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Problems
Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2021
Russian TST 2021
Part of
Russian Team Selection Tests
Subcontests
(3)
P2
2
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Excircle geometry
The
A
A{}
A
-excircle
ω
A
\omega_A{}
ω
A
of the triangle
A
B
C
ABC
A
BC
touches the side of the
B
C
BC
BC
at point
A
1
A_1
A
1
and the extensions of the sides
A
B
AB
A
B
and
A
C
AC
A
C
are at points
C
1
C_1
C
1
and
B
1
B_1
B
1
respectively. Let
P
P{}
P
be the middle of the segment
B
1
C
1
B_1C_1
B
1
C
1
. The line
A
1
P
A_1P
A
1
P
intersects
ω
A
\omega_A{}
ω
A
a second time at point
X
X{}
X
. The tangents to the circumcircle of the triangle
A
B
C
ABC
A
BC
at point
A
A{}
A
and to
ω
A
\omega_A{}
ω
A
at point
X
X{}
X
intersect at point
R
R
R
. Prove that
R
P
=
R
X
RP = RX
RP
=
RX
.
Number of lattice points inside triangle
The natural numbers
t
t{}
t
and
q
q{}
q
are given. For an integer
s
s{}
s
, we denote by
f
(
s
)
f(s)
f
(
s
)
the number of lattice points lying in the triangle with vertices
(
0
;
−
t
/
q
)
,
(
0
;
t
/
q
)
(0;-t/q), (0; t/q)
(
0
;
−
t
/
q
)
,
(
0
;
t
/
q
)
and
(
t
;
t
s
/
q
)
(t; ts/q)
(
t
;
t
s
/
q
)
. Suppose that
q
q{}
q
divides
r
s
−
1
rs-1{}
rs
−
1
. Prove that
f
(
r
)
=
f
(
s
)
f(r) = f(s)
f
(
r
)
=
f
(
s
)
.
P3
3
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Dangerous numbers
Given an integer
m
>
1
m > 1
m
>
1
, we call the number
x
x{}
x
dangerous if
x
x{}
x
divides the number
y
y{}
y
, which is obtained by writing the digits of
x
x{}
x
in base
m
m{}
m
in reverse order, with
x
≠
y
x\neq y
x
=
y
. Prove that if there exists a three-digit (in base
m
m
m
) dangerous number for a given
m
m
m
, then there exists a two-digit (in base
m
m
m
) dangerous number.
Graph coloring problem
Given a natural number
n
⩾
2
n\geqslant 2
n
⩾
2
, find the smallest possible number of edges in a graph that has the following property: for any coloring of the vertices of the graph in
n
n{}
n
colors, there is a vertex that has at least two neighbors of the same color as itself.
Another n-variable polynomial
Given an integer
n
⩾
3
n \geqslant 3
n
⩾
3
the polynomial
f
(
x
1
,
…
,
x
n
)
f(x_1, \ldots, x_n)
f
(
x
1
,
…
,
x
n
)
with integer coefficients is called good if
f
(
0
,
…
,
0
)
=
0
f(0,\ldots, 0) = 0
f
(
0
,
…
,
0
)
=
0
and
f
(
x
1
,
…
,
x
n
)
=
f
(
x
π
1
,
…
,
x
π
n
)
,
f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),
f
(
x
1
,
…
,
x
n
)
=
f
(
x
π
1
,
…
,
x
π
n
)
,
for any permutation of
π
\pi
π
of the numbers
1
,
…
,
n
1,\ldots, n
1
,
…
,
n
. Denote by
J
\mathcal{J}
J
the set of polynomials of the form
p
1
q
1
+
⋯
+
p
m
q
m
,
p_1q_1+\cdots+p_mq_m,
p
1
q
1
+
⋯
+
p
m
q
m
,
where
m
m
m
is a positive integer and
q
1
,
…
,
q
m
q_1,\ldots , q_m
q
1
,
…
,
q
m
are polynomials with integer coefficients, and
p
1
,
…
,
p
m
p_1,\ldots , p_m
p
1
,
…
,
p
m
are good polynomials. Find the smallest natural number
D
D{}
D
such that each monomial of degree
D
D{}
D
lies in the set
J
\mathcal{J}
J
.
P1
3
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Representing positive integers in a certain form
Do there exist infinitely many positive integers not expressible in the form
(
a
+
b
)
+
log
2
(
b
+
c
)
−
2
c
+
a
,
(a+b)+\log_2(b+c)-2^{c+a},
(
a
+
b
)
+
lo
g
2
(
b
+
c
)
−
2
c
+
a
,
where
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive integers?
Vending machine
A machine accepts coins of
k
k{}
k
values
1
=
a
1
<
⋯
<
a
k
1 = a_1 <\cdots < a_k
1
=
a
1
<
⋯
<
a
k
and sells
k
k{}
k
different drinks with prices
0
<
b
1
<
⋯
<
b
k
0<b_1 < \cdots < b_k
0
<
b
1
<
⋯
<
b
k
. It is known that if we start inserting coins into the machine in an arbitrary way, sooner or later the total value of the coins will be equal to the price of a drink. For which sets of numbers
(
a
1
,
…
,
a
k
;
b
1
,
…
,
b
k
)
(a_1,\ldots,a_k;b_1,\ldots,b_k)
(
a
1
,
…
,
a
k
;
b
1
,
…
,
b
k
)
does this property hold?
Excircle geometry 2
A point
P
P{}
P
is considered on the incircle of the triangle
A
B
C
ABC
A
BC
. We draw the tangent segments from
P
P{}
P
to the three excircles of
A
B
C
ABC
A
BC
. Prove that from the obtained three tangent segments it is possible to make a right triangle if and only if the point
P
P{}
P
lies on one of the lines connecting two of the midpoints of the sides of
A
B
C
ABC
A
BC
.