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Contests
International Contests
Balkan MO Shortlist
2016 Balkan MO Shortlist
2016 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(15)
A4
1
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min of (a^{-n}+b)/(1-a)+(b^{-n}+c)/(1-b)+(c^{-n}+a)/(1-c) if a+b+c=1, a,b,c>0
The positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the equality
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
. For every natural number
n
n
n
find the minimal possible value of the expression
E
=
a
−
n
+
b
1
−
a
+
b
−
n
+
c
1
−
b
+
c
−
n
+
a
1
−
c
E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}
E
=
1
−
a
a
−
n
+
b
+
1
−
b
b
−
n
+
c
+
1
−
c
c
−
n
+
a
A2
1
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1/(x^2+y+z)+1/(y^2+z+x)+1/(z^2+x+y) <= 1 if x/yz+y/zx+z/xy<=1, for x,y,z>0
For all
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
satisfying
x
y
z
+
y
z
x
+
z
x
y
≤
x
+
y
+
z
\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\le x+y+z
yz
x
+
z
x
y
+
x
y
z
≤
x
+
y
+
z
, prove that
1
x
2
+
y
+
z
+
1
y
2
+
z
+
x
+
1
z
2
+
x
+
y
≤
1
\frac{1}{x^2+y+z}+\frac{1}{y^2+z+x}+\frac{1}{z^2+x+y} \le 1
x
2
+
y
+
z
1
+
y
2
+
z
+
x
1
+
z
2
+
x
+
y
1
≤
1
A5
1
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a+b+c+d=2 and ab+bc+cd+da+ac+bd = 0, find min, max of abcd
Let
a
,
b
,
c
a, b,c
a
,
b
,
c
and
d
d
d
be real numbers such that
a
+
b
+
c
+
d
=
2
a + b + c + d = 2
a
+
b
+
c
+
d
=
2
and
a
b
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
=
0
ab + bc + cd + da + ac + bd = 0
ab
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
=
0
. Find the minimum value and the maximum value of the product
a
b
c
d
abcd
ab
c
d
.
A6
1
Hide problems
f(f(x) + y) = f(x) + 3x + yf(y) , for x,y>0, f: (0,+\infty)\to(0,+\infty),
Prove that there is no function from positive real numbers to itself,
f
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)
f : (0,+\infty)\to(0,+\infty)
f
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)
such that:
f
(
f
(
x
)
+
y
)
=
f
(
x
)
+
3
x
+
y
f
(
y
)
f(f(x) + y) = f(x) + 3x + yf(y)
f
(
f
(
x
)
+
y
)
=
f
(
x
)
+
3
x
+
y
f
(
y
)
,for every
x
,
y
∈
(
0
,
+
∞
)
x,y \in (0,+\infty)
x
,
y
∈
(
0
,
+
∞
)
by Greece, Athanasios Kontogeorgis (aka socrates)
A8
1
Hide problems
f : Z \to Z, f(g(n)) - g(f(n)) is independent on n for any g : Z\to Z.
Find all functions
f
:
Z
→
Z
f : Z \to Z
f
:
Z
→
Z
for which
f
(
g
(
n
)
)
−
g
(
f
(
n
)
)
f(g(n)) - g(f(n))
f
(
g
(
n
))
−
g
(
f
(
n
))
is independent on
n
n
n
for any
g
:
Z
→
Z
g : Z \to Z
g
:
Z
→
Z
.
A1
1
Hide problems
\sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge 4/3 (ab+bc+ca)
Let
a
,
b
,
c
a, b,c
a
,
b
,
c
be positive real numbers. Prove that
a
3
b
+
a
3
c
+
b
3
c
+
b
3
a
+
c
3
a
+
c
3
b
≥
4
3
(
a
b
+
b
c
+
c
a
)
\sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)
a
3
b
+
a
3
c
+
b
3
c
+
b
3
a
+
c
3
a
+
c
3
b
≥
3
4
(
ab
+
b
c
+
c
a
)
N3
1
Hide problems
diophantine (x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)
Find all the integer solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of the equation
(
x
+
y
+
z
)
5
=
80
x
y
z
(
x
2
+
y
2
+
z
2
)
(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)
(
x
+
y
+
z
)
5
=
80
x
yz
(
x
2
+
y
2
+
z
2
)
,
N2
1
Hide problems
d(n) is the largest divisor of the odd number n, different from n
Find all odd natural numbers
n
n
n
such that
d
(
n
)
d(n)
d
(
n
)
is the largest divisor of the number
n
n
n
different from
n
n
n
. (
d
(
n
)
d(n)
d
(
n
)
is the number of divisors of the number n including
1
1
1
and
n
n
n
).
N1
1
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1^{\phi (n)} + 2^{\phi (n)} +... + n^{\phi (n)} coprime with n
Find all natural numbers
n
n
n
for which
1
ϕ
(
n
)
+
2
ϕ
(
n
)
+
.
.
.
+
n
ϕ
(
n
)
1^{\phi (n)} + 2^{\phi (n)} +... + n^{\phi (n)}
1
ϕ
(
n
)
+
2
ϕ
(
n
)
+
...
+
n
ϕ
(
n
)
is coprime with
n
n
n
.
C1
1
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number n is a d-digit palindrome in all number bases b_1,b_2,..., b_K
Let positive integers
K
K
K
and
d
d
d
be given. Prove that there exists a positive integer
n
n
n
and a sequence of
K
K
K
positive integers
b
1
,
b
2
,
.
.
.
,
b
K
b_1,b_2,..., b_K
b
1
,
b
2
,
...
,
b
K
such that the number
n
n
n
is a
d
d
d
-digit palindrome in all number bases
b
1
,
b
2
,
.
.
.
,
b
K
b_1,b_2,..., b_K
b
1
,
b
2
,
...
,
b
K
.
C2
1
Hide problems
2016 costumers in a shop on a particular day, maximal wanted
There are
2016
2016
2016
costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.) Find the maximal
k
k
k
such that the following holds: There are
k
k
k
customers such that either all of them were in the shop at a speci c time instance or no two of them were both in the shop at any time instance.
G3
1
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orthocenter wanted, circumcenters and BF = CE = BC given
Given that
A
B
C
ABC
A
BC
is a triangle where
A
B
<
A
C
AB < AC
A
B
<
A
C
. On the half-lines
B
A
BA
B
A
and
C
A
CA
C
A
we take points
F
F
F
and
E
E
E
respectively such that
B
F
=
C
E
=
B
C
BF = CE = BC
BF
=
CE
=
BC
. Let
M
,
N
M,N
M
,
N
and
H
H
H
be the mid-points of the segments
B
F
,
C
E
BF,CE
BF
,
CE
and
B
C
BC
BC
respectively and
K
K
K
and
O
O
O
be the circumcenters of the triangles
A
B
C
ABC
A
BC
and
M
N
H
MNH
MN
H
respectively. We assume that
O
K
OK
O
K
cuts
B
E
BE
BE
and
H
N
HN
H
N
at the points
A
1
A_1
A
1
and
B
1
B_1
B
1
respectively and that
C
1
C_1
C
1
is the point of intersection of
H
N
HN
H
N
and
F
E
FE
FE
. If the parallel line from
A
1
A_1
A
1
to
O
C
1
OC_1
O
C
1
cuts the line
F
E
FE
FE
at
D
D
D
and the perpendicular from
A
1
A_1
A
1
to the line
D
B
1
DB_1
D
B
1
cuts
F
E
FE
FE
at the point
M
1
M_1
M
1
, prove that
E
E
E
is the orthocenter of the triangle
A
1
O
M
1
A_1OM_1
A
1
O
M
1
.
A7
1
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BMO Shortlist 2016
Find all integers
n
≥
2
n\geq 2
n
≥
2
for which there exist the real numbers
a
k
,
1
≤
k
≤
n
a_k, 1\leq k \leq n
a
k
,
1
≤
k
≤
n
, which are satisfying the following conditions:
∑
k
=
1
n
a
k
=
0
,
∑
k
=
1
n
a
k
2
=
1
and
n
⋅
(
∑
k
=
1
n
a
k
3
)
=
2
(
b
n
−
1
)
,
where
b
=
max
1
≤
k
≤
n
{
a
k
}
.
\sum_{k=1}^n a_k=0, \sum_{k=1}^n a_k^2=1 \text{ and } \sqrt{n}\cdot \Bigr(\sum_{k=1}^n a_k^3\Bigr)=2(b\sqrt{n}-1), \text{ where } b=\max_{1\leq k\leq n} \{a_k\}.
k
=
1
∑
n
a
k
=
0
,
k
=
1
∑
n
a
k
2
=
1
and
n
⋅
(
k
=
1
∑
n
a
k
3
)
=
2
(
b
n
−
1
)
,
where
b
=
1
≤
k
≤
n
max
{
a
k
}
.
N5
1
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Polynomial taking downhill values
A positive integer is called downhill if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial
P
(
x
)
P(x)
P
(
x
)
with rational coefficients takes on an integer value for each downhill positive integer
x
x
x
. Is it necessarily true that
P
(
x
)
P(x)
P
(
x
)
takes on an integer value for each integer
x
x
x
?
G1
1
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Bulgarian National Olympiad 2018 Day 2 P4
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral ,circumscribed about a circle. Let
M
M
M
be a point on the side
A
B
AB
A
B
. Let
I
1
I_{1}
I
1
,
I
2
I_{2}
I
2
and
I
3
I_{3}
I
3
be the incentres of triangles
A
M
D
AMD
A
M
D
,
C
M
D
CMD
CM
D
and
B
M
C
BMC
BMC
respectively. Prove that
I
1
I
2
I
3
M
I_{1}I_{2}I_{3}M
I
1
I
2
I
3
M
is circumscribed.