MathDB
Problems
Contests
National and Regional Contests
Belarus Contests
Belarus Team Selection Test
2019 Belarus Team Selection Test
2019 Belarus Team Selection Test
Part of
Belarus Team Selection Test
Subcontests
(16)
8.2
1
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FE in integers
Let
Z
\mathbb Z
Z
be the set of all integers. Find all functions
f
:
Z
→
Z
f:\mathbb Z\to\mathbb Z
f
:
Z
→
Z
satisfying the following conditions: 1.
f
(
f
(
x
)
)
=
x
f
(
x
)
−
x
2
+
2
f(f(x))=xf(x)-x^2+2
f
(
f
(
x
))
=
x
f
(
x
)
−
x
2
+
2
for all
x
∈
Z
x\in\mathbb Z
x
∈
Z
; 2.
f
f
f
takes all integer values.(I. Voronovich)
7.1
1
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Just angles
The internal bisectors of angles
∠
D
A
B
\angle DAB
∠
D
A
B
and
∠
B
C
D
\angle BCD
∠
BC
D
of a quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at the point
X
1
X_1
X
1
, and the external bisectors of these angles intersect at the point
X
2
X_2
X
2
. The internal bisectors of angles
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
C
D
A
\angle CDA
∠
C
D
A
intersect at the point
Y
1
Y_1
Y
1
, and the external bisectors of these angles intersect at the point
Y
2
Y_2
Y
2
. Prove that the angle between the lines
X
1
X
2
X_1X_2
X
1
X
2
and
Y
1
Y
2
Y_1Y_2
Y
1
Y
2
equals the angle between the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
.(A. Voidelevich)
6.2
1
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A process with numbers
The numbers
1
,
2
,
…
,
49
,
50
1,2,\ldots,49,50
1
,
2
,
…
,
49
,
50
are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
from the board, replaces them by their sum
a
+
b
+
c
a+b+c
a
+
b
+
c
and writes
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
(a+b)(b+c)(c+a)
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all
24
24
24
numbers written in the notebook. Let
A
A
A
and
B
B
B
be the maximum and the minimum possible sums that Ann san obtain. Find the value of
A
B
\frac{A}{B}
B
A
.(I. Voronovich)
6.1
1
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Something about Archimedes' lemma
Two circles
Ω
\Omega
Ω
and
Γ
\Gamma
Γ
are internally tangent at the point
B
B
B
. The chord
A
C
AC
A
C
of
Γ
\Gamma
Γ
is tangent to
Ω
\Omega
Ω
at the point
L
L
L
, and the segments
A
B
AB
A
B
and
B
C
BC
BC
intersect
Ω
\Omega
Ω
at the points
M
M
M
and
N
N
N
. Let
M
1
M_1
M
1
and
N
1
N_1
N
1
be the reflections of
M
M
M
and
N
N
N
about the line
B
L
BL
B
L
; and let
M
2
M_2
M
2
and
N
2
N_2
N
2
be the reflections of
M
M
M
and
N
N
N
about the line
A
C
AC
A
C
. The lines
M
1
M
2
M_1M_2
M
1
M
2
and
N
1
N
2
N_1N_2
N
1
N
2
intersect at the point
K
K
K
. Prove that the lines
B
K
BK
B
K
and
A
C
AC
A
C
are perpendicular.(M. Karpuk)
5.3
1
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Plump polygons
A polygon (not necessarily convex) on the coordinate plane is called plump if it satisfies the following conditions:
∙
\bullet
∙
coordinates of vertices are integers;
∙
\bullet
∙
each side forms an angle of
0
∘
0^\circ
0
∘
,
9
0
∘
90^\circ
9
0
∘
, or
4
5
∘
45^\circ
4
5
∘
with the abscissa axis;
∙
\bullet
∙
internal angles belong to the interval
[
9
0
∘
,
27
0
∘
]
[90^\circ, 270^\circ]
[
9
0
∘
,
27
0
∘
]
. Prove that if a square of each side length of a plump polygon is even, then such a polygon can be cut into several convex plump polygons.(A. Yuran)
5.2
1
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Moving points and lines
Let
A
A
1
AA_1
A
A
1
be the bisector of a triangle
A
B
C
ABC
A
BC
. Points
D
D
D
and
F
F
F
are chosen on the line
B
C
BC
BC
such that
A
1
A_1
A
1
is the midpoint of the segment
D
F
DF
D
F
. A line
l
l
l
, different from
B
C
BC
BC
, passes through
A
1
A_1
A
1
and intersects the lines
A
B
AB
A
B
and
A
C
AC
A
C
at points
B
1
B_1
B
1
and
C
1
C_1
C
1
, respectively. Find the locus of the points of intersection of the lines
B
1
D
B_1D
B
1
D
and
C
1
F
C_1F
C
1
F
for all possible positions of
l
l
l
.(M. Karpuk)
5.1
1
Hide problems
A natural question about division
A function
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
, where
N
\mathbb N
N
is the set of positive integers, satisfies the following condition: for any positive integers
m
m
m
and
n
n
n
(
m
>
n
m>n
m
>
n
) the number
f
(
m
)
−
f
(
n
)
f(m)-f(n)
f
(
m
)
−
f
(
n
)
is divisible by
m
−
n
m-n
m
−
n
. Is the function
f
f
f
necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial
p
(
x
)
p(x)
p
(
x
)
with real coefficients such that
f
(
n
)
=
p
(
n
)
f(n)=p(n)
f
(
n
)
=
p
(
n
)
for all positive integers
n
n
n
?)(Folklore)
2.4
1
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Colorings of a table
Cells of
11
×
11
11\times 11
11
×
11
table are colored with
n
n
n
colors (each cell is colored with exactly one color). For each color, the total amount of the cells of this color is not less than
7
7
7
and not greater than
13
13
13
. Prove that there exists at least one row or column which contains cells of at least four different colors.(N. Sedrakyan)
2.3
1
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A process with boxes containing stones
1019
1019
1019
stones are placed into two non-empty boxes. Each second Alex chooses a box with an even amount of stones and shifts half of these stones into another box. Prove that for each
k
k
k
,
1
≤
k
≤
1018
1\le k\le1018
1
≤
k
≤
1018
, at some moment there will be a box with exactly
k
k
k
stones.(O. Izhboldin)
2.2
1
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A circle containing nine-point cener
Let
O
O
O
be the circumcenter and
H
H
H
be the orthocenter of an acute-angled triangle
A
B
C
ABC
A
BC
. Point
T
T
T
is the midpoint of the segment
A
O
AO
A
O
. The perpendicular bisector of
A
O
AO
A
O
intersects the line
B
C
BC
BC
at point
S
S
S
. Prove that the circumcircle of the triangle
A
S
T
AST
A
ST
bisects the segment
O
H
OH
O
H
.(M. Berindeanu, RMC 2018 book)
2.1
1
Hide problems
Existence problem about quadratic trinomial
Given a quadratic trinomial
p
(
x
)
p(x)
p
(
x
)
with integer coefficients such that
p
(
x
)
p(x)
p
(
x
)
is not divisible by
3
3
3
for all integers
x
x
x
. Prove that there exist polynomials
f
(
x
)
f(x)
f
(
x
)
and
h
(
x
)
h(x)
h
(
x
)
with integer coefficients such that
p
(
x
)
⋅
f
(
x
)
+
3
h
(
x
)
=
x
6
+
x
4
+
x
2
+
1.
p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1.
p
(
x
)
⋅
f
(
x
)
+
3
h
(
x
)
=
x
6
+
x
4
+
x
2
+
1.
(I. Gorodnin)
1.4
1
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Inequality with a sequence
Let the sequence
(
a
n
)
(a_n)
(
a
n
)
be constructed in the following way: a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots. Prove that
a
180
>
19
a_{180}>19
a
180
>
19
.(Folklore)
1.3
1
Hide problems
Playing with equation in integers
Given the equation
a
b
⋅
b
c
=
c
a
a^b\cdot b^c=c^a
a
b
⋅
b
c
=
c
a
in positive integers
a
a
a
,
b
b
b
, and
c
c
c
. (i) Prove that any prime divisor of
a
a
a
divides
b
b
b
as well. (ii) Solve the equation under the assumption
b
≥
a
b\ge a
b
≥
a
. (iii) Prove that the equation has infinitely many solutions.(I. Voronovich)
1.2
1
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Inequality with medians
Points
M
M
M
and
N
N
N
are the midpoints of the sides
B
C
BC
BC
and
A
D
AD
A
D
, respectively, of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Is it possible that
A
B
+
C
D
>
max
(
A
M
+
D
M
,
B
N
+
C
N
)
?
AB+CD>\max(AM+DM,BN+CN)?
A
B
+
C
D
>
max
(
A
M
+
D
M
,
BN
+
CN
)?
(Folklore)
1.1
1
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FE with interesting statement
Does there exist a function
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
such that
f
(
f
(
n
+
1
)
)
=
f
(
f
(
n
)
)
+
2
n
−
1
f(f(n+1))=f(f(n))+2^{n-1}
f
(
f
(
n
+
1
))
=
f
(
f
(
n
))
+
2
n
−
1
for any positive integer
n
n
n
? (As usual,
N
\mathbb N
N
stands for the set of positive integers.)(I. Gorodnin)
8.3
1
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Classical problem
Prove that for
n
>
1
n>1
n
>
1
,
n
n
n
does not divide
2
n
−
1
+
1
2^{n-1}+1
2
n
−
1
+
1