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Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2005 Bosnia and Herzegovina Junior BMO TST
2005 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
3
1
Hide problems
In which position of this sequence is 2005/ 2004 ?
Rational numbers are written in the following sequence:
1
1
,
2
1
,
1
2
,
3
1
,
2
2
,
1
3
,
4
1
,
3
2
,
2
3
,
1
4
,
.
.
.
\frac{1}{1},\frac{2}{1},\frac{1}{2},\frac{3}{1},\frac{2}{2},\frac{1}{3},\frac{4}{1},\frac{3}{2},\frac{2}{3},\frac{1}{4}, . . .
1
1
,
1
2
,
2
1
,
1
3
,
2
2
,
3
1
,
1
4
,
2
3
,
3
2
,
4
1
,
...
In which position of this sequence is
2005
2004
\frac{2005}{2004}
2004
2005
?
2
1
Hide problems
2 + 2\sqrt{1 + 28n^2} is an integer, then it is the square of an integer
Let n be a positive integer. Prove the following statement: ”If
2
+
2
1
+
28
n
2
2 + 2\sqrt{1 + 28n^2}
2
+
2
1
+
28
n
2
is an integer, then it is the square of an integer.”
1
1
Hide problems
min max of w = 2x − 3y + 4z when 3x + 5y + 7z = 10 and x + 2y + 5z = 6
Non-negative real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfy the following relations:
3
x
+
5
y
+
7
z
=
10
3x + 5y + 7z = 10
3
x
+
5
y
+
7
z
=
10
and
x
+
2
y
+
5
z
=
6
x + 2y + 5z = 6
x
+
2
y
+
5
z
=
6
.Find the minimum and maximum of
w
=
2
x
−
3
y
+
4
z
w = 2x - 3y + 4z
w
=
2
x
−
3
y
+
4
z
.
4
1
Hide problems
equal segments in trapezoid
The sum of the angles on the bigger base of a trapezoid is
9
0
o
90^o
9
0
o
. Prove that the line segment whose ends are the midpoints of the bases, is equal to the line segment whose ends are the midpoints of the diagonals.