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Problems
Contests
International Contests
Balkan MO Shortlist
2011 Balkan MO Shortlist
2011 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(10)
N1
1
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Prove the given congruency
Given an odd number
n
>
1
n >1
n
>
1
, let \begin{align*} S =\{ k \mid 1 \le k < n , \gcd(k,n) =1 \} \end{align*} and let \begin{align*} T = \{ k \mid k \in S , \gcd(k+1,n) =1 \} \end{align*} For each
k
∈
S
k \in S
k
∈
S
, let
r
k
r_k
r
k
be the remainder left by
k
∣
S
∣
−
1
n
\frac{k^{|S|}-1}{n}
n
k
∣
S
∣
−
1
upon division by
n
n
n
. Prove \begin{align*} \prod _{k \in T} \left( r_k - r_{n-k} \right) \equiv |S| ^{|T|} \pmod{n} \end{align*}
C3
1
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partition the set of positive integers into two class
Is it possible to partition the set of positive integer numbers into two classes, none of which contains an infinite arithmetic sequence (with a positive ratio)? What is we impose the extra condition that in each class
C
\mathcal{C}
C
of the partition, the set of difference \begin{align*} \left\{ \min \{ n \in \mathcal{C} \mid n >m \} -m \mid m \in \mathcal{C} \right \} \end{align*} be bounded?
N2
1
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divisibility with p when p=17^{2n}+4 for some natural n
Let
n
∈
N
n \in \mathbb{N}
n
∈
N
such that
p
=
1
7
2
n
+
4
p=17^{2n}+4
p
=
1
7
2
n
+
4
is a prime. Show \begin{align*} p \mid 7^{\tfrac{p-1}{2}} +1 \end{align*}
A2
1
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maximum value of product of non-negative real numbers
Given an integer
n
≥
3
n \geq 3
n
≥
3
, determine the maximum value of product of
n
n
n
non-negative real numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2, \ldots , x_n
x
1
,
x
2
,
…
,
x
n
when subjected to the condition \begin{align*} \sum_{k=1}^n \frac{x_k}{1+x_k} =1 \end{align*}
A4
1
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Inequality with condition xyz=3(x+y+z)
Let
x
,
y
,
z
∈
R
+
x,y,z \in \mathbb{R}^+
x
,
y
,
z
∈
R
+
satisfying
x
y
z
=
3
(
x
+
y
+
z
)
xyz=3(x+y+z)
x
yz
=
3
(
x
+
y
+
z
)
. Prove, that \begin{align*} \sum \frac{1}{x^2(y+1)} \geq \frac{3}{4(x+y+z)} \end{align*}
G4
1
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AA_1xAA_2 + BB_1xBB_2 + CC_1xCC_2 >=1/9 (AB^2 + BC^2 + CA^2)
Given a triangle
A
B
C
ABC
A
BC
, the line parallel to the side
B
C
BC
BC
and tangent to the incircle of the triangle meets the sides
A
B
AB
A
B
and
A
C
AC
A
C
at the points
A
1
A_1
A
1
and
A
2
A_2
A
2
, the points
B
1
,
B
2
B_1, B_2
B
1
,
B
2
and
C
1
,
C
2
C_1, C_2
C
1
,
C
2
are de ned similarly. Show that
A
A
1
⋅
A
A
2
+
B
B
1
⋅
B
B
2
+
C
C
1
⋅
C
C
2
≥
1
9
(
A
B
2
+
B
C
2
+
C
A
2
)
AA_1 \cdot AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2 \ge \frac19 (AB^2 + BC^2 + CA^2)
A
A
1
⋅
A
A
2
+
B
B
1
⋅
B
B
2
+
C
C
1
⋅
C
C
2
≥
9
1
(
A
B
2
+
B
C
2
+
C
A
2
)
G2
1
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<BAN =<CAM wanted, symmedian , circumcircles, angle bisectors related
Let
A
B
C
ABC
A
BC
be a triangle and let
O
O
O
be its circumcentre. The internal and external bisectrices of the angle
B
A
C
BAC
B
A
C
meet the line
B
C
BC
BC
at points
D
D
D
and
E
E
E
, respectively. Let further
M
M
M
and
L
L
L
respectively denote the midpoints of the segments
B
C
BC
BC
and
D
E
DE
D
E
. The circles
A
B
C
ABC
A
BC
and
A
L
O
ALO
A
L
O
meet again at point
N
N
N
. Show that the angles
B
A
N
BAN
B
A
N
and
C
A
M
CAM
C
A
M
are equal.
G3
1
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AFED is cyclic iff A-median meets EF on the circumcircle
Given a triangle
A
B
C
ABC
A
BC
, let
D
D
D
be the midpoint of the side
A
C
AC
A
C
and let
M
M
M
be the point that divides the segment
B
D
BD
B
D
in the ratio
1
/
2
1/2
1/2
; that is,
M
B
/
M
D
=
1
/
2
MB/MD=1/2
MB
/
M
D
=
1/2
. The rays
A
M
AM
A
M
and
C
M
CM
CM
meet the sides
B
C
BC
BC
and
A
B
AB
A
B
at points
E
E
E
and
F
F
F
, respectively. Assume the two rays perpendicular:
A
M
⊥
C
M
AM\perp CM
A
M
⊥
CM
. Show that the quadrangle
A
F
E
D
AFED
A
FE
D
is cyclic if and only if the median from
A
A
A
in triangle
A
B
C
ABC
A
BC
meets the line
E
F
EF
EF
at a point situated on the circle
A
B
C
ABC
A
BC
.
G1
1
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The angles COQ and DOQ are equal
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrangle such that
A
B
=
A
C
=
B
D
AB=AC=BD
A
B
=
A
C
=
B
D
(vertices are labelled in circular order). The lines
A
C
AC
A
C
and
B
D
BD
B
D
meet at point
O
O
O
, the circles
A
B
C
ABC
A
BC
and
A
D
O
ADO
A
D
O
meet again at point
P
P
P
, and the lines
A
P
AP
A
P
and
B
C
BC
BC
meet at the point
Q
Q
Q
. Show that the angles
C
O
Q
COQ
COQ
and
D
O
Q
DOQ
D
OQ
are equal.
A3
1
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Equation implies inequality for real k greater than 1
Let
n
n
n
be an integer number greater than
2
2
2
, let
x
1
,
x
2
,
…
,
x
n
x_{1},x_{2},\ldots ,x_{n}
x
1
,
x
2
,
…
,
x
n
be
n
n
n
positive real numbers such that
∑
i
=
1
n
1
x
i
+
1
=
1
\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1
i
=
1
∑
n
x
i
+
1
1
=
1
and let
k
k
k
be a real number greater than
1
1
1
. Show that:
∑
i
=
1
n
1
x
i
k
+
1
≥
n
(
n
−
1
)
k
+
1
\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}
i
=
1
∑
n
x
i
k
+
1
1
≥
(
n
−
1
)
k
+
1
n
and determine the cases of equality.