
,
P
2
P
3
,
...
,
P
n
P
1
such that
∙
\bullet
∙
the points
P
1
,
P
2
,
.
.
.
,
P
n
P_1,P_2,...,P_n
P
1
,
P
2
,
...
,
P
n
are coplanar and no three of them are collinear,
∙
\bullet
∙
each of the
n
n
n
line segments intersects at least one of the other line segments at a point other than an endpoint,
∙
\bullet
∙
all of the angles at
P
1
,
P
2
,
.
.
.
,
P
n
P_1, P_2,..., P_n
P
1
,
P
2
,
...
,
P
n
are congruent ,
∙
\bullet
∙
all of the
n
n
n
line segments
P
1
P
2
,
P
2
P
3
,
.
.
.
,
P
n
P
1
P_1P_2, P_2P_3, ..., P_nP_1
P
1
P
2
,
P
2
P
3
,
...
,
P
n
P
1
are congruent, and
∙
\bullet
∙
the path
P
1
P
2
.
.
.
P
n
P
1
P_1P_2...P_nP_1
P
1
P
2
...
P
n
P
1
turns counterclockwise at an angle less than
18
0
o
180^o
18
0
o
at each vertex. There are no regular
3
3
3
-pointed,
4
4
4
-pointed, or
6
6
6
-pointed stars. All regular
5
5
5
-pointed star are similar, but there are two non-similar regular
7
7
7
-pointed stars. Find all possible values of
n
n
n
such that there are exactly
29
29
29
non-similar regular
n
n
n
-pointed stars.
largest integer k such that k divides n^{55} - n
Find the largest integer
k
k
k
such that
k
k
k
divides
n
55
−
n
n^{55} - n
n
55
−
n
for all integer
n
n
n
.
angle chasing given AB = ID and AH = OH , orthocenter, incenter, circumcenter
A
B
C
ABC
A
BC
is a triangle,
H
H
H
its orthocenter,
I
I
I
its incenter,
O
O
O
its circumcenter and
ω
\omega
ω
its circumcircle. Line
C
I
CI
C
I
intersects circle
ω
\omega
ω
at point
D
D
D
different from
C
C
C
. Assume that
A
B
=
I
D
AB = ID
A
B
=
I
D
and
A
H
=
O
H
AH = OH
A
H
=
O
H
. Find the angles of triangle
A
B
C
ABC
A
BC
.
2
3
Hide problems
f(X) is irreducible. when p >\sum_{i=1}^n |a_i|
Let
f
(
X
)
=
a
n
X
n
+
a
n
−
1
X
n
−
1
+
.
.
.
+
a
1
X
+
p
f(X) = a_nX^n + a_{n-1}X^{n-1} + ...+ a_1X + p
f
(
X
)
=
a
n
X
n
+
a
n
−
1
X
n
−
1
+
...
+
a
1
X
+
p
be a polynomial of integer coefficients where
p
p
p
is a prime number. Assume that
p
>
∑
i
=
1
n
∣
a
i
∣
p >\sum_{i=1}^n |a_i|
p
>
∑
i
=
1
n
∣
a
i
∣
. Prove that
f
(
X
)
f(X)
f
(
X
)
is irreducible.
sum a^3/(a^2 + ab + b^2 ) >= (a + b + c)/3
For positive real numbers
a
,
b
a, b
a
,
b
and
c
c
c
, prove that
a
3
a
2
+
a
b
+
b
2
+
b
3
b
2
+
b
c
+
c
2
+
c
3
c
2
+
c
a
+
a
2
≥
a
+
b
+
c
3
\frac{a^3}{a^2 + ab + b^2} +\frac{b^3}{b^2 + bc + c^2} +\frac{c^3}{ c^2 + ca + a^2} \ge\frac{ a + b + c}{3}
a
2
+
ab
+
b
2
a
3
+
b
2
+
b
c
+
c
2
b
3
+
c
2
+
c
a
+
a
2
c
3
≥
3
a
+
b
+
c
convex cyclic non-regular polygon with all equal internal angles
Find all values of
n
n
n
for which there exists a convex cyclic non-regular polygon with
n
n
n
vertices such that the measures of all its internal angles are equal.
1
3
Hide problems
some distinct numbers from the set S = {2,...,111}
Tarik wants to choose some distinct numbers from the set
S
=
{
2
,
.
.
.
,
111
}
S = \{2,...,111\}
S
=
{
2
,
...
,
111
}
in such a way that each of the chosen numbers cannot be written as the product of two other distinct chosen numbers. What is the maximum number of numbers Tarik can choose ?
f(x)f(y) = f(xy) + f (x/y)
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
which satisfy
f
(
3
3
x
)
=
3
f
(
x
)
−
2
3
3
x
f \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x
f
(
3
3
x
)
=
3
f
(
x
)
−
3
2
3
x
and
f
(
x
)
f
(
y
)
=
f
(
x
y
)
+
f
(
x
y
)
f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right)
f
(
x
)
f
(
y
)
=
f
(
x
y
)
+
f
(
y
x
)
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
, with
y
≠
0
y \ne 0
y
=
0
AB = AC if AP = AQ, circumcircle related
An acute triangle
A
B
C
ABC
A
BC
is inscribed in circle
ω
\omega
ω
centered at
O
O
O
. Line
B
O
BO
BO
and side
A
C
AC
A
C
meet at
B
1
B_1
B
1
. Line
C
O
CO
CO
and side
A
B
AB
A
B
meet at
C
1
C_1
C
1
. Line
B
1
C
1
B_1C_1
B
1
C
1
meets circle
ω
\omega
ω
at
P
P
P
and
Q
Q
Q
. If
A
P
=
A
Q
AP = AQ
A
P
=
A
Q
, prove that
A
B
=
A
C
AB = AC
A
B
=
A
C
.