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Bosnia And Herzegovina - Regional Olympiad
2015 Bosnia And Herzegovina - Regional Olympiad
2015 Bosnia And Herzegovina - Regional Olympiad
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Bosnia And Herzegovina - Regional Olympiad
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Regional Olympiad - FBH 2015 Grade 10 Problem 4
On competition there were
67
67
67
students. They were solving
6
6
6
problems. Student who solves
k
k
k
th problem gets
k
k
k
points, while student who solves incorrectly
k
k
k
th problem gets
−
k
-k
−
k
points.
a
)
a)
a
)
Prove that there exist two students with exactly the same answers to problems
b
)
b)
b
)
Prove that there exist at least
4
4
4
students with same number of points
Regional Olympiad - FBH 2015 Grade 9 Problem 4
Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist
5
5
5
different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box
Regional Olympiad - FBH 2015 Grade 12 Problem 4
It is given set
A
=
{
1
,
2
,
3
,
.
.
.
,
2
n
−
1
}
A=\{1,2,3,...,2n-1\}
A
=
{
1
,
2
,
3
,
...
,
2
n
−
1
}
. From set
A
A
A
, at least
n
−
1
n-1
n
−
1
numbers are expelled such that:
a
)
a)
a
)
if number
a
∈
A
a \in A
a
∈
A
is expelled, and if
2
a
∈
A
2a \in A
2
a
∈
A
then
2
a
2a
2
a
must be expelled
b
)
b)
b
)
if
a
,
b
∈
A
a,b \in A
a
,
b
∈
A
are expelled, and
a
+
b
∈
A
a+b \in A
a
+
b
∈
A
then
a
+
b
a+b
a
+
b
must be also expelled Which numbers must be expelled such that sum of numbers remaining in set stays minimal
3
4
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