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Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2015 Saudi Arabia Pre-TST
2015 Saudi Arabia Pre-TST
Part of
Saudi Arabia Pre-TST + Training Tests
Subcontests
(12)
3.3
1
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a^2_n$ divides a_{n-1}a_{n+1}
Let
(
a
n
)
n
≥
0
(a_n)_{n\ge0}
(
a
n
)
n
≥
0
be a sequence of positive integers such that
a
n
2
a^2_n
a
n
2
divides
a
n
−
1
a
n
+
1
a_{n-1}a_{n+1}
a
n
−
1
a
n
+
1
, for all
n
≥
1
n \ge 1
n
≥
1
. Prove that if there exists an integer
k
≥
2
k \ge 2
k
≥
2
such that
a
k
a_k
a
k
and
a
1
a_1
a
1
are relatively prime, then
a
1
a_1
a
1
divides
a
0
a_0
a
0
.(Malik Talbi)
3.4
1
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22 chairs in a round table
There are
22
22
22
chairs in a round table. Find the minimum n such that for any group of
n
n
n
people sitting in the table, we always can find two people with exactly
2
2
2
or
8
8
8
chairs between them.(Le Anh Vinh)
3.2
1
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(X^2-12X +11)^4+23 not product of 3 integer polynomials
Prove that the polynomial
P
(
X
)
=
(
X
2
−
12
X
+
11
)
4
+
23
P(X) = (X^2-12X +11)^4+23
P
(
X
)
=
(
X
2
−
12
X
+
11
)
4
+
23
can not be written as the product of three non-constant polynomials with integer coefficients.(Le Anh Vinh)
2.4
1
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a_i \equiv i^2 mod 12 for all 1 <= i <= 11
How many sequences of integers
1
≤
a
1
≤
a
2
≤
.
.
.
≤
a
11
≤
2015
1 \le a_1 \le a_2\le ... \le a_{11 }\le 2015
1
≤
a
1
≤
a
2
≤
...
≤
a
11
≤
2015
that satisfy
a
i
≡
i
2
a_i \equiv i^2
a
i
≡
i
2
(mod
12
12
12
) for all
1
≤
i
≤
11
1 \le i \le 11
1
≤
i
≤
11
are there?(Le Anh Vinh)
2.3
1
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14^x - 3^y = 2015
Find all integer solutions of the equation
1
4
x
−
3
y
=
2015
14^x - 3^y = 2015
1
4
x
−
3
y
=
2015
.(Malik Talbi)
2.2
1
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f(x + y^2 - f(y)) = f(x)
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
that satisfy
f
(
x
+
y
2
−
f
(
y
)
)
=
f
(
x
)
f(x + y^2 - f(y)) = f(x)
f
(
x
+
y
2
−
f
(
y
))
=
f
(
x
)
for all
x
,
y
∈
R
x,y \in R
x
,
y
∈
R
.(Vo Quoc Ba Can)
1.4
1
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green or blue on each unit square of a 8x 8 table
We color each unit square of a
8
×
8
8\times 8
8
×
8
table into green or blue such that there are
a
a
a
green unit squares in each 3 \times 3 square and
b
b
b
green unit squares in each
2
×
4
2 \times 4
2
×
4
rectangle. Find all possible values of
(
a
,
b
)
(a, b)
(
a
,
b
)
.(Le Anh Vinh)
1.3
1
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x^2y^5 - 2^x5^y = 2015 + 4xy
Find all integer solutions of the equation
x
2
y
5
−
2
x
5
y
=
2015
+
4
x
y
x^2y^5 - 2^x5^y = 2015 + 4xy
x
2
y
5
−
2
x
5
y
=
2015
+
4
x
y
.(Malik Talbi)
1.2
1
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no of integer poolynomials 0<= P(x) < 72$ for all xin {0, 1, 2, 3, 4}
How many polynomials
P
P
P
of integer coefficients and degree at most
4
4
4
satisfy
0
≤
P
(
x
)
<
72
0 \le P(x) < 72
0
≤
P
(
x
)
<
72
for all
x
∈
{
0
,
1
,
2
,
3
,
4
}
x\in \{0, 1, 2, 3, 4\}
x
∈
{
0
,
1
,
2
,
3
,
4
}
?Harvard-MIT Mathematics Tournament 2011
3.1
1
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collinear wanted, incenter, circumcircle, angle bisectors related
Let
A
B
C
ABC
A
BC
be a triangle,
I
I
I
its incenter, and
D
D
D
a point on the arc
B
C
BC
BC
of the circumcircle of
A
B
C
ABC
A
BC
not containing
A
A
A
. The bisector of the angle
∠
A
D
B
\angle ADB
∠
A
D
B
intesects the segment
A
B
AB
A
B
at
E
E
E
. The bisector of the angle
∠
C
D
A
\angle CDA
∠
C
D
A
intesects the segment
A
C
AC
A
C
at
F
F
F
. Prove that the points
E
,
F
,
I
E, F,I
E
,
F
,
I
are collinear.(Malik Talbi)
2.1
1
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A and circumcenters of ABC, DEF are collinear
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
a point on the side
B
C
BC
BC
. The tangent line to the circumcircle of the triangle
A
B
D
ABD
A
B
D
at the point
D
D
D
intersects the side
A
C
AC
A
C
at
E
E
E
. The tangent line to the circumcircle of the triangle
A
C
D
ACD
A
C
D
at the the point
D
D
D
intersects the side
A
B
AB
A
B
at
F
F
F
. Prove that the point
A
A
A
and the circumcenters of the triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
are collinear.(Malik Talbi)
1.1
1
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(con)cyclic wanted, symmetrics wrt sides
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
a point on the side
B
C
BC
BC
. Point
E
E
E
is the symmetric of
D
D
D
with respect to
A
B
AB
A
B
. Point
F
F
F
is the symmetric of
E
E
E
with respect to
A
C
AC
A
C
. Point
P
P
P
is the intersection of line
D
F
DF
D
F
with line
A
C
AC
A
C
. Prove that the quadrilateral
A
E
D
P
AEDP
A
E
D
P
is cyclic.(Malik Talbi)