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Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2002 Bosnia Herzegovina Team Selection Test
2002 Bosnia Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(3)
3
1
Hide problems
Solve for x,y,z in integers given p,q are distinct primes
Let
p
p
p
and
q
q
q
be different prime numbers. Solve the following system in integers:
z
+
p
x
+
z
−
p
y
=
q
,
z
+
p
y
−
z
−
p
x
=
q
.
\frac{z+ p}x+\frac{z-p}y= q,\\ \frac{z+ p}y -\frac{z-p}x= q.
x
z
+
p
+
y
z
−
p
=
q
,
y
z
+
p
−
x
z
−
p
=
q
.
2
2
Hide problems
Show that 2MN=BM+CN
Triangle
A
B
C
ABC
A
BC
is given in a plane. Draw the bisectors of all three of its angles. Then draw the line that connects the points where the bisectors of angles
A
B
C
ABC
A
BC
and
A
C
B
ACB
A
CB
meet the opposite sides of the triangle. Through the point of intersection of this line and the bisector of angle
B
A
C
BAC
B
A
C
, draw another line parallel to
B
C
BC
BC
. Let this line intersect
A
B
AB
A
B
in
M
M
M
and
A
C
AC
A
C
in
N
N
N
. Prove that
2
M
N
=
B
M
+
C
N
2MN = BM+CN
2
MN
=
BM
+
CN
.
The vertices of the convex quad ABCD are integer points
The vertices of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
and the intersection point
S
S
S
of its diagonals are integer points in the plane. Let
P
P
P
be the area of
A
B
C
D
ABCD
A
BC
D
and
P
1
P_1
P
1
the area of triangle
A
B
S
ABS
A
BS
. Prove that \sqrt{P} \ge \sqrt{P_1}+\frac{\sqrt2}2
1
2
Hide problems
Find 'a' for which max(x)-min(x)=8
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers that satisfy
x
+
y
+
z
=
3
and
x
y
+
y
z
+
z
x
=
a
x+y+z= 3 \ \ \text{ and } \ \ xy+yz+zx= a
x
+
y
+
z
=
3
and
x
y
+
yz
+
z
x
=
a
where
a
a
a
is a real parameter. Find the value of
a
a
a
for which the difference between the maximum and minimum possible values of
x
x
x
equals
8
8
8
.
Inequality with a^2+b^2+c^2=1; P4 : B&H TST 2002
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers such that
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
. Prove that
a
2
1
+
2
b
c
+
b
2
1
+
2
c
a
+
c
2
1
+
2
a
b
≥
3
5
\frac{a^2}{1+2bc}+\frac{b^2}{1+2ca}+\frac{c^2}{1+2ab} \ge \frac35
1
+
2
b
c
a
2
+
1
+
2
c
a
b
2
+
1
+
2
ab
c
2
≥
5
3