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International Contests
Balkan MO Shortlist
2015 Balkan MO Shortlist
2015 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(19)
G7
1
Hide problems
lines concurrent with circumcircle
Let scalene triangle
A
B
C
ABC
A
BC
have orthocentre
H
H
H
and circumcircle
Γ
\Gamma
Γ
.
A
H
AH
A
H
meets
Γ
\Gamma
Γ
at
D
D
D
distinct from
A
A
A
.
B
H
BH
B
H
and
C
H
CH
C
H
meet
C
A
CA
C
A
and
A
B
AB
A
B
at
E
E
E
and
F
F
F
respectively, and
E
F
EF
EF
meets
B
C
BC
BC
at
P
P
P
. The tangents to
Γ
\Gamma
Γ
at
B
B
B
and
C
C
C
meet at
T
T
T
. Show that
A
P
AP
A
P
and
D
T
DT
D
T
are concurrent on the circumcircle of
A
F
E
AFE
A
FE
.
G6
1
Hide problems
3 circles given, equal segments wanted
Let
A
B
AB
A
B
be a diameter of a circle
(
ω
)
(\omega)
(
ω
)
with centre
O
O
O
. From an arbitrary point
M
M
M
on
A
B
AB
A
B
such that
M
A
<
M
B
MA < MB
M
A
<
MB
we draw the circles
(
ω
1
)
(\omega_1)
(
ω
1
)
and
(
ω
2
)
(\omega_2)
(
ω
2
)
with diameters
A
M
AM
A
M
and
B
M
BM
BM
respectively. Let
C
D
CD
C
D
be an exterior common tangent of
(
ω
1
)
,
(
ω
2
)
(\omega_1), (\omega_2)
(
ω
1
)
,
(
ω
2
)
such that
C
C
C
belongs to
(
ω
1
)
(\omega_1)
(
ω
1
)
and
D
D
D
belongs to
(
ω
2
)
(\omega_2)
(
ω
2
)
. The point
E
E
E
is diametrically opposite to
C
C
C
with respect to
(
ω
1
)
(\omega_1)
(
ω
1
)
and the tangent to
(
ω
1
)
(\omega_1)
(
ω
1
)
at the point
E
E
E
intersects
(
ω
2
)
(\omega_2)
(
ω
2
)
at the points
F
,
G
F, G
F
,
G
. If the line of the common chord of the circumcircles of the triangles
C
E
D
CED
CE
D
and
C
F
G
CFG
CFG
intersects the circle
(
ω
)
(\omega)
(
ω
)
at the points
K
,
L
K, L
K
,
L
and the circle
(
ω
2
)
(\omega_2)
(
ω
2
)
at the point
N
N
N
(with
N
N
N
closer to
L
L
L
), then prove that
K
C
=
N
L
KC = NL
K
C
=
N
L
.
C3
1
Hide problems
min no of cells, with known cover by 1x10 dominos in 1000x1000 chessboard
A chessboard
1000
×
1000
1000 \times 1000
1000
×
1000
is covered by dominoes
1
×
10
1 \times 10
1
×
10
that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few
N
N
N
cells of the chessboard, for which we know the position of the dominoes that cover them. Which is the minimum
N
N
N
such that after the choice of
N
N
N
and knowing the dominoed that cover them, we can be sure and for the rest of the cover?(Bulgaria)
C2
1
Hide problems
Is it true that k_i<k_j ? combo game with n2^{n-1} questions for 2^n integers
Isaak and Jeremy play the following game. Isaak says to Jeremy that he thinks a few
2
n
2^n
2
n
integers
k
1
,
.
.
,
k
2
n
k_1,..,k_{2^n}
k
1
,
..
,
k
2
n
. Jeremy asks questions of the form: ''Is it true that
k
i
<
k
j
k_i<k_j
k
i
<
k
j
?'' in which Isaak answers by always telling the truth. After
n
2
n
−
1
n2^{n-1}
n
2
n
−
1
questions, Jeramy must decide whether numbers of Isaak are all distinct each other or not. Prove that Jeremy has bo way to be ''sure'' for his final decision.(UK)
N7
1
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1 positive integer anagram of the sum of 3 other positive integers
Positive integer
m
m
m
shall be called anagram of positive
n
n
n
if every digit
a
a
a
appears as many times in the decimal representation of
m
m
m
as it appears in the decimal representation of
n
n
n
also. Is it possible to find
4
4
4
different positive integers such that each of the four to be anagram of the sum of the other
3
3
3
?(Bulgaria)
N1
1
Hide problems
1^2 ,3^2 ,\ldots,(2n-1)^2 on the blackboard, erasing 3 each time
Let
d
d
d
be an even positive integer. John writes the numbers
1
2
,
3
2
,
…
,
(
2
n
−
1
)
2
1^2 ,3^2 ,\ldots,(2n-1)^2
1
2
,
3
2
,
…
,
(
2
n
−
1
)
2
on the blackboard and then chooses three of them, let them be
a
1
,
a
2
,
a
3
{a_1}, {a_2}, {a_3}
a
1
,
a
2
,
a
3
, erases them and writes the number
1
+
∑
1
≤
i
<
j
≤
3
∣
a
i
−
a
j
∣
1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|
1
+
1
≤
i
<
j
≤
3
∑
∣
a
i
−
a
j
∣
He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers
1
2
,
3
2
,
…
,
(
2
n
−
1
)
2
1^2 ,3^2 ,\ldots,(2n-1)^2
1
2
,
3
2
,
…
,
(
2
n
−
1
)
2
.(Albania)
N2
1
Hide problems
a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, prove n^2 / a_n for infinite n
Sequence
(
a
n
)
n
≥
0
(a_n)_{n\geq 0}
(
a
n
)
n
≥
0
is defined as
a
0
=
0
,
a
1
=
1
,
a
2
=
2
,
a
3
=
6
a_{0}=0, a_1=1, a_2=2, a_3=6
a
0
=
0
,
a
1
=
1
,
a
2
=
2
,
a
3
=
6
, and
a
n
+
4
=
2
a
n
+
3
+
a
n
+
2
−
2
a
n
+
1
−
a
n
,
n
≥
0
a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0
a
n
+
4
=
2
a
n
+
3
+
a
n
+
2
−
2
a
n
+
1
−
a
n
,
n
≥
0
. Prove that
n
2
n^2
n
2
divides
a
n
a_n
a
n
for infinite
n
n
n
.(Romania)
N3
1
Hide problems
no (a_s-a_t)/(s-t) is an integer - if p is prime divisor of s-t, then p/(a-1)
Let
a
a
a
be a positive integer. For all positive integer n, we define
a
n
=
1
+
a
+
a
2
+
…
+
a
n
−
1
.
a_n=1+a+a^2+\ldots+a^{n-1}.
a
n
=
1
+
a
+
a
2
+
…
+
a
n
−
1
.
Let
s
,
t
s,t
s
,
t
be two different positive integers with the following property: If
p
p
p
is prime divisor of
s
−
t
s-t
s
−
t
, then
p
p
p
divides
a
−
1
a-1
a
−
1
. Prove that number
a
s
−
a
t
s
−
t
\frac{a_{s}-a_{t}}{s-t}
s
−
t
a
s
−
a
t
is an integer.(FYROM)
N5
1
Hide problems
v_2 (\prod_{n=1}^{2^m}\binom{2n}{n} )=m2^{m-1}+1
For a positive integer
s
s
s
, denote with
v
2
(
s
)
v_2(s)
v
2
(
s
)
the maximum power of
2
2
2
that divides
s
s
s
. Prove that for any positive integer
m
m
m
that:
v
2
(
∏
n
=
1
2
m
(
2
n
n
)
)
=
m
2
m
−
1
+
1.
v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.
v
2
(
n
=
1
∏
2
m
(
n
2
n
)
)
=
m
2
m
−
1
+
1.
(FYROM)
N4
1
Hide problems
a,b relative prime and divide x^3 + y^3 => (a+b - 1) divides (x^3+y^3)
Find all pairs of positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
with the following property: If
a
,
b
a,b
a
,
b
are relative prime and positive divisors of
x
3
+
y
3
x^3 + y^3
x
3
+
y
3
, then
a
+
b
−
1
a+b - 1
a
+
b
−
1
is divisor of
x
3
+
y
3
x^3+y^3
x
3
+
y
3
.(Cyprus)
A6
1
Hide problems
exists P\in R[x]$ of degree 2014^{2015} such that f(P)=2015 ?
For a polynomials
P
∈
R
[
x
]
P\in \mathbb{R}[x]
P
∈
R
[
x
]
, denote
f
(
P
)
=
n
f(P)=n
f
(
P
)
=
n
if
n
n
n
is the smallest positive integer for which is valid
(
∀
x
∈
R
)
(
P
(
P
(
…
P
⏟
n
(
x
)
)
…
)
>
0
)
,
(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),
(
∀
x
∈
R
)
(
n
P
(
P
(
…
P
(
x
))
…
)
>
0
)
,
and
f
(
P
)
=
0
f(P)=0
f
(
P
)
=
0
if such n doeas not exist. Exists polyomial
P
∈
R
[
x
]
P\in \mathbb{R}[x]
P
∈
R
[
x
]
of degree
201
4
2015
2014^{2015}
201
4
2015
such that
f
(
P
)
=
2015
f(P)=2015
f
(
P
)
=
2015
?(Serbia)
A5
1
Hide problems
if (a^{m+1}-1)/(a^m-1)=(b^{n+1}-1)/(b^n-1)=c then a^m c^n > b^n c^{m}
Let
m
,
n
m, n
m
,
n
be positive integers and
a
,
b
a, b
a
,
b
positive real numbers different from
1
1
1
such thath
m
>
n
m > n
m
>
n
and
a
m
+
1
−
1
a
m
−
1
=
b
n
+
1
−
1
b
n
−
1
=
c
\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c
a
m
−
1
a
m
+
1
−
1
=
b
n
−
1
b
n
+
1
−
1
=
c
. Prove that
a
m
c
n
>
b
n
c
m
a^m c^n > b^n c^{m}
a
m
c
n
>
b
n
c
m
(Turkey)
A4
1
Hide problems
(x+y) f(2yf(x)+f(y))=x^3 f(yf(x)) for all x,y\in R^+
Find all functions
f
:
R
+
→
R
+
f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}
f
:
R
+
→
R
+
such that
(
x
+
y
)
f
(
2
y
f
(
x
)
+
f
(
y
)
)
=
x
3
f
(
y
f
(
x
)
)
,
∀
x
,
y
∈
R
+
.
(x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.
(
x
+
y
)
f
(
2
y
f
(
x
)
+
f
(
y
))
=
x
3
f
(
y
f
(
x
))
,
∀
x
,
y
∈
R
+
.
(Albania)
A3
1
Hide problems
m_a (b/a-1)(c/a-1 )+m_b (a/b-1)(c/b-1)+m_c (a/c-1)(b/c-1 ) >= 0
Let a
,
b
,
c
,b,c
,
b
,
c
be sidelengths of a triangle and
m
a
,
m
b
,
m
c
m_a,m_b,m_c
m
a
,
m
b
,
m
c
the medians at the corresponding sides. Prove that
m
a
(
b
a
−
1
)
(
c
a
−
1
)
+
m
b
(
a
b
−
1
)
(
c
b
−
1
)
+
m
c
(
a
c
−
1
)
(
b
c
−
1
)
≥
0.
m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) +m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.
m
a
(
a
b
−
1
)
(
a
c
−
1
)
+
m
b
(
b
a
−
1
)
(
b
c
−
1
)
+
m
c
(
c
a
−
1
)
(
c
b
−
1
)
≥
0.
(FYROM)
A2
1
Hide problems
1/a + b +1/(a+c)+1/(b+c) \leq r/16Rs+ s/16Rr + 11/8s
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be sidelengths of a triangle and
r
,
R
,
s
r,R,s
r
,
R
,
s
be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that:
1
a
+
b
+
1
a
+
c
+
1
b
+
c
≤
r
16
R
s
+
s
16
R
r
+
11
8
s
\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c} \leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}
a
+
b
1
+
a
+
c
1
+
b
+
c
1
≤
16
R
s
r
+
16
R
r
s
+
8
s
11
(Albania)
G1
1
Hide problems
\sin a _A \sin a _B \sina _C\leq \frac{3\sqrt{6}}{32}
In an acute angled triangle
A
B
C
ABC
A
BC
, let
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
be the altitudes. Ray
C
′
B
′
C'B'
C
′
B
′
intersects the circumcircle at
B
′
′
B''
B
′′
andl let
α
A
\alpha_A
α
A
be the angle
A
B
B
′
′
^
\widehat{ABB''}
A
B
B
′′
. Similarly are defined the angles
α
B
\alpha_B
α
B
and
α
C
\alpha_C
α
C
. Prove that
sin
α
A
sin
α
B
sin
α
C
≤
3
6
32
\displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}
sin
α
A
sin
α
B
sin
α
C
≤
32
3
6
(Romania)
G2
1
Hide problems
collinear points, starting with the circumcircle
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
ω
\omega
ω
. Point
D
D
D
lies on the arc
B
C
BC
BC
of
ω
\omega
ω
and is different than
B
,
C
B,C
B
,
C
and the midpoint of arc
B
C
BC
BC
. Tangent of
Γ
\Gamma
Γ
at
D
D
D
intersects lines
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
, respectively. Lines
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
intersect at
E
E
E
. Line
A
A
′
AA'
A
A
′
intersects the circle
ω
\omega
ω
again at
F
F
F
. Prove that points
D
,
E
,
F
D,E,F
D
,
E
,
F
are collinear.(Saudi Arabia)
G3
1
Hide problems
obtuse-angled set, every triangle has one angle > 91^o
A set of points of the plane is called obtuse-angled if every three of it's points are not collinear and every triangle with vertices inside the set has one angle
>
9
1
o
>91^o
>
9
1
o
. Is it correct that every finite obtuse-angled set can be extended to an infinite obtuse-angled set?(UK)
G5
1
Hide problems
2015 Bulgaria Team Selection Test Round 1, Problem 2
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is given with
A
D
∦
B
C
AD \nparallel BC
A
D
∦
BC
. The midpoints of
A
D
AD
A
D
and
B
C
BC
BC
are denoted by
M
M
M
and
N
N
N
, respectively. The line
M
N
MN
MN
intersects the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
in points
K
K
K
and
L
L
L
, respectively. Prove that the circumcircles of the triangles
A
K
M
AKM
A
K
M
and
B
N
L
BNL
BN
L
have common point on the line
A
B
AB
A
B
.( Proposed by Emil Stoyanov ) http://estoyanov.net/wp-content/uploads/2015/09/est.png