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Contests
National and Regional Contests
Albania Contests
Albania Team Selection Test
2012 Albania Team Selection Test
2012 Albania Team Selection Test
Part of
Albania Team Selection Test
Subcontests
(5)
4
1
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Albania IMO TST Problem 4
Find all couples of natural numbers
(
a
,
b
)
(a,b)
(
a
,
b
)
not relatively prime (
gcd
(
a
,
b
)
≠
1
\gcd(a,b)\neq\ 1
g
cd
(
a
,
b
)
=
1
) such that
gcd
(
a
,
b
)
+
9
lcm
[
a
,
b
]
+
9
(
a
+
b
)
=
7
a
b
.
\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.
g
cd
(
a
,
b
)
+
9
lcm
[
a
,
b
]
+
9
(
a
+
b
)
=
7
ab
.
2
1
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Albania IMO TST Geometry
It is given an acute triangle
A
B
C
ABC
A
BC
,
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
where the feet of altitude from
A
A
A
its
H
H
H
. In the extensions of the sides
A
B
AB
A
B
and
A
C
AC
A
C
(in the direction of
B
B
B
and
C
C
C
) we take the points
P
P
P
and
Q
Q
Q
respectively such that
H
P
=
H
Q
HP=HQ
H
P
=
H
Q
and the points
B
,
C
,
P
,
Q
B,C,P,Q
B
,
C
,
P
,
Q
are concyclic. Find the ratio
H
P
H
A
\tfrac{HP}{HA}
H
A
H
P
.
3
1
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Albania IMO TST 2012 equation
It is given the equation
x
4
−
2
a
x
3
+
a
(
a
+
1
)
x
2
−
2
a
x
+
a
2
=
0
x^4-2ax^3+a(a+1)x^2-2ax+a^2=0
x
4
−
2
a
x
3
+
a
(
a
+
1
)
x
2
−
2
a
x
+
a
2
=
0
. a) Find the greatest value of
a
a
a
, such that this equation has at least one real root. b) Find all the values of
a
a
a
, such that the equation has at least one real root.
5
1
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Albania IMO TST functional equation
Let
f
:
R
+
→
R
+
f:\mathbb R^+ \to \mathbb R^+
f
:
R
+
→
R
+
be a function such that:
x
,
y
>
0
f
(
x
+
f
(
y
)
)
=
y
f
(
x
y
+
1
)
.
x,y > 0 \qquad f(x+f(y)) = yf(xy+1).
x
,
y
>
0
f
(
x
+
f
(
y
))
=
y
f
(
x
y
+
1
)
.
a) Show that
(
y
−
1
)
∗
(
f
(
y
)
−
1
)
≤
0
(y-1)*(f(y)-1) \le 0
(
y
−
1
)
∗
(
f
(
y
)
−
1
)
≤
0
for
y
>
0
y>0
y
>
0
. b) Find all such functions that require the given condition.
1
1
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Albania IMO TST inequality
Find the greatest value of the expression
1
x
2
−
4
x
+
9
+
1
y
2
−
4
y
+
9
+
1
z
2
−
4
z
+
9
\frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9}
x
2
−
4
x
+
9
1
+
y
2
−
4
y
+
9
1
+
z
2
−
4
z
+
9
1
where
x
x
x
,
y
y
y
,
z
z
z
are nonnegative real numbers such that
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
.