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Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2021 Bosnia and Herzegovina Junior BMO TST
2021 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
2
1
Hide problems
p^2 +qt =(p + t)^n, p^2 + qr = t^4, diophantine with primes
Let
p
,
q
,
r
p, q, r
p
,
q
,
r
be prime numbers and
t
,
n
t, n
t
,
n
be natural numbers such that
p
2
+
q
t
=
(
p
+
t
)
n
p^2 +qt =(p + t)^n
p
2
+
qt
=
(
p
+
t
)
n
and
p
2
+
q
r
=
t
4
p^2 + qr = t^4
p
2
+
q
r
=
t
4
. a) Show that
n
<
3
n < 3
n
<
3
. b) Determine all the numbers
p
,
q
,
r
,
t
,
n
p, q, r, t, n
p
,
q
,
r
,
t
,
n
that satisfy the given conditions.
4
1
Hide problems
beautiful 3xn board with integers from 1 to n
Let
n
n
n
be a nonzero natural number and let
S
=
{
1
,
2
,
.
.
.
,
n
}
S = \{1, 2, . . . , n\}
S
=
{
1
,
2
,
...
,
n
}
. A
3
×
n
3 \times n
3
×
n
board is called beautiful if it can be completed with numbers from the set
S
S
S
like this as long as the following conditions are met:
∙
\bullet
∙
on each line, each number from the set S appears exactly once,
∙
\bullet
∙
on each column the sum of the products of two numbers on that column is divisible by
n
n
n
(that is, if the numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are written on a column, it must be
a
b
+
b
c
+
c
a
ab + bc + ca
ab
+
b
c
+
c
a
be divisible by
n
n
n
). For which values of the natural number
n
n
n
are there beautiful tables ¸and for which values do not exist? Justify your answer.
3
1
Hide problems
<ACB=90^o if AD = BD , <ACD = 3 <BAC, AM=//MD, CM//AB,
In the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
,
A
D
=
B
D
AD = BD
A
D
=
B
D
and
∠
A
C
D
=
3
∠
B
A
C
\angle ACD = 3 \angle BAC
∠
A
C
D
=
3∠
B
A
C
. Let
M
M
M
be the midpoint of side
A
D
AD
A
D
. If the lines
C
M
CM
CM
and
A
B
AB
A
B
are parallel, prove that the angle
∠
A
C
B
\angle ACB
∠
A
CB
is right.
1
1
Hide problems
ab + c + d = 3, bc + d + a = 5, cd + a + b = 2, da + b + c = 6
Determine all real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
for which
a
b
+
c
+
d
=
3
ab + c + d = 3
ab
+
c
+
d
=
3
b
c
+
d
+
a
=
5
bc + d + a = 5
b
c
+
d
+
a
=
5
c
d
+
a
+
b
=
2
cd + a + b = 2
c
d
+
a
+
b
=
2
d
a
+
b
+
c
=
6
da + b + c = 6
d
a
+
b
+
c
=
6