MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
2022 Kyiv City MO
2022 Kyiv City MO Round 2
2022 Kyiv City MO Round 2
Part of
2022 Kyiv City MO
Subcontests
(4)
Problem 3
3
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Very cute algebra
Nonzero real numbers
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
satisfy the following condition:
x
1
−
1
x
2
=
x
2
−
1
x
3
=
…
=
x
n
−
1
−
1
x
n
=
x
n
−
1
x
1
x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}
x
1
−
x
2
1
=
x
2
−
x
3
1
=
…
=
x
n
−
1
−
x
n
1
=
x
n
−
x
1
1
Determine all
n
n
n
for which
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
have to be equal.(Proposed by Oleksii Masalitin, Anton Trygub)
Equal ratio implies tangency
Let
A
H
A
,
B
H
B
,
C
H
C
AH_A, BH_B, CH_C
A
H
A
,
B
H
B
,
C
H
C
be the altitudes of triangle
A
B
C
ABC
A
BC
. Prove that if
H
B
C
A
C
=
H
C
A
A
B
\frac{H_BC}{AC} = \frac{H_CA}{AB}
A
C
H
B
C
=
A
B
H
C
A
, then the line symmetric to
B
C
BC
BC
with respect to line
H
B
H
C
H_BH_C
H
B
H
C
is tangent to the circumscribed circle of triangle
H
B
H
C
A
H_BH_CA
H
B
H
C
A
.(Proposed by Mykhailo Bondarenko)
Prefix sums of permutation
Find the largest
k
k
k
for which there exists a permutation
(
a
1
,
a
2
,
…
,
a
2022
)
(a_1, a_2, \ldots, a_{2022})
(
a
1
,
a
2
,
…
,
a
2022
)
of integers from
1
1
1
to
2022
2022
2022
such that for at least
k
k
k
distinct
i
i
i
with
1
≤
i
≤
2022
1 \le i \le 2022
1
≤
i
≤
2022
the number
a
1
+
a
2
+
…
+
a
i
1
+
2
+
…
+
i
\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}
1
+
2
+
…
+
i
a
1
+
a
2
+
…
+
a
i
is an integer larger than
1
1
1
.(Proposed by Oleksii Masalitin)
Problem 4
3
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Game of stones
Fedir and Mykhailo have three piles of stones: the first contains
100
100
100
stones, the second
101
101
101
, the third
102
102
102
. They are playing a game, going in turns, Fedir makes the first move. In one move player can select any two piles of stones, let's say they have
a
a
a
and
b
b
b
stones left correspondently, and remove
g
c
d
(
a
,
b
)
gcd(a, b)
g
c
d
(
a
,
b
)
stones from each of them. The player after whose move some pile becomes empty for the first time wins. Who has a winning strategy?As a reminder,
g
c
d
(
a
,
b
)
gcd(a, b)
g
c
d
(
a
,
b
)
denotes the greatest common divisor of
a
,
b
a, b
a
,
b
.(Proposed by Oleksii Masalitin)
NT: Smallest degree of polynomial
Prime
p
>
2
p>2
p
>
2
and a polynomial
Q
Q
Q
with integer coefficients are such that there are no integers
1
≤
i
<
j
≤
p
−
1
1 \le i < j \le p-1
1
≤
i
<
j
≤
p
−
1
for which
(
Q
(
j
)
−
Q
(
i
)
)
(
j
Q
(
j
)
−
i
Q
(
i
)
)
(Q(j)-Q(i))(jQ(j)-iQ(i))
(
Q
(
j
)
−
Q
(
i
))
(
j
Q
(
j
)
−
i
Q
(
i
))
is divisible by
p
p
p
. What is the smallest possible degree of
Q
Q
Q
?(Proposed by Anton Trygub)
Crazy geometry on incircles
Let
A
B
C
D
ABCD
A
BC
D
be the cyclic quadrilateral. Suppose that there exists some line
l
l
l
parallel to
B
D
BD
B
D
which is tangent to the inscribed circles of triangles
A
B
C
,
C
D
A
ABC, CDA
A
BC
,
C
D
A
. Show that
l
l
l
passes through the incenter of
B
C
D
BCD
BC
D
or through the incenter of
D
A
B
DAB
D
A
B
.(Proposed by Fedir Yudin)
Problem 2
4
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Problem 1
4
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