MathDB
Problems
Contests
National and Regional Contests
Albania Contests
Albania Team Selection Test
2013 Albania Team Selection Test
2013 Albania Team Selection Test
Part of
Albania Team Selection Test
Subcontests
(5)
5
1
Hide problems
Find the couples
Let
k
k
k
be a natural number.Find all the couples of natural numbers
(
n
,
m
)
(n,m)
(
n
,
m
)
such that :
(
2
k
)
!
=
2
n
∗
m
(2^k)!=2^n*m
(
2
k
)!
=
2
n
∗
m
4
1
Hide problems
Find the angle
It is given a triangle
A
B
C
ABC
A
BC
whose circumcenter is
O
O
O
and orthocenter
H
H
H
. If
A
O
=
A
H
AO=AH
A
O
=
A
H
find the angle
B
A
C
^
\hat{BAC}
B
A
C
^
of that triangle.
2
1
Hide problems
Inequality
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers such that
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
.Find with proof that
x
=
3
x=3
x
=
3
is the minimal value for which the following inequality holds:
a
x
+
b
x
+
c
x
+
d
x
≥
1
a
+
1
b
+
1
c
+
1
d
a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}
a
x
+
b
x
+
c
x
+
d
x
≥
a
1
+
b
1
+
c
1
+
d
1
1
1
Hide problems
Find the number
Find the 3-digit number whose ratio with the sum of its digits it's minimal.
3
1
Hide problems
Albania IMO Selection
Solve the function
f
:
ℜ
→
ℜ
f: \Re \to \Re
f
:
ℜ
→
ℜ
:
f
(
x
3
)
+
f
(
y
3
)
=
(
x
+
y
)
(
f
(
x
2
)
+
f
(
y
2
)
−
f
(
x
y
)
)
f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))
f
(
x
3
)
+
f
(
y
3
)
=
(
x
+
y
)
(
f
(
x
2
)
+
f
(
y
2
)
−
f
(
x
y
))