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Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2010 Saudi Arabia Pre-TST
2010 Saudi Arabia Pre-TST
Part of
Saudi Arabia Pre-TST + Training Tests
Subcontests
(16)
3.4
1
Hide problems
2 parameter equation, 1/1(x + a)^2 - b^2} +...
Let
a
a
a
and
b
b
b
be real numbers such that
a
+
b
≠
0
a + b \ne 0
a
+
b
=
0
. Solve the equation
1
(
x
+
a
)
2
−
b
2
+
1
(
x
+
b
)
2
−
a
2
=
1
x
2
−
(
a
+
b
)
2
+
1
x
2
−
(
a
−
b
)
2
\frac{1}{(x + a)^2 - b^2} +\frac{1}{(x +b)^2 - a^2}=\frac{1}{x^2 -(a + b)^2}+\frac{1}{x^2-(a -b)^2}
(
x
+
a
)
2
−
b
2
1
+
(
x
+
b
)
2
−
a
2
1
=
x
2
−
(
a
+
b
)
2
1
+
x
2
−
(
a
−
b
)
2
1
4.3
1
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sum (a-2)/(a+1) >= 0 if abc = 8, a,b,c>0 2010 Saudi Arabia Pre-TST 4.3
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
8
abc = 8
ab
c
=
8
. Prove that
a
−
2
a
+
1
+
b
−
2
b
+
1
+
c
−
2
c
+
1
≤
0
\frac{a-2}{a+1}+\frac{b-2}{b+1}+\frac{c-2}{c+1} \le 0
a
+
1
a
−
2
+
b
+
1
b
−
2
+
c
+
1
c
−
2
≤
0
4.4
1
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x^4 + 2z^3 - y =\sqrt3 - 1/4 , y^4 + 2y^3 - x = - \sqrt3 - 1/4
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers that satisfy the system of equations
{
x
4
+
2
z
3
−
y
=
3
−
1
4
y
4
+
2
y
3
−
x
=
−
3
−
1
4
\begin{cases} x^4 + 2z^3 - y =\sqrt3 - \dfrac14 \\ y^4 + 2y^3 - x = - \sqrt3 - \dfrac14 \end{cases}
⎩
⎨
⎧
x
4
+
2
z
3
−
y
=
3
−
4
1
y
4
+
2
y
3
−
x
=
−
3
−
4
1
4.1
1
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a + bc=2010, b + ca = 250 diophantine
Find all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers for which
{
a
+
b
c
=
2010
b
+
c
a
=
250
\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}
{
a
+
b
c
=
2010
b
+
c
a
=
250
3.1
1
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(a-b+c)(1/a -1/b +1/c ) >=1 if a >= b>= c > 0 2010 Saudi Arabia Pre-TST 3.1
Let
a
≥
b
≥
c
>
0
a \ge b \ge c > 0
a
≥
b
≥
c
>
0
. Prove that
(
a
−
b
+
c
)
(
1
a
−
1
b
+
1
c
)
≥
1
(a-b+c)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right) \ge 1
(
a
−
b
+
c
)
(
a
1
−
b
1
+
c
1
)
≥
1
3.2
1
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among any 9 divisors of 30^{2010} there are 2 with product is a perfect square
Prove that among any nine divisors of
3
0
2010
30^{2010}
3
0
2010
there are two whose product is a perfect square.
2.3
1
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a_{n + 1} =\sqrt{a_n^2 + 1}
Let
a
0
a_0
a
0
be a positive integer and
a
n
+
1
=
a
n
2
+
1
a_{n + 1} =\sqrt{a_n^2 + 1}
a
n
+
1
=
a
n
2
+
1
, for all
n
≥
0
n \ge 0
n
≥
0
.1) Prove that for all
a
0
a_0
a
0
the sequence contains infinitely many integers and infinitely many irrational numbers. 2) Is there an
a
0
a_0
a
0
for which
a
2010
a_{2010}
a
2010
is an integer?
2.2
1
Hide problems
n consecutive integers whose sum of squares is a prime
Find all
n
n
n
for which there are
n
n
n
consecutive integers whose sum of squares is a prime.
2.1
1
Hide problems
x + y +z = 2010, x^2 + y^2 + z^2 - xy - yz - zx =3
Find all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of positive integers such that
{
x
+
y
+
z
=
2010
x
2
+
y
2
+
z
2
−
x
y
−
y
z
−
z
x
=
3
\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}
{
x
+
y
+
z
=
2010
x
2
+
y
2
+
z
2
−
x
y
−
yz
−
z
x
=
3
1.3
1
Hide problems
b divides a^n - 1, multiple of 7^{2010} of the form 99... 9
1) Let
a
a
a
and
b
b
b
be relatively prime positive integers. Prove that there is a positive integer
n
n
n
such that
1
≤
n
≤
b
1 \le n \le b
1
≤
n
≤
b
and
b
b
b
divides
a
n
−
1
a^n - 1
a
n
−
1
. 2) Prove that there is a multiple of
7
2010
7^{2010}
7
2010
of the form
99...9
99... 9
99...9
(
n
n
n
nines), for some positive integer
n
n
n
not exceeding
7
2010
7^{2010}
7
2010
.
1.2
1
Hide problems
n(n + 2010) is a perfect square
Find all integers
n
n
n
for which
n
(
n
+
2010
)
n(n + 2010)
n
(
n
+
2010
)
is a perfect square.
1.1
1
Hide problems
2010 by each of the first 8 primes exactly once
Using each of the first eight primes exactly once and several algebraic operations, obtain the result
2010
2010
2010
.
4.2
1
Hide problems
exists triangle with sides \sqrt{a^2-a + 1}, \sqrt{a^2+a + 1}, \sqrt{4a^2 + 3}
Let
a
a
a
be a real number. 1) Prove that there is a triangle with side lengths
a
2
−
a
+
1
\sqrt{a^2-a + 1}
a
2
−
a
+
1
,
a
2
+
a
+
1
\sqrt{a^2+a + 1}
a
2
+
a
+
1
, and
4
a
2
+
3
\sqrt{4a^2 + 3}
4
a
2
+
3
. 2) Prove that the area of this triangle does not depend on
a
a
a
.
3.3
1
Hide problems
AB = mn/(m+n) if AC = m, AD = n in regular heptagon ABCDEFG
Let
A
B
C
D
E
F
G
ABCDEFG
A
BC
D
EFG
be a regular heptagon. If
A
C
=
m
AC = m
A
C
=
m
and
A
D
=
n
AD = n
A
D
=
n
, prove that
A
B
=
m
n
m
+
n
AB =\frac{mn}{m+n}
A
B
=
m
+
n
mn
.
2.4
1
Hide problems
x^3- (a^2 + b^2 + c^2)x -2abc = 0 for quadrilateral inscribed in semicircle
Let
A
M
N
B
AMNB
A
MNB
be a quadrilateral inscribed in a semicircle of diameter
A
B
=
x
AB = x
A
B
=
x
. Denote
A
M
=
a
AM = a
A
M
=
a
,
M
N
=
b
MN = b
MN
=
b
,
N
B
=
c
NB = c
NB
=
c
. Prove that
x
3
−
(
a
2
+
b
2
+
c
2
)
x
−
2
a
b
c
=
0
x^3- (a^2 + b^2 + c^2)x -2abc = 0
x
3
−
(
a
2
+
b
2
+
c
2
)
x
−
2
ab
c
=
0
.
1.4
1
Hide problems
M, G, N are collinear iff MB/MA + NC/NA =1
In triangle
A
B
C
ABC
A
BC
with centroid
G
G
G
, let
M
∈
(
A
B
)
M \in (AB)
M
∈
(
A
B
)
and
N
∈
(
A
C
)
N \in (AC)
N
∈
(
A
C
)
be points on two of its sides. Prove that points
M
,
G
,
N
M, G, N
M
,
G
,
N
are collinear if and only if
M
B
M
A
+
N
C
N
A
=
1
\frac{MB}{MA}+\frac{NC}{NA}=1
M
A
MB
+
N
A
NC
=
1
.