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Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia GMO TST
2014 Saudi Arabia GMO TST
2014 Saudi Arabia GMO TST
Part of
Saudi Arabia GMO TST
Subcontests
(4)
1
3
Hide problems
(con)cyclic wanted, projections of vertices on medians related
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
<
∠
B
≤
∠
C
\angle A < \angle B \le \angle C
∠
A
<
∠
B
≤
∠
C
,
M
M
M
and
N
N
N
the midpoints of sides
C
A
CA
C
A
and
A
B
AB
A
B
, respectively, and
P
P
P
and
Q
Q
Q
the projections of
B
B
B
and
C
C
C
on the medians
C
N
CN
CN
and
B
M
BM
BM
, respectively. Prove that the quadrilateral
M
N
P
Q
MNPQ
MNPQ
is cyclic.
AP x AQ = AB x AC , circle pasing through B,C of a ABC, angle bisectors
Let
A
,
B
,
C
A, B,C
A
,
B
,
C
be colinear points in this order,
ω
\omega
ω
an arbitrary circle passing through
B
B
B
and
C
C
C
, and
l
l
l
an arbitrary line different from
B
C
BC
BC
, passing through A and intersecting
ω
\omega
ω
at
M
M
M
and
N
N
N
. The bisectors of the angles
∠
C
M
B
\angle CMB
∠
CMB
and
∠
C
N
B
\angle CNB
∠
CNB
intersect
B
C
BC
BC
at
P
P
P
and
Q
Q
Q
. Prove that
A
P
⋅
A
Q
=
A
B
⋅
A
C
AP\cdot AQ = AB \cdot AC
A
P
⋅
A
Q
=
A
B
⋅
A
C
.
5^a + 3^b - 2^c = 32
Find all ordered triples
(
a
,
b
,
c
)
(a,b, c)
(
a
,
b
,
c
)
of positive integers which satisfy
5
a
+
3
b
−
2
c
=
32
5^a + 3^b - 2^c = 32
5
a
+
3
b
−
2
c
=
32
4
3
Hide problems
concyclic wanted, angle bisectors related
Let
A
B
C
ABC
A
BC
be a triangle,
D
D
D
the midpoint of side
B
C
BC
BC
and
E
E
E
the intersection point of the bisector of angle
∠
B
A
C
\angle BAC
∠
B
A
C
with side
B
C
BC
BC
. The perpendicular bisector of
A
E
AE
A
E
intersects the bisectors of angles
∠
C
B
A
\angle CBA
∠
CB
A
and
∠
C
D
A
\angle CDA
∠
C
D
A
at
M
M
M
and
N
N
N
, respectively. The bisectors of angles
∠
C
B
A
\angle CBA
∠
CB
A
and
∠
C
D
A
\angle CDA
∠
C
D
A
intersect at
P
P
P
. Prove that points
A
,
M
,
N
,
P
A, M, N, P
A
,
M
,
N
,
P
are concyclic.
a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n
Let
a
1
≥
a
2
≥
.
.
.
≥
a
n
>
0
a_1 \ge a_2 \ge ... \ge a_n > 0
a
1
≥
a
2
≥
...
≥
a
n
>
0
be real numbers. Prove that
a
1
a
2
(
a
1
−
a
2
)
+
a
2
a
3
(
a
2
−
a
3
)
+
.
.
.
+
a
n
−
1
a
n
(
a
n
−
1
−
a
n
)
≥
a
1
a
n
(
a
1
−
a
n
)
a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n(a_1 - a_n)
a
1
a
2
(
a
1
−
a
2
)
+
a
2
a
3
(
a
2
−
a
3
)
+
...
+
a
n
−
1
a
n
(
a
n
−
1
−
a
n
)
≥
a
1
a
n
(
a
1
−
a
n
)
|x- z|/ |y - z |= 2 or |y - z|/ |x- z |= 2
Let
X
X
X
be a set of rational numbers satisfying the following two conditions: (a) The set
X
X
X
contains at least two elements, (b) For any
x
,
y
x, y
x
,
y
in
X
X
X
, if
x
≠
y
x \ne y
x
=
y
then there exists
z
z
z
in
X
X
X
such that either
∣
x
−
z
y
−
z
∣
=
2
\left| \frac{x- z}{y - z} \right|= 2
y
−
z
x
−
z
=
2
or
∣
y
−
z
x
−
z
∣
=
2
\left| \frac{y -z}{x - z} \right|= 2
x
−
z
y
−
z
=
2
. Prove that
X
X
X
contains infinitely many elements.
2
3
Hide problems
p divides 1^n + 2^n +... + (p + 1)^n.
Let
p
p
p
be a prime number. Prove that there exist infinitely many positive integers
n
n
n
such that
p
p
p
divides
1
n
+
2
n
+
.
.
.
+
(
p
+
1
)
n
.
1^n + 2^n +... + (p + 1)^n.
1
n
+
2
n
+
...
+
(
p
+
1
)
n
.
a_p / b_p = 1 +1/2+ .. +1/(p^2 -1)
Let
p
≥
2
p \ge 2
p
≥
2
be a prime number and
a
p
b
p
=
1
+
1
2
+
.
.
+
1
p
2
−
1
\frac{a_p}{b_p}= 1 +\frac12+ .. +\frac{1}{p^2 -1}
b
p
a
p
=
1
+
2
1
+
..
+
p
2
−
1
1
, where
a
p
a_p
a
p
and
b
p
b_p
b
p
are two relatively prime positive integers. Compute gcd
(
p
,
b
p
)
(p, b_p)
(
p
,
b
p
)
.
integer polynomials p(X, Y ) with minimal degree
Let
S
=
{
f
(
a
,
b
)
∣
a
,
b
=
1
,
2
,
3
,
4
S = \{f(a, b) | a, b = 1,2,3, 4
S
=
{
f
(
a
,
b
)
∣
a
,
b
=
1
,
2
,
3
,
4
and
a
≠
b
}
a \ne b\}
a
=
b
}
, and consider all nonzero polynomials
p
(
X
,
Y
)
p(X,Y )
p
(
X
,
Y
)
with integer coefficients such that
p
(
a
,
b
)
=
0
p(a, b) = 0
p
(
a
,
b
)
=
0
for every element
(
a
,
b
)
(a,b)
(
a
,
b
)
in
S
S
S
. (a) What is the minimal degree of such polynomial
p
(
X
,
Y
)
p(X, Y )
p
(
X
,
Y
)
? (b) Determine all such polynomials
p
(
X
,
Y
)
p(X, Y )
p
(
X
,
Y
)
with minimal degree.
3
3
Hide problems
divide a square into finitely many white and green rectangles
Turki has divided a square into finitely many white and green rectangles, each with sides parallel to the sides of the square. Within each white rectangle, he writes down its width divided by its height. Within each green rectangle, he writes down its height divided by its width. Finally, he calculates
S
S
S
, the sum of these numbers. If the total area of white rectangles equals the total area of green rectangles, determine the minimum possible value of
S
S
S
.
concurrency wanted, circle concentric with incircle
Let
A
B
C
ABC
A
BC
be a triangle,
I
I
I
its incenter, and
ω
\omega
ω
a circle of center
I
I
I
. Points
A
′
,
B
′
,
C
′
A',B', C'
A
′
,
B
′
,
C
′
are on
ω
\omega
ω
such that rays
I
A
′
,
I
B
′
,
I
C
′
,
IA', IB', IC',
I
A
′
,
I
B
′
,
I
C
′
,
starting from
I
I
I
intersect perpendicularly sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Prove that lines
A
A
′
,
B
B
′
,
C
C
′
AA', BB', CC'
A
A
′
,
B
B
′
,
C
C
′
are concurrent.
BC x DE = CD x PQ wanted, if BP xQE = PQ^ 2 inside a cyclic pentagon
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a cyclic pentagon such that the diagonals
A
C
AC
A
C
and
A
D
AD
A
D
intersect
B
E
BE
BE
at
P
P
P
and
Q
Q
Q
, respectively, with
B
P
⋅
Q
E
=
P
Q
2
BP \cdot QE = PQ^2
BP
⋅
QE
=
P
Q
2
. Prove that
B
C
⋅
D
E
=
C
D
⋅
P
Q
BC \cdot DE = CD \cdot PQ
BC
⋅
D
E
=
C
D
⋅
PQ
.