MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2020
Russian TST 2020
Part of
Russian Team Selection Tests
Subcontests
(3)
P3
1
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Nice Miquel-like geometry
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, the lines
A
B
AB
A
B
and
D
C
DC
D
C
intersect at point
P
P{}
P
and the lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at point
Q
Q{}
Q
. The points
E
E{}
E
and
F
F{}
F
are inside the quadrilateral
A
B
C
D
ABCD
A
BC
D
such that the circles
(
A
B
E
)
,
(
C
D
E
)
,
(
B
C
F
)
,
(
A
D
F
)
(ABE), (CDE), (BCF),(ADF)
(
A
BE
)
,
(
C
D
E
)
,
(
BCF
)
,
(
A
D
F
)
intersect at one point
K
K{}
K
. Prove that the circles
(
P
K
F
)
(PKF)
(
P
K
F
)
and
(
Q
K
E
)
(QKE)
(
Q
K
E
)
intersect a second time on the line
P
Q
PQ
PQ
.
P1
2
Hide problems
Finances
There are coins worth
1
,
2
,
…
,
b
1, 2, \ldots , b
1
,
2
,
…
,
b
rubles, blue bills with worth
a
a{}
a
rubles and red bills worth
a
+
b
a + b
a
+
b
rubles. Ilya wants to exchange a certain amount into coins and blue bills, and use no more than
a
−
1
a-1
a
−
1
coins. Pasha wants to exchange the same amount in coins and red bills, but use no more than
a
a{}
a
coins. Prove that they have equally many ways of doing so.
Functional equation
Determine all functions
f
:
R
+
→
R
+
f:\mathbb{R}^+\to\mathbb{R}^+
f
:
R
+
→
R
+
satisfying
x
f
(
x
f
(
2
y
)
)
=
y
+
x
y
f
(
x
)
xf(xf(2y))=y+xyf(x)
x
f
(
x
f
(
2
y
))
=
y
+
x
y
f
(
x
)
for all
x
,
y
>
0
x,y>0
x
,
y
>
0
.
P2
3
Hide problems
Delightful sets of vertices
There are 10,000 vertices in a graph, with at least one edge coming out of each vertex. Call a set
S
S{}
S
of vertices delightful if no two of its vertices are connected by an edge, but any vertex not from
S
S{}
S
is connected to at least one vertex from
S
S{}
S
. For which smallest
m
m
m
is there necessarily a delightful set of at most
m
m
m
vertices?
Inequality with gcd
Given a natural number
n
n{}
n
find the smallest
λ
\lambda
λ
such that
gcd
(
x
(
x
+
1
)
⋯
(
x
+
n
−
1
)
,
y
(
y
+
1
)
⋯
(
y
+
n
−
1
)
)
⩽
(
x
−
y
)
λ
,
\gcd(x(x + 1)\cdots(x + n - 1), y(y + 1)\cdots(y + n - 1)) \leqslant (x-y)^\lambda,
g
cd
(
x
(
x
+
1
)
⋯
(
x
+
n
−
1
)
,
y
(
y
+
1
)
⋯
(
y
+
n
−
1
))
⩽
(
x
−
y
)
λ
,
for any positive integers
y
y{}
y
and
x
⩾
y
+
n
x \geqslant y + n
x
⩾
y
+
n
.
Inscribed octagon, geometry
Octagon
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A_1A_2A_3A_4A_5A_6A_7A_8
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
is inscribed in a circle
Ω
\Omega
Ω
with center
O
O
O
. It is known that
A
1
A
2
∥
A
5
A
6
A_1A_2\|A_5A_6
A
1
A
2
∥
A
5
A
6
,
A
3
A
4
∥
A
7
A
8
A_3A_4\|A_7A_8
A
3
A
4
∥
A
7
A
8
and
A
2
A
3
∥
A
5
A
8
A_2A_3\|A_5A_8
A
2
A
3
∥
A
5
A
8
. The circle
ω
12
\omega_{12}
ω
12
passes through
A
1
A_1
A
1
,
A
2
A_2
A
2
and touches
A
1
A
6
A_1A_6
A
1
A
6
; circle
ω
34
\omega_{34}
ω
34
passes through
A
3
A_3
A
3
,
A
4
A_4
A
4
and touches
A
3
A
8
A_3A_8
A
3
A
8
; the circle
ω
56
\omega_{56}
ω
56
passes through
A
5
A_5
A
5
,
A
6
A_6
A
6
and touches
A
5
A
2
A_5A_2
A
5
A
2
; the circle
ω
78
\omega_{78}
ω
78
passes through
A
7
A_7
A
7
,
A
8
A_8
A
8
and touches
A
7
A
4
A_7A_4
A
7
A
4
. The common external tangent to
ω
12
\omega_{12}
ω
12
and
ω
34
\omega_{34}
ω
34
cross the line passing through
A
1
A
6
∩
A
3
A
8
{A_1A_6}\cap{A_3A_8}
A
1
A
6
∩
A
3
A
8
and
A
5
A
2
∩
A
7
A
4
{A_5A_2}\cap{A_7A_4}
A
5
A
2
∩
A
7
A
4
at the point
X
X
X
. Prove that one of the common tangents to
ω
56
\omega_{56}
ω
56
and
ω
78
\omega_{78}
ω
78
passes through
X
X
X
.