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Problems
Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2022
Russian TST 2022
Part of
Russian Team Selection Tests
Subcontests
(3)
P1
1
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Trigonometric identity
Let
a
a{}
a
and
b
b{}
b
be positive integers. Prove that for any real number
x
x{}
x
∑
j
=
0
a
(
a
j
)
(
2
cos
(
(
2
j
−
a
)
x
)
)
b
=
∑
j
=
0
b
(
b
j
)
(
2
cos
(
(
2
j
−
b
)
x
)
)
a
.
\sum_{j=0}^a\binom{a}{j}\big(2\cos((2j-a)x)\big)^b=\sum_{j=0}^b\binom{b}{j}\big(2\cos((2j-b)x)\big)^a.
j
=
0
∑
a
(
j
a
)
(
2
cos
((
2
j
−
a
)
x
)
)
b
=
j
=
0
∑
b
(
j
b
)
(
2
cos
((
2
j
−
b
)
x
)
)
a
.
P3
4
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P2
2
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Interesting FE
Determine all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
satisfying
f
(
x
y
+
f
(
x
)
)
+
f
(
y
)
=
x
f
(
y
)
+
f
(
x
+
y
)
,
f(xy+f(x))+f(y)=xf(y)+f(x+y),
f
(
x
y
+
f
(
x
))
+
f
(
y
)
=
x
f
(
y
)
+
f
(
x
+
y
)
,
for all real numbers
x
,
y
x,y
x
,
y
.
Quadrilateral and incircles
The quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in the circle
Γ
\Gamma
Γ
. Let
I
B
I_B
I
B
and
I
D
I_D
I
D
be the centers of the circles
ω
B
\omega_B
ω
B
and
ω
D
\omega_D
ω
D
inscribed in the triangles
A
B
C
ABC
A
BC
and
A
D
C
ADC
A
D
C
, respectively. A common external tangent to
ω
B
\omega_B
ω
B
and
ω
D
\omega_D
ω
D
intersects
Γ
\Gamma
Γ
at
K
K
K
and
L
L{}
L
. Prove that
I
B
,
I
D
,
K
I_B,I_D,K
I
B
,
I
D
,
K
and
L
L{}
L
lie on the same circle.