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Bosnia And Herzegovina - Regional Olympiad
2017 Bosnia And Herzegovina - Regional Olympiad
2017 Bosnia And Herzegovina - Regional Olympiad
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Bosnia And Herzegovina - Regional Olympiad
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4
4
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Regional Olympiad - FBH 2017 Grade 10 Problem 3
Find prime numbers
p
p
p
,
q
q
q
,
r
r
r
and
s
s
s
, pairwise distinct, such that their sum is prime number and numbers
p
2
+
q
r
p^2+qr
p
2
+
q
r
and
p
2
+
q
s
p^2+qs
p
2
+
q
s
are perfect squares
Regional Olympiad - FBH 2017 Grade 9 Problem 3
Does there exist positive integer
n
n
n
such that sum of all digits of number
n
(
4
n
+
1
)
n(4n+1)
n
(
4
n
+
1
)
is equal to
2017
2017
2017
Regional Olympiad - FBH 2017 Grade 11 Problem 3
Let
S
S
S
be a set of
6
6
6
positive real numbers such that
(
a
,
b
∈
S
)
(
a
>
b
)
⇒
a
+
b
∈
S
\left(a,b \in S \right) \left(a>b \right) \Rightarrow a+b \in S
(
a
,
b
∈
S
)
(
a
>
b
)
⇒
a
+
b
∈
S
or
a
−
b
∈
S
a-b \in S
a
−
b
∈
S
Prove that if we sort these numbers in ascending order, then they form an arithmetic progression
2
4
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1
3
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Regional Olympiad - FBH 2017 Grade 9 Problem 1
Let
a
a
a
,
b
b
b
and
c
c
c
be real numbers such that
a
b
c
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≠
0
abc(a+b)(b+c)(c+a)\neq0
ab
c
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
=
0
and
(
a
+
b
+
c
)
(
1
a
+
1
b
+
1
c
)
=
1007
1008
(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}
(
a
+
b
+
c
)
(
a
1
+
b
1
+
c
1
)
=
1008
1007
Prove that
a
b
(
a
+
c
)
(
b
+
c
)
+
b
c
(
b
+
a
)
(
c
+
a
)
+
c
a
(
c
+
b
)
(
a
+
b
)
=
2017
\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017
(
a
+
c
)
(
b
+
c
)
ab
+
(
b
+
a
)
(
c
+
a
)
b
c
+
(
c
+
b
)
(
a
+
b
)
c
a
=
2017
Regional Olympiad - FBH 2017 Grade 10 Problem 1
If
a
a
a
is real number such that
x
1
x_1
x
1
and
x
2
x_2
x
2
,
x
1
≠
x
2
x_1\neq x_2
x
1
=
x
2
, are real numbers and roots of equation
x
2
−
x
+
a
=
0
x_2-x+a=0
x
2
−
x
+
a
=
0
. Prove that
∣
x
1
2
−
x
2
2
∣
=
1
\mid {x_1}^2-{x_2}^2 \mid =1
∣
x
1
2
−
x
2
2
∣=
1
iff
∣
x
1
3
−
x
2
3
∣
=
1
\mid {x_1}^3-{x_2}^3 \mid =1
∣
x
1
3
−
x
2
3
∣=
1
Regional Olympiad - FBH 2017 Grade 11 Problem 1
In terms of real parameter
a
a
a
solve inequality:
log
a
x
+
∣
a
+
log
a
x
∣
⋅
log
x
a
≥
a
log
x
a
\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}
lo
g
a
x
+
∣
a
+
lo
g
a
x
∣
⋅
lo
g
x
a
≥
a
lo
g
x
a
in set of real numbers