MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2014 Saudi Arabia Pre-TST
2014 Saudi Arabia Pre-TST
Part of
Saudi Arabia Pre-TST + Training Tests
Subcontests
(16)
4.3
1
Hide problems
2013 pairwise different numbers game
Fatima and Asma are playing the following game. First, Fatima chooses
2013
2013
2013
pairwise different numbers, called
a
1
,
a
2
,
.
.
.
,
a
2013
a_1, a_2, ..., a_{2013}
a
1
,
a
2
,
...
,
a
2013
. Then, Asma tries to know the value of each number
a
1
,
a
2
,
.
.
.
,
a
2013
a_1, a_2, ..., a_{2013}
a
1
,
a
2
,
...
,
a
2013
.. At each time, Asma chooses
1
≤
i
<
j
≤
2013
1 \le i < j \le 2013
1
≤
i
<
j
≤
2013
and asks Fatima ''What is the set
{
a
i
,
a
j
}
\{a_i,a_j\}
{
a
i
,
a
j
}
?'' (For example, if Asma asks what is the set
{
a
i
,
a
j
}
\{a_i,a_j\}
{
a
i
,
a
j
}
, and
a
1
=
17
a_1 = 17
a
1
=
17
and
a
2
=
13
a_2 = 13
a
2
=
13
, Fatima will answer
{
13.17
}
\{13. 17\}
{
13.17
}
). Find the least number of questions Asma needs to ask, to know the value of all the numbers
a
1
,
a
2
,
.
.
.
,
a
2013
a_1, a_2, ..., a_{2013}
a
1
,
a
2
,
...
,
a
2013
.
4.2
1
Hide problems
(x^2 + 1)^6 /2^7 +1/2 x^5 - x^3 + x for x>=0
Given
x
≥
0
x \ge 0
x
≥
0
, prove that
(
x
2
+
1
)
6
2
7
+
1
2
≥
x
5
−
x
3
+
x
\frac{(x^2 + 1)^6}{2^7}+\frac12 \ge x^5 - x^3 + x
2
7
(
x
2
+
1
)
6
+
2
1
≥
x
5
−
x
3
+
x
4.1
1
Hide problems
n > \sqrt{p}-1 when p | (n^6 -1)
Let
p
p
p
be a prime number and
n
≥
2
n \ge 2
n
≥
2
a positive integer, such that
p
∣
(
n
6
−
1
)
p | (n^6 -1)
p
∣
(
n
6
−
1
)
. Prove that
n
>
p
−
1
n > \sqrt{p}-1
n
>
p
−
1
.
3.4
1
Hide problems
last digits of n^3 are ...201320132013.
Prove that there exists a positive integer
n
n
n
such that the last digits of
n
3
n^3
n
3
are
.
.
.
201320132013
...201320132013
...201320132013
.
3.2
1
Hide problems
min of x^2 + xy + y^2 / + 2^6 /(x + y)+ 3^4 /x^3 for x,y>0
Let
x
,
y
x, y
x
,
y
be positive real numbers. Find the minimum of
x
2
+
x
y
+
y
2
2
+
2
6
x
+
y
+
3
4
x
3
x^2 + xy +\frac{y^2}{2}+\frac{2^6}{x + y}+\frac{3^4}{x^3}
x
2
+
x
y
+
2
y
2
+
x
+
y
2
6
+
x
3
3
4
3.1
1
Hide problems
14 students in a 3 hour test consisting on 15 short problems
There are
14
14
14
students who have particiated to a
3
3
3
hour test consisting on
15
15
15
short problems. Each student has solved a different number of problems and each problem has been solved by a different number of students. Prove that there exists a student who has solved exactly
5
5
5
problems.
1.1
1
Hide problems
\sqrt{a^2_j-a^2_k < \frac{1}{\sqrt{n} +\sqrt{n - 1}}
Let
a
1
,
a
2
,
.
.
.
,
a
2
n
a_1, a_2,...,a_{2n}
a
1
,
a
2
,
...
,
a
2
n
be positive real numbers such that
a
i
+
a
n
+
i
=
1
a_i + a_{n+i} = 1
a
i
+
a
n
+
i
=
1
, for all
i
=
1
,
.
.
.
,
n
i = 1,...,n
i
=
1
,
...
,
n
. Prove that there exist two different integers
1
≤
j
,
k
≤
2
n
1 \le j, k \le 2n
1
≤
j
,
k
≤
2
n
for which
a
j
2
−
a
k
2
<
1
n
+
n
−
1
\sqrt{a^2_j-a^2_k} < \frac{1}{\sqrt{n} +\sqrt{n - 1}}
a
j
2
−
a
k
2
<
n
+
n
−
1
1
2.2
1
Hide problems
P(x)=prod a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}
Let
a
1
,
a
2
,
a
3
,
a
4
,
a
5
a_1, a_2, a_3, a_4, a_5
a
1
,
a
2
,
a
3
,
a
4
,
a
5
be nonzero real numbers. Prove that the polynomial
P
(
x
)
=
∏
k
=
0
4
a
k
+
1
x
4
+
a
k
+
2
x
3
+
a
k
+
3
x
2
+
a
k
+
4
x
+
a
k
+
5
P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}
P
(
x
)
=
k
=
0
∏
4
a
k
+
1
x
4
+
a
k
+
2
x
3
+
a
k
+
3
x
2
+
a
k
+
4
x
+
a
k
+
5
, where
a
5
+
i
=
a
i
a_{5+i} = a_i
a
5
+
i
=
a
i
for
i
=
1
,
2
,
3
,
4
i = 1,2, 3,4
i
=
1
,
2
,
3
,
4
, has a root with negative real part.
2.3
1
Hide problems
1/1x2, 1/2x3, 1/ 3x4, ..., 1/2013 x 2014 on a circle
The
2013
2013
2013
numbers \frac{1}{1\times 2}, \frac{1}{2\times 3},\frac{1}{3\times 4},...,\frac{1}{2013 \times 2014} are arranged randomly on a circle. (a) Prove that there exist ten consecutive numbers on the circle whose sum is less than
1
4000
\frac{1}{4000}
4000
1
. (b) Prove that there exist ten consecutive numbers on the circle whose sum is less than
1
10000
\frac{1}{10000}
10000
1
.
2.1
1
Hide problems
2014 divides 53n^{55}- 57n^{53} + 4n
Prove that
2014
2014
2014
divides
53
n
55
−
57
n
53
+
4
n
53n^{55}- 57n^{53} + 4n
53
n
55
−
57
n
53
+
4
n
for all integer
n
n
n
.
1.3
1
Hide problems
1 - 5^n + 5^{2n+1} is a perfect square
Find all positive integers
n
n
n
for which
1
−
5
n
+
5
2
n
+
1
1 - 5^n + 5^{2n+1}
1
−
5
n
+
5
2
n
+
1
is a perfect square.
1.4
1
Hide problems
bw color the cells of an nx n chessboard
Majid wants to color the cells of an
n
×
n
n\times n
n
×
n
chessboard into white and black so that each
2
×
2
2\times 2
2
×
2
subsquare contains two white cells and two black cells. In how many ways can Majid color this n\times n chessboard?
4.4
1
Hide problems
cyclic wanted, AF = FM = JD = DK = LE = EA, incircle related
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an acute triangle, with
∠
A
>
∠
B
≥
∠
C
\angle A> \angle B \ge \angle C
∠
A
>
∠
B
≥
∠
C
. Let
D
,
E
D, E
D
,
E
and
F
F
F
be the tangency points between the incircle of triangle and sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Let
J
J
J
be a point on
(
B
D
)
(BD)
(
B
D
)
,
K
K
K
a point on
(
D
C
)
(DC)
(
D
C
)
,
L
L
L
a point on
(
E
C
)
(EC)
(
EC
)
and
M
M
M
a point on
(
F
B
)
(FB)
(
FB
)
, such that
A
F
=
F
M
=
J
D
=
D
K
=
L
E
=
E
A
.
AF = FM = JD = DK = LE = EA.
A
F
=
FM
=
J
D
=
DK
=
L
E
=
E
A
.
Let
P
P
P
be the intersection point between
A
J
AJ
A
J
and
K
M
KM
K
M
and let
Q
Q
Q
be the intersection point between
A
K
AK
A
K
and
J
L
JL
J
L
. Prove that
P
J
K
Q
PJKQ
P
J
K
Q
is cyclic.
3.3
1
Hide problems
isosceles wanted, incenters and perpendicular bisector related
Let
A
B
C
ABC
A
BC
be a triangle and
I
I
I
its incenter. The line
A
I
AI
A
I
intersects the side
B
C
BC
BC
at
D
D
D
and the perpendicular bisector of
B
C
BC
BC
at
E
E
E
. Let
J
J
J
be the incenter of triangle
C
D
E
CDE
C
D
E
. Prove that triangle
C
I
J
CIJ
C
I
J
is isosceles.
2.4
1
Hide problems
express angles of DEO in terms of angles of ABC, perpendicular bisectors
Let
A
B
C
ABC
A
BC
be an acute triangle with
∠
A
<
∠
B
≤
∠
C
\angle A < \angle B \le \angle C
∠
A
<
∠
B
≤
∠
C
, and
O
O
O
its circumcenter. The perpendicular bisector of side
A
B
AB
A
B
intersects side
A
C
AC
A
C
at
D
D
D
. The perpendicular bisector of side
A
C
AC
A
C
intersects side
A
B
AB
A
B
at
E
E
E
. Express the angles of triangle
D
E
O
DEO
D
EO
in terms of the angles of triangle
A
B
C
ABC
A
BC
.
1.2
1
Hide problems
ratio of areas 5/12 wanted, midpoints related
Let
D
D
D
be the midpoint of side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
and
E
E
E
the midpoint of median
A
D
AD
A
D
. Line
B
E
BE
BE
intersects side
C
A
CA
C
A
at
F
F
F
. Prove that the area of quadrilateral
C
D
E
F
CDEF
C
D
EF
is
5
12
\frac{5}{12}
12
5
the area of triangle
A
B
C
ABC
A
BC
.