MathDB
Problems
Contests
National and Regional Contests
Kosovo Contests
Kosovo Team Selection Test
2012 Kosovo Team Selection Test
2012 Kosovo Team Selection Test
Part of
Kosovo Team Selection Test
Subcontests
(5)
5
1
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infinitly natural solution
Prove that the equation
4
n
=
1
x
+
1
y
+
1
z
\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}
n
4
=
x
1
+
y
1
+
z
1
has infinitly many natural solutions
4
1
Hide problems
1,0,1,0,1,0,3,5,...
Each term in a sequence
1
,
0
,
1
,
0
,
1
,
0...
1,0,1,0,1,0...
1
,
0
,
1
,
0
,
1
,
0...
starting with the seventh is the sum of the last 6 terms mod 10 .Prove that the sequence
.
.
.
,
0
,
1
,
0
,
1
,
0
,
1...
...,0,1,0,1,0,1...
...
,
0
,
1
,
0
,
1
,
0
,
1...
never occurs
3
1
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4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are the sides of a triangle and
m
a
,
m
b
,
m
c
m_a , m_b, m_c
m
a
,
m
b
,
m
c
are the medians prove that
4
(
m
a
2
+
m
b
2
+
m
c
2
)
=
3
(
a
2
+
b
2
+
c
2
)
4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)
4
(
m
a
2
+
m
b
2
+
m
c
2
)
=
3
(
a
2
+
b
2
+
c
2
)
2
1
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a^2+b^2+c^2=90
Find all three digit numbers, for which the sum of squares of each digit is
90
90
90
.
1
1
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Prove that the student was wrong during the counting.
A student had
18
18
18
papers. He seleced some of these papers, then he cut each of them in
18
18
18
pieces.He took these pieces and selected some of them, which he again cut in
18
18
18
pieces each.The student took this procedure untill he got tired .After a time he counted the pieces and got
2012
2012
2012
pieces .Prove that the student was wrong during the counting.