MathDB
Problems
Contests
National and Regional Contests
Kosovo Contests
Kosovo Team Selection Test
2011 Kosovo Team Selection Test
2011 Kosovo Team Selection Test
Part of
Kosovo Team Selection Test
Subcontests
(5)
5
1
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Easy Functional Equation with one variable
Find all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
such that
∀
x
∉
{
−
1
,
1
}
\forall x\notin\{-1,1\}
∀
x
∈
/
{
−
1
,
1
}
holds:
f
(
x
−
3
x
+
1
)
+
f
(
3
+
x
1
−
x
)
=
x
\displaystyle{f\Big(\frac{x-3}{x+1}\Big)+f\Big(\frac{3+x}{1-x}\Big)=x}
f
(
x
+
1
x
−
3
)
+
f
(
1
−
x
3
+
x
)
=
x
4
1
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Prove that the left number has two same digits
From the number
7
1996
7^{1996}
7
1996
we delete its first digit, and then add the same digit to the remaining number. This process continues until the left number has ten digits. Show that the left number has two same digits.
3
1
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Set Problem (KSV IMO TST 2011)
Let
n
n
n
be a natural number, for which we define
S
(
n
)
=
{
1
+
g
+
g
2
+
.
.
.
+
g
n
−
1
∣
g
∈
N
,
g
≥
2
}
S(n)=\{1+g+g^2+...+g^{n-1}|g\in{\mathbb{N}},g\geq2\}
S
(
n
)
=
{
1
+
g
+
g
2
+
...
+
g
n
−
1
∣
g
∈
N
,
g
≥
2
}
a
)
a)
a
)
Prove that:
S
(
3
)
∩
S
(
4
)
=
∅
S(3)\cap S(4)=\varnothing
S
(
3
)
∩
S
(
4
)
=
∅
b
)
b)
b
)
Determine:
S
(
3
)
∩
S
(
5
)
S(3)\cap S(5)
S
(
3
)
∩
S
(
5
)
2
1
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Convex quadrilateral (KSV IMO TST 2011)
Prove that the lines joining the middle-points of non-adjacent sides of an convex quadrilateral and the line joining the middle-points of diagonals, are concurrent. Prove that the intersection point is the middle point of the three given segments.
1
1
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Cyclic inequality (KSV IMO TST Question 1)
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real positive numbers. Prove that the following inequality holds: { \sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2} }