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National and Regional Contests
Belarus Contests
Belarus Team Selection Test
2024 Belarus Team Selection Test
2024 Belarus Team Selection Test
Part of
Belarus Team Selection Test
Subcontests
(16)
4.4
1
Hide problems
Placing checkers in circles
Given positive integers
n
n
n
and
k
≤
n
k \leq n
k
≤
n
. Consider an equilateral triangular board with side
n
n
n
, which consists of circles: in the first (top) row there is one circle, in the second row there are two circles,
…
\ldots
…
, in the bottom row there are
n
n
n
circles (see the figure below). Let us place checkers on this board so that any line parallel to a side of the triangle (there are
3
n
3n
3
n
such lines) contains no more than
k
k
k
checkers. Denote by
T
(
k
,
n
)
T(k, n)
T
(
k
,
n
)
the largest possible number of checkers in such a placement. https://i.ibb.co/bJjjK1M/Image2.jpg a) Prove that the following upper bound is true:
T
(
k
,
n
)
≤
⌊
k
(
2
n
+
1
)
3
⌋
T(k,n) \leq \lfloor \frac{k(2n+1)}{3} \rfloor
T
(
k
,
n
)
≤
⌊
3
k
(
2
n
+
1
)
⌋
b) Find
T
(
1
,
n
)
T(1,n)
T
(
1
,
n
)
and
T
(
2
,
n
)
T(2,n)
T
(
2
,
n
)
D. Zmiaikou
3.1
1
Hide problems
Inequality on radii of incircles
Triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
, having a common incircle of radius
R
R
R
, intersect at points
X
1
,
X
2
,
…
,
X
6
X_1, X_2, \ldots , X_6
X
1
,
X
2
,
…
,
X
6
and form six triangles (see the figure below). Let
r
1
,
r
2
,
…
,
r
6
r_1, r_2,\ldots, r_6
r
1
,
r
2
,
…
,
r
6
be the radii of the inscribed circles of these triangles, and let
R
1
,
R
2
,
…
,
R
6
R_1, R_2, \ldots , R_6
R
1
,
R
2
,
…
,
R
6
be the radii of the inscribed circles of the triangles
A
X
1
F
,
F
X
2
B
,
B
X
3
D
,
D
X
4
C
,
C
X
5
E
AX_1F, FX_2B, BX_3D, DX_4C, CX_5E
A
X
1
F
,
F
X
2
B
,
B
X
3
D
,
D
X
4
C
,
C
X
5
E
and
E
X
6
A
EX_6A
E
X
6
A
respectively. https://i.ibb.co/BspgdHB/Image.jpg Prove that
∑
i
=
1
6
1
r
i
<
6
R
+
∑
i
=
1
6
1
R
i
\sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i}
i
=
1
∑
6
r
i
1
<
R
6
+
i
=
1
∑
6
R
i
1
U. Maksimenkau
4.3
1
Hide problems
Easy looking geometry not very easy to solve
An isosceles triangle
A
B
C
ABC
A
BC
is given(
A
B
=
B
C
AB=BC
A
B
=
BC
). Point
D
D
D
lies inside of it such that
∠
A
D
C
=
150
\angle ADC=150
∠
A
D
C
=
150
,
E
E
E
lies on
C
D
CD
C
D
such that
A
E
=
A
B
AE=AB
A
E
=
A
B
. It turned out that
∠
E
B
C
+
∠
B
A
E
=
60
\angle EBC+\angle BAE=60
∠
EBC
+
∠
B
A
E
=
60
. Prove that
∠
B
D
C
+
∠
C
A
E
=
90
\angle BDC+\angle CAE=90
∠
B
D
C
+
∠
C
A
E
=
90
D. Vasilyev
4.2
1
Hide problems
Quadratic polynomial divides [0,1] in two parts
Let
f
(
x
)
=
x
2
+
b
x
+
c
f(x)=x^2+bx+c
f
(
x
)
=
x
2
+
b
x
+
c
, where
b
,
c
∈
R
b,c \in \mathbb{R}
b
,
c
∈
R
and
b
>
0
b>0
b
>
0
Do there exist disjoint sets
A
A
A
and
B
B
B
, whose union is
[
0
,
1
]
[0,1]
[
0
,
1
]
and
f
(
A
)
=
B
f(A)=B
f
(
A
)
=
B
, where
f
(
X
)
=
{
f
(
x
)
,
x
∈
X
}
f(X)=\{f(x), x \in X\}
f
(
X
)
=
{
f
(
x
)
,
x
∈
X
}
D. Zmiaikou
4.1
1
Hide problems
System of equations on integers
Six integers
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d,e,f
a
,
b
,
c
,
d
,
e
,
f
satisfy:
{
a
c
e
+
3
e
b
d
−
3
b
c
f
+
3
a
d
f
=
5
b
c
e
+
a
c
f
−
a
d
e
+
3
b
d
f
=
2
\begin{cases} ace+3ebd-3bcf+3adf=5 \\ bce+acf-ade+3bdf=2 \end{cases}
{
a
ce
+
3
e
b
d
−
3
b
c
f
+
3
a
df
=
5
b
ce
+
a
c
f
−
a
d
e
+
3
b
df
=
2
Find all possible values of
a
b
c
d
e
abcde
ab
c
d
e
D. Bazyleu
3.4
1
Hide problems
Find points with sames integer distances as given
Points
A
1
,
…
A
n
A_1, \ldots A_n
A
1
,
…
A
n
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points
B
1
,
…
,
B
n
B_1, \ldots ,B_n
B
1
,
…
,
B
n
with integer coordinates such that
A
i
A
j
=
B
i
B
j
A_iA_j=B_iB_j
A
i
A
j
=
B
i
B
j
for every pair
1
≤
i
≤
j
≤
n
1 \leq i \leq j \leq n
1
≤
i
≤
j
≤
n
N. Sheshko, D. Zmiaikou
3.3
1
Hide problems
A game coloring [0,1] segment
Olya and Tolya are playing a game on
[
0
,
1
]
[0,1]
[
0
,
1
]
segment. In the beginning it is white. In the first round Tolya chooses a number
0
≤
l
≤
1
0 \leq l \leq 1
0
≤
l
≤
1
, and then Olya chooses a subsegment of
[
0
,
1
]
[0,1]
[
0
,
1
]
of length
l
l
l
and recolors every its point to the opposite color(white to black, black to white). In the next round players change roles, etc. The game lasts
2024
2024
2024
rounds. Let
L
L
L
be the sum of length of white segments after the end of the game. If
L
>
1
2
L > \frac{1}{2}
L
>
2
1
Olya wins, otherwise Tolya wins. Which player has a strategy to guarantee his win? A. Naradzetski
3.2
1
Hide problems
Easiest FE ever
Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that for any reals
x
≠
y
x \neq y
x
=
y
the following equality is true:
f
(
x
+
y
)
2
=
f
(
x
+
y
)
+
f
(
x
)
+
f
(
y
)
f(x+y)^2=f(x+y)+f(x)+f(y)
f
(
x
+
y
)
2
=
f
(
x
+
y
)
+
f
(
x
)
+
f
(
y
)
D. Zmiaikou
2.4
1
Hide problems
Visiting Belarus and Armenia
There are
k
k
k
cities in Belarus and
k
k
k
cities in Armenia, between some cities there are non-directed flights. From any Belarusian city there are exactly
n
n
n
flights to Armenian cities, and for every pair of Armenian cities exactly two Belarusian cities have flights to both of the Armenian cities. a) Prove that from every Armenian city there are exactly
n
n
n
flights to Belarusian cities. b) Prove that there exists a flight route in which every city is visited at most once and that consists of at least
⌊
(
n
+
1
)
2
4
⌋
\lfloor \frac{(n+1)^2}{4} \rfloor
⌊
4
(
n
+
1
)
2
⌋
cities in both countries. D. Gorovoy
2.3
1
Hide problems
XQ passes through orthocenter
A right triangle
A
B
C
ABC
A
BC
(
∠
A
=
90
\angle A=90
∠
A
=
90
) is inscribed in a circle
ω
\omega
ω
. Tangent to
ω
\omega
ω
at
A
A
A
intersects
B
C
BC
BC
at
P
P
P
,
B
B
B
lies between
P
P
P
and
C
C
C
. Let
M
M
M
be the midpoint of the minor arc
A
B
AB
A
B
.
M
P
MP
MP
intersects
ω
\omega
ω
at
Q
Q
Q
. Point
X
X
X
lies on a ray
P
A
PA
P
A
such that
∠
X
C
B
=
90
\angle XCB=90
∠
XCB
=
90
. Prove that line
X
Q
XQ
XQ
passes through the orthocenter of the triangle
A
B
O
ABO
A
BO
Mayya Golitsyna
2.2
1
Hide problems
Counting polynomials that reduce to other polynomials
A positive integer
n
n
n
is given. Consider all polynomials
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
…
+
a
0
P(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_0
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
…
+
a
0
, whose coefficients are nonnegative integers, not exceeding
100
100
100
. Call
P
P
P
reducible if it can be factored into two non-constant polynomials with nonnegative integer coeffiecients, and irreducible otherwise. Prove that the number of irreducible polynomials is at least twice as big as the number of reducible polynomials. D. Zmiaikou
2.1
1
Hide problems
A sequence of reals which produce perfect squares
A sequence
{
y
i
}
\{y_i\}
{
y
i
}
is given, where
y
0
=
−
1
4
,
y
1
=
0
y_0=-\frac{1}{4},y_1=0
y
0
=
−
4
1
,
y
1
=
0
. For every positive integer
n
n
n
the following equality holds:
y
n
−
1
+
y
n
+
1
=
4
y
n
+
1
y_{n-1}+y_{n+1}=4y_n+1
y
n
−
1
+
y
n
+
1
=
4
y
n
+
1
Prove that for every positive integer
n
n
n
the number
2
y
2
n
+
3
2
2y_{2n}+\frac{3}{2}
2
y
2
n
+
2
3
a) is a positive integer b) is a square of a positive integer D. Zmiaikou
1.4
1
Hide problems
Changing permutations to get the position
Two permutations of
1
,
…
,
n
1,\ldots, n
1
,
…
,
n
are written on the board:
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
b
1
,
…
,
b
n
b_1,\ldots,b_n
b
1
,
…
,
b
n
A move consists of one of the following two operations: 1) Change the first row to
b
a
1
,
…
,
b
a
n
b_{a_1},\ldots,b_{a_n}
b
a
1
,
…
,
b
a
n
2) Change the second row to
a
b
1
,
…
,
a
b
n
a_{b_1},\ldots,a_{b_n}
a
b
1
,
…
,
a
b
n
The starting position is:
2134
…
n
2134\ldots n
2134
…
n
234
…
n
1
234\ldots n1
234
…
n
1
Is it possible by finitely many moves to get:
2314
…
n
2314\ldots n
2314
…
n
234
…
n
1
234 \ldots n1
234
…
n
1
? D. Zmiaikou
1.3
1
Hide problems
Inequality with reals at least one third
Prove that for any real numbers
a
,
b
,
c
,
d
≥
1
3
a,b,c,d \geq \frac{1}{3}
a
,
b
,
c
,
d
≥
3
1
the following inequality holds:
a
6
b
4
+
c
3
+
b
6
c
4
+
d
3
+
c
6
d
4
+
a
3
+
d
6
a
4
+
b
3
≥
a
+
b
+
c
+
d
4
\sqrt{\frac{a^6}{b^4+c^3}+\frac{b^6}{c^4+d^3}+\frac{c^6}{d^4+a^3}+\frac{d^6}{a^4+b^3}}\geq \frac{a+b+c+d}{4}
b
4
+
c
3
a
6
+
c
4
+
d
3
b
6
+
d
4
+
a
3
c
6
+
a
4
+
b
3
d
6
≥
4
a
+
b
+
c
+
d
D. Zmiaikou
1.2
1
Hide problems
Center lies on TK
An acute-angled triangle
A
B
C
ABC
A
BC
with an altitude
A
D
AD
A
D
and orthocenter
H
H
H
are given.
A
D
AD
A
D
intersects the circumcircle of
A
B
C
ABC
A
BC
ω
\omega
ω
at
P
P
P
.
K
K
K
is a point on segment
B
C
BC
BC
such that
K
C
=
B
D
KC=BD
K
C
=
B
D
. The circumcircle of
K
P
H
KPH
K
P
H
intersects
ω
\omega
ω
at
Q
Q
Q
and
B
C
BC
BC
at
N
N
N
. A line perpendicular to
P
Q
PQ
PQ
and passing through
N
N
N
intersects
A
D
AD
A
D
at
T
T
T
. Prove that the center of
ω
\omega
ω
lies on line
T
K
TK
T
K
. U. Maksimenkau
1.1
1
Hide problems
Choosing number from 1 to 1000
Find the minimal positive integer
n
n
n
such that no matter what
n
n
n
distinct numbers from
1
1
1
to
1000
1000
1000
you choose, such that no two are divisible by a square of the same prime, one of the chosen numbers is a square of prime. D. Zmiaikou