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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia JBMO TST
2012 Serbia JBMO TST
2012 Serbia JBMO TST
Part of
Serbia JBMO TST
Subcontests
(4)
4
1
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Using only a ruler and a compass, find the origin of the coordinate system.
In a coordinate system there are drawn the graphs of the functions
y
=
a
x
+
b
y=ax+b
y
=
a
x
+
b
and
y
=
b
x
+
a
,
(
a
≠
b
)
y=bx+a, (a\neq b)
y
=
b
x
+
a
,
(
a
=
b
)
. Their intersection is marked with red and their intersections with the
O
y
Oy
O
y
axis are marked with blue. Everything is erased except the marked points. Using only a ruler and a compass, find the origin of the coordinate system.
3
1
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a, \overline{bcd}, \overline{aef}, \overline{cfg}, \overline{hci},
Let
a
,
b
c
d
‾
,
a
e
f
‾
,
c
f
g
‾
,
h
c
i
‾
,
d
e
a
‾
,
i
f
d
‾
,
j
g
f
‾
,
b
f
e
g
‾
,
…
a, \overline{bcd}, \overline{aef}, \overline{cfg}, \overline{hci}, \overline{dea}, \overline{ifd}, \overline{jgf}, \overline{bfeg},\ldots
a
,
b
c
d
,
a
e
f
,
c
f
g
,
h
c
i
,
d
e
a
,
i
fd
,
j
g
f
,
b
f
e
g
,
…
be an increasing arithmetic progression. Find the
16
16
16
th term of this sequence.
2
1
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$x^2+y^2+z^2-xy-yz-zx=3$
Show that the equation
x
2
+
y
2
+
z
2
−
x
y
−
y
z
−
z
x
=
3
x^2+y^2+z^2-xy-yz-zx=3
x
2
+
y
2
+
z
2
−
x
y
−
yz
−
z
x
=
3
has an infinity solutions over nonnegative integers.
1
1
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Find all $4$-digit numbers $\overline{abba}$
Find all
4
4
4
-digit numbers
a
b
b
a
‾
\overline{abba}
abba
that are equal to the product of some consecutive prime numbers.