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Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2005 Serbia Team Selection Test
2005 Serbia Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(6)
3
2
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Nice problem about polynomial
Find all polynomial with real coefficients such that: P(x^2+1)=P(x)^2+1
TST BMO 2005 serbia
problem 3: (a) Show that there exists a multiple of 2005 whose sum of (decimal) digits equals 2. (b) Let
x
n
x_n
x
n
denote the number obtained by writing natural numbers from
1
1
1
to
n
n
n
one after another (for example,
x
1
=
1
,
x
2
=
12
,
.
.
.
,
x
13
=
12345678910111213
x_1 = 1, x_2 = 12,...,x_{13} = 12345678910111213
x
1
=
1
,
x
2
=
12
,
...
,
x
13
=
12345678910111213
). Prove that the sequence
x
1
,
x
2
,
.
.
.
x_1,x_2,...
x
1
,
x
2
,
...
contains infinitely many terms that are divisiblenby 2005.
5
1
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simple ineq....
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals such that
a
b
c
=
1
abc=1
ab
c
=
1
.Prove the inequality
a
a
2
+
2
+
b
b
2
+
2
+
c
c
2
+
2
≤
1
\frac{a}{a^2+2}+\frac{b}{b^2+2}+\frac{c}{c^2+2}\leq 1
a
2
+
2
a
+
b
2
+
2
b
+
c
2
+
2
c
≤
1
6
1
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a hard one
We say that
n
n
n
squares in a
n
×
n
n\times n
n
×
n
board are scattered if no two of them are in the same row or column.In every square of this board is witten a natural number so that the sum of numbrs in
n
n
n
scattered squares is always the same and no row or no column contains two equal numbers .It turned out that the numbers on the main diagonal are arranged in the increasing order ,and that their product is the smallest among all products of
n
n
n
scattered numbers .Prove that scattered numbers with the greatest product are exactly those on the other diagonal.
2
2
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Convex angle in the plane, unique point
A convex angle
x
O
y
xOy
x
O
y
and a point
M
M
M
inside it are given in the plane. Prove that there is a unique point
P
P
P
in the plane with the following property: - For any line
l
l
l
through
M
M
M
, meeting the rays
x
x
x
and
y
y
y
(or their extensions) at
X
X
X
and
Y
Y
Y
, the angle
X
P
Y
XPY
XP
Y
is not obtuse.
serbia BMO 2005
p
r
o
b
l
e
m
2
problem2
p
ro
b
l
e
m
2
:Determine the number of 100-digit numbers whose all digits are odd, and in which every two consecutive digits differ by 2
1
2
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A trig one with absurb
Prove that there is n rational number
r
r
r
such that
c
o
s
r
π
=
3
5
cosr\pi=\frac{3}{5}
cosr
π
=
5
3
serbia BMO
problem 1 :A sequence is defined by
x
1
=
1
,
x
2
=
4
x_1 = 1, x_2 = 4
x
1
=
1
,
x
2
=
4
and
x
n
+
2
=
4
x
n
+
1
−
x
n
x_{n+2} = 4x_{n+1} -x_n
x
n
+
2
=
4
x
n
+
1
−
x
n
for
n
≥
1
n \geq 1
n
≥
1
. Find all natural numbers
m
m
m
such that the number
3
x
n
2
+
m
3x_n^2 + m
3
x
n
2
+
m
is a perfect square for all natural numbers
n
n
n
4
1
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angle inequality involving centroid of triangle
Let
T
T
T
be the centroid of triangle
A
B
C
ABC
A
BC
. Prove that
1
sin
∠
T
A
C
+
1
sin
∠
T
B
C
≥
4
\frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4
sin
∠
T
A
C
1
+
sin
∠
TBC
1
≥
4