MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2003 Junior Macedonian Mathematical Olympiad
2003 Junior Macedonian Mathematical Olympiad
Part of
JBMO TST - Macedonia
Subcontests
(5)
Problem 2
1
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Coins in bags and pockets
There are
2003
2003
2003
coins distributed in several bags. The bags are then distributed in several pockets. It is known that the total number of bags is greater than the number of coins in each of the pockets. Is it true that the total number of pockets is greater than the number of coins in some of the bags?
Problem 5
1
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Covering a board with 1x2 horizontal dominoes and 3x1 vertical trominoes
Is it possible to cover a
2003
×
2003
2003 \times 2003
2003
×
2003
chessboard (without overlap) using only horizontal
1
×
2
1 \times 2
1
×
2
dominoes and only vertical
3
×
1
3 \times 1
3
×
1
trominoes?
Problem 4
1
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Three variable inequality with Jensen like condition
Let
x
x
x
,
y
y
y
and
z
z
z
be positive real numbers such that
x
+
y
+
z
=
1
x+y+z = 1
x
+
y
+
z
=
1
. Prove the inequality:
x
2
1
+
y
+
y
2
1
+
z
+
z
2
1
+
x
≤
1
\frac{x^2}{1+y}+\frac{y^2}{1+z} +\frac{z^2}{1+x} \leq 1
1
+
y
x
2
+
1
+
z
y
2
+
1
+
x
z
2
≤
1
Problem 3
1
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Geometric inequality for radii of incircle and circumcircle
Let
A
B
C
ABC
A
BC
be a given triangle. The circumcircle of the triangle has radius
R
R
R
, the incircle has radius
r
r
r
, the longest side of the triangle is
a
a
a
, while the shortest altitude is
h
h
h
. Show that:
R
r
>
a
h
\frac{R}{r} > \frac{a}{h}
r
R
>
h
a
.
Problem 1
1
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6^n-1 is never a divisor of 7^n-1
Show that for every positive integer
n
n
n
the number
7
n
−
1
7^n-1
7
n
−
1
is not divisible by
6
n
−
1
6^n-1
6
n
−
1
.