MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2011 Saudi Arabia Pre-TST
2011 Saudi Arabia Pre-TST
Part of
Saudi Arabia Pre-TST + Training Tests
Subcontests
(20)
3
1
Hide problems
\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n} is an integer
Find all integers
n
≥
2
n \ge 2
n
≥
2
for which
3
n
+
4
n
+
5
n
+
8
n
+
1
0
n
n
\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}
n
3
n
+
4
n
+
5
n
+
8
n
+
1
0
n
is an integer.
2
1
Hide problems
binomial (x y)= 1432
Find all positive integers
x
x
x
and
y
y
y
such that
(
x
y
)
=
1432
{x \choose y} = 1432
(
y
x
)
=
1432
4.3
2
Hide problems
x_n <\1/\sqrt{n! H_n}
Let
n
≥
2
n \ge 2
n
≥
2
be a positive integer and let
x
n
x_n
x
n
be a positive real root to the equation
x
(
x
+
1
)
.
.
.
(
x
+
n
)
=
1
x(x+1)...(x + n) = 1
x
(
x
+
1
)
...
(
x
+
n
)
=
1
. Prove that
x
n
<
1
n
!
H
n
x_n <\frac{1}{\sqrt{n! H_n}}
x
n
<
n
!
H
n
1
where
H
n
=
1
+
1
2
+
.
.
.
+
1
n
H_n = 1+\frac12+...+\frac{1}{n}
H
n
=
1
+
2
1
+
...
+
n
1
.
x_1x_2...x_n >= (n -1)^n if sum 1/(1+x_k)=1
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
be positive real numbers for which
1
1
+
x
1
+
1
1
+
x
2
+
.
.
.
+
1
1
+
x
n
=
1
\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1
1
+
x
1
1
+
1
+
x
2
1
+
...
+
1
+
x
n
1
=
1
Prove that
x
1
x
2
.
.
.
x
n
≥
(
n
−
1
)
n
x_1x_2...x_n \ge (n -1)^n
x
1
x
2
...
x
n
≥
(
n
−
1
)
n
.
4.1
2
Hide problems
distance between 2 neighboring stations on Earth
A Geostationary Earth Orbit is situated directly above the equator and has a period equal to the Earth’s rotational period. It is at the precise distance of
22
,
236
22,236
22
,
236
miles above the Earth that a satellite can maintain an orbit with a period of rotation around the Earth exactly equal to
24
24
24
hours. Be cause the satellites revolve at the same rotational speed of the Earth, they appear stationary from the Earth surface. That is why most station antennas (satellite dishes) do not need to move once they have been properly aimed at a tar get satellite in the sky. In an international project, a total of ten stations were equally spaced on this orbit (at the precise distance of
22
,
236
22,236
22
,
236
miles above the equator). Given that the radius of the Earth is
3960
3960
3960
miles, find the exact straight distance between two neighboring stations. Write your answer in the form
a
+
b
c
a + b\sqrt{c}
a
+
b
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are integers and
c
>
0
c > 0
c
>
0
is square-free.
|PM-PN|/PC is constant , MC _|_ NC, semicircle of diameter AB and center C
On a semicircle of diameter
A
B
AB
A
B
and center
C
C
C
, consider variable points
M
M
M
and
N
N
N
such that
M
C
⊥
N
C
MC \perp NC
MC
⊥
NC
. The circumcircle of triangle
M
N
C
MNC
MNC
intersects
A
B
AB
A
B
for the second time at
P
P
P
. Prove that
∣
P
M
−
P
N
∣
P
C
\frac{|PM-PN|}{PC}
PC
∣
PM
−
PN
∣
constant and find its value.
3.2
2
Hide problems
any parallelogram can be dissected in n>=4 cyclic quadrilaterals.
Prove that for each
n
≥
4
n \ge 4
n
≥
4
a parallelogram can be dissected in
n
n
n
cyclic quadrilaterals.
a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}
Find all pairs of nonnegative integers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that
a
+
2
b
−
b
2
=
2
a
+
a
2
+
∣
2
a
+
1
−
2
b
∣
a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}
a
+
2
b
−
b
2
=
2
a
+
a
2
+
∣2
a
+
1
−
2
b
∣
.
3.1
2
Hide problems
sin^3 a/ sin b +cos^3 a/cos b >= 2/cos(a - b)
Prove that
sin
3
a
sin
b
+
cos
3
a
cos
b
≥
1
cos
(
a
−
b
)
\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)}
sin
b
sin
3
a
+
cos
b
cos
3
a
≥
cos
(
a
−
b
)
1
for all
a
a
a
and
b
b
b
in the interval
(
0
,
π
/
2
)
(0, \pi/2)
(
0
,
π
/2
)
.
2011^{2012} divides n! if 2011^{2011} divides n!
Let
n
n
n
be a positive integer such that
201
1
2011
2011^{2011}
201
1
2011
divides
n
!
n!
n
!
. Prove that
201
1
2012
2011^{2012}
201
1
2012
divides
n
!
n!
n
!
.
4.4
2
Hide problems
2011 is not a divisor of ab -cd if a+b+c+d = 2011
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive integers such that
a
+
b
+
c
+
d
=
2011
a+b+c+d = 2011
a
+
b
+
c
+
d
=
2011
. Prove that
2011
2011
2011
is not a divisor of
a
b
−
c
d
ab - cd
ab
−
c
d
.
MP = MQ wanted, midpoint of AH, circumcenter, orthocenter
In a triangle
A
B
C
ABC
A
BC
, let
O
O
O
be the circumcenter,
H
H
H
the orthocenter, and
M
M
M
the midpoint of the segment
A
H
AH
A
H
. The perpendicular at
M
M
M
onto
O
M
OM
OM
intersects lines
A
B
AB
A
B
and
A
C
AC
A
C
at
P
P
P
and
Q
Q
Q
, respectively. Prove that
M
P
=
M
Q
MP = MQ
MP
=
MQ
.
3.4
2
Hide problems
+ y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1 diophantine
Find all quadruples
(
x
,
y
,
z
,
w
)
(x,y,z,w)
(
x
,
y
,
z
,
w
)
of integers satisfying the system of equations
x
+
y
+
z
+
w
=
x
y
+
y
z
+
z
x
+
w
2
−
w
=
x
y
z
−
w
3
=
−
1
x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1
x
+
y
+
z
+
w
=
x
y
+
yz
+
z
x
+
w
2
−
w
=
x
yz
−
w
3
=
−
1
x^3 + y^3 = n! + 4
Find all positive integers
n
n
n
for which the equation
x
3
+
y
3
=
n
!
+
4
x^3 + y^3 = n! + 4
x
3
+
y
3
=
n
!
+
4
has solutions in integers.
2.3
2
Hide problems
(x+y)^2/( x^3 + xy^2 - x^2y - y^3) not an integer
Let
x
,
y
x, y
x
,
y
be distinct positive integers. Prove that the number
(
x
+
y
)
2
x
3
+
x
y
2
−
x
2
y
−
y
3
\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}
x
3
+
x
y
2
−
x
2
y
−
y
3
(
x
+
y
)
2
is not an integer
f = g^2 if f = aX^2 + bX+ c is a perfect square for every n
Let
f
=
a
X
2
+
b
X
+
c
∈
Z
[
X
]
f = aX^2 + bX+ c \in Z[X]
f
=
a
X
2
+
b
X
+
c
∈
Z
[
X
]
be a polynomial such that for every positive integer
n
n
n
,
f
(
n
)
f(n )
f
(
n
)
is a perfect square. Prove that
f
=
g
2
f = g^2
f
=
g
2
for some polynomial
g
∈
Z
[
X
]
g \in Z[X]
g
∈
Z
[
X
]
.
2.2
2
Hide problems
2(a^3 + b^3 + c^3 + abc) >= (a+b)(b + c)(c + a) for a,b,c>0
Prove that for any positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
,
2
(
a
3
+
b
3
+
c
3
+
a
b
c
)
≥
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
2(a^3 + b^3 + c^3 + abc) \ge (a+b)(b + c)(c + a)
2
(
a
3
+
b
3
+
c
3
+
ab
c
)
≥
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
.
s_m = x_0 + x_1 + x_2 + ... + x_m is power of 2 , x_n = 2^n-n
Consider the sequence
x
n
=
2
n
−
n
x_n = 2^n-n
x
n
=
2
n
−
n
,
n
=
0
,
1
,
2
,
.
.
.
n = 0,1 ,2 ,...
n
=
0
,
1
,
2
,
...
. Find all integers
m
≥
0
m \ge 0
m
≥
0
such that
s
m
=
x
0
+
x
1
+
x
2
+
.
.
.
+
x
m
s_m = x_0 + x_1 + x_2 + ... + x_m
s
m
=
x
0
+
x
1
+
x
2
+
...
+
x
m
is a power of
2
2
2
.
2.1
2
Hide problems
17 soldiers patrol in equilateral military base
The shape of a military base is an equilateral triangle of side
10
10
10
kilometers. Security constraints make cellular phone communication possible only within
2.5
2.5
2.5
kilometers. Each of
17
17
17
soldiers patrols the base randomly and tries to contact all others. Prove that at each moment at least two soldiers can communicate.
an interval does not contain any integer
Let
n
n
n
be a positive integer. Prove that the interval
I
n
=
(
1
+
8
n
+
1
2
,
1
+
8
n
+
9
2
)
I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)
I
n
=
(
2
1
+
8
n
+
1
,
2
1
+
8
n
+
9
)
does not contain any integer.
1.3
2
Hide problems
27^n- 2^n is perfect square
Find all positive integers
n
n
n
such that
2
7
n
−
2
n
27^n- 2^n
2
7
n
−
2
n
is a perfect square.
4CF<= CB, 4CF = CB => AE is bisector of <DAF, AD=DC=CB<AB, <ADE=<AEF
The quadrilateral
A
B
C
D
ABCD
A
BC
D
has
A
D
=
D
C
=
C
B
<
A
B
AD = DC = CB < AB
A
D
=
D
C
=
CB
<
A
B
and
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
. Points
E
E
E
and
F
F
F
lie on the sides
C
D
CD
C
D
and
B
C
BC
BC
such that
∠
A
D
E
=
∠
A
E
F
\angle ADE = \angle AEF
∠
A
D
E
=
∠
A
EF
. Prove that: (a)
4
C
F
≤
C
B
4CF \le CB
4
CF
≤
CB
. (b) If
4
C
F
=
C
B
4CF = CB
4
CF
=
CB
, then
A
E
AE
A
E
is the angle bisector of
∠
D
A
F
\angle DAF
∠
D
A
F
.
1.2
2
Hide problems
2011x^3 +ax^2 +bx+c = 0 if a+ b + c = 2010 \cdot 2011
Find all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of integers such that
a
+
b
+
c
=
2010
⋅
2011
a+ b + c = 2010 \cdot 2011
a
+
b
+
c
=
2010
⋅
2011
and the solutions to the equation
2011
x
3
+
a
x
2
+
b
x
+
c
=
0
2011x^3 +ax^2 +bx+c = 0
2011
x
3
+
a
x
2
+
b
x
+
c
=
0
are all nonzero integers.
q_1^4+q_2^4+q_3^4+q_4^4+q_5^ is product of 2 even consecutive
Find all primes
q
1
,
q
2
,
q
3
,
q
4
,
q
5
q_1, q_2, q_3, q_4, q_5
q
1
,
q
2
,
q
3
,
q
4
,
q
5
such that
q
1
4
+
q
2
4
+
q
3
4
+
q
4
4
+
q
5
4
q_1^4+q_2^4+q_3^4+q_4^4+q_5^4
q
1
4
+
q
2
4
+
q
3
4
+
q
4
4
+
q
5
4
is the product of two consecutive even integers.
1.1
2
Hide problems
equal sums in 2 sets with consecutive positive integers
Set
A
A
A
consists of
7
7
7
consecutive positive integers less than
2011
2011
2011
, while set
B
B
B
consists of
11
11
11
consecutive positive integers. If the sum of the numbers in
A
A
A
is equal to the sum of the numbers in
B
B
B
, what is the maximum possible element that
A
A
A
could contain?
8(a+b+c) (a/b+b/c +c/a) < = 9 (1+a/b) (1+b/c) (1+c/a)
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers. Prove that
8
(
a
+
b
+
c
)
(
a
b
+
b
c
+
c
a
)
≤
9
(
1
+
a
b
)
(
1
+
b
c
)
(
1
+
c
a
)
8(a+b+c) \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \le 9 \left(1+\frac{a}{b} \right)\left(1+\frac{b}{c} \right)\left(1+\frac{c}{a} \right)
8
(
a
+
b
+
c
)
(
b
a
+
c
b
+
a
c
)
≤
9
(
1
+
b
a
)
(
1
+
c
b
)
(
1
+
a
c
)
1
1
Hide problems
AB^2/PC + AC^2/PB >= BC^3/(PA^2 + PB x PC), <BAC=90^o
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
9
0
o
\angle A = 90^o
∠
A
=
9
0
o
and let
P
P
P
be a point on the hypotenuse
B
C
BC
BC
. Prove that
A
B
2
P
C
+
A
C
2
P
B
≥
B
C
3
P
A
2
+
P
B
⋅
P
C
\frac{AB^2}{PC}+\frac{AC^2}{PB} \ge \frac{BC^3}{PA^2 + PB \cdot PC}
PC
A
B
2
+
PB
A
C
2
≥
P
A
2
+
PB
⋅
PC
B
C
3
4
1
Hide problems
fixed length of square
Points
A
,
B
,
C
,
D
A ,B ,C ,D
A
,
B
,
C
,
D
lie on a line in this order. Draw parallel lines
a
a
a
and
b
b
b
through
A
A
A
and
B
B
B
, respectively, and parallel lines
c
c
c
and
d
d
d
through
C
C
C
and
D
D
D
, respectively, such that their points of intersection are vertices of a square. Prove that the side length of this square does not depend on the length of segment
B
C
BC
BC
.
4.2
2
Hide problems
computational geo with cyclic pentagon
Pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in a circle. Distances from point
E
E
E
to lines
A
B
AB
A
B
,
B
C
BC
BC
and
C
D
CD
C
D
are equal to
a
,
b
a, b
a
,
b
and
c
c
c
, respectively. Find the distance from point
E
E
E
to line
A
D
AD
A
D
.
sum a_k (k!)^{2010}= (2011 !)^{2010}
Find positive integers
a
1
<
a
2
<
.
.
.
<
a
2010
a_1 < a_2<... <a_{2010}
a
1
<
a
2
<
...
<
a
2010
such that
a
1
(
1
!
)
2010
+
a
2
(
2
!
)
2010
+
.
.
.
+
a
2010
(
2010
!
)
2010
=
(
2011
!
)
2010
.
a_1(1!)^{2010} + a_2(2!)^{2010} + ... + a_{2010}(2010!)^{2010} = (2011 !)^{2010}.
a
1
(
1
!
)
2010
+
a
2
(
2
!
)
2010
+
...
+
a
2010
(
2010
!
)
2010
=
(
2011
!
)
2010
.
3.3
2
Hide problems
angles wanted in isosceles AB, BB'+ B'A = BC
In the isosceles triangle
A
B
C
ABC
A
BC
, with
A
B
=
A
C
AB = AC
A
B
=
A
C
, the angle bisector of
∠
B
\angle B
∠
B
intersects side
A
C
AC
A
C
at
B
′
B'
B
′
. Suppose that
B
B
′
+
B
′
A
=
B
C
B B' + B'A = BC
B
B
′
+
B
′
A
=
BC
. Find the angles of the triangle.
AP x BP x CP >= 8PL x PM x PN
Let
P
P
P
be a point in the interior of triangle
A
B
C
ABC
A
BC
. Lines
A
P
AP
A
P
,
B
P
BP
BP
,
C
P
CP
CP
intersect sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
L
L
L
,
M
M
M
,
N
N
N
, respectively. Prove that
A
P
⋅
B
P
⋅
C
P
≥
8
P
L
⋅
P
M
⋅
P
N
.
AP \cdot BP \cdot CP \ge 8PL \cdot PM \cdot PN.
A
P
⋅
BP
⋅
CP
≥
8
P
L
⋅
PM
⋅
PN
.
2.4
2
Hide problems
SM//CL wanted, ABCD rectangle , <DAC = 60^o
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle of center
O
O
O
, such that
∠
D
A
C
=
6
0
o
\angle DAC = 60^o
∠
D
A
C
=
6
0
o
. The angle bisector of
∠
D
A
C
\angle DAC
∠
D
A
C
meets
D
C
DC
D
C
at
S
S
S
. Lines
O
S
OS
OS
and
A
D
AD
A
D
meet at
L
L
L
and lines
B
L
BL
B
L
and
A
C
AC
A
C
meet at
M
M
M
. Prove that lines
S
M
SM
SM
and
C
L
CL
C
L
are parallel.
triangle of medians of ABC similar to ABC criterion
Let
A
B
C
ABC
A
BC
be a triangle with medians
m
a
m_a
m
a
,
m
b
m_b
m
b
,
m
c
m_c
m
c
. Prove that: (a) There is a triangle with side lengths
m
a
m_a
m
a
,
m
b
m_b
m
b
,
m
c
m_c
m
c
. (b) This triangle is similar to
A
B
C
ABC
A
BC
if and only if the squares of the side lengths of triangle
A
B
C
ABC
A
BC
form an arithmetical sequence.
1.4
2
Hide problems
BT = 2PT if <BAT = < BCS = 10^o in 40^o-70^o-70^o triangle
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
AB=AC
A
B
=
A
C
and
∠
B
A
C
=
4
0
o
\angle BAC = 40^o
∠
B
A
C
=
4
0
o
. Points
S
S
S
and
T
T
T
lie on the sides
A
B
AB
A
B
and
B
C
BC
BC
, such that
∠
B
A
T
=
∠
B
C
S
=
1
0
o
\angle BAT = \angle BCS = 10^o
∠
B
A
T
=
∠
BCS
=
1
0
o
. Lines
A
T
AT
A
T
and
C
S
CS
CS
meet at
P
P
P
. Prove that
B
T
=
2
P
T
BT = 2PT
BT
=
2
PT
.
sum 2^{k-1}/f_k <C for fermat numbers
Let
f
n
=
2
2
n
+
1
f_n = 2^{2^n}+ 1
f
n
=
2
2
n
+
1
,
n
=
1
,
2
,
3
,
.
.
.
n = 1,2,3,...
n
=
1
,
2
,
3
,
...
, be the Fermat’s numbers. Find the least real number
C
C
C
such that
1
f
1
+
2
f
2
+
2
2
f
3
+
.
.
.
+
2
n
−
1
f
n
<
C
\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C
f
1
1
+
f
2
2
+
f
3
2
2
+
...
+
f
n
2
n
−
1
<
C
for all positive integers
n
n
n