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Contests
National and Regional Contests
North Macedonia Contests
North Macedonia National Olympiads
1999 North Macedonia National Olympiad
1999 North Macedonia National Olympiad
Part of
North Macedonia National Olympiads
Subcontests
(5)
5
1
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a+b+c+1/abc \ge 4\sqrt3 if a^2 +b^2 +c^2 = 1 with a,b,c >0
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive numbers with
a
2
+
b
2
+
c
2
=
1
a^2 +b^2 +c^2 = 1
a
2
+
b
2
+
c
2
=
1
, prove that
a
+
b
+
c
+
1
a
b
c
≥
4
3
a+b+c+\frac{1}{abc} \ge 4\sqrt3
a
+
b
+
c
+
ab
c
1
≥
4
3
4
1
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100 straight lines on plane with exaclty 1999 intersection points
Do there exist
100
100
100
straight lines on a plane such that they intersect each other in exactly
1999
1999
1999
points?
3
1
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angle bisector wanted, two tangents given
Let the two tangents from a point
A
A
A
outside a circle
k
k
k
touch
k
k
k
at
M
M
M
and
N
N
N
. A line
p
p
p
through
A
A
A
intersects
k
k
k
at
B
B
B
and
C
C
C
, and
D
D
D
is the midpoint of
M
N
MN
MN
. Prove that
M
N
MN
MN
bisects the angle
B
D
C
BDC
B
D
C
2
1
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13 balls, of which one has different weight, 3 weighings
We are given
13
13
13
apparently equal balls, all but one having the same weight (the remaining one has a different weight). Is it posible to determine the ball with the different weight in
3
3
3
weighings?
1
1
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in set of 21 real numbers sum of any 10 is less than sum of rest 11
In a set of
21
21
21
real numbers, the sum of any
10
10
10
numbers is less than the sum of the remaining
11
11
11
numbers. Prove that all the numbers are positive.