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Contests
International Contests
International Zhautykov Olympiad
2021 International Zhautykov Olympiad
2021 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(6)
5
1
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Another Party with 99 guests
On a party with
99
99
99
guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are
99
99
99
chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair
c
c
c
. If some chair adjacent to
c
c
c
is already occupied, the same host orders one guest on such chair to stand up (if both chairs adjacent to
c
c
c
are occupied, the host chooses exactly one of them). All orders are carried out immediately. Ann makes the first move; her goal is to fulfill, after some move of hers, that at least
k
k
k
chairs are occupied. Determine the largest
k
k
k
for which Ann can reach the goal, regardless of Bob's play.
6
1
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Number of Polynomial Q such that P(x) | P(Q(x))
Let
P
(
x
)
P(x)
P
(
x
)
be a nonconstant polynomial of degree
n
n
n
with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials
Q
(
x
)
Q(x)
Q
(
x
)
of degree less than
n
n
n
with rational coefficients such that
P
(
x
)
P(x)
P
(
x
)
divides
P
(
Q
(
x
)
)
P(Q(x))
P
(
Q
(
x
))
a) is finite b) does not exceed
n
n
n
.
4
1
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IZHO P4 (Day 2, Problem 1)
Let there be an incircle of triangle
A
B
C
ABC
A
BC
, and 3 circles each inscribed between incircle and angles of
A
B
C
ABC
A
BC
. Let
r
,
r
1
,
r
2
,
r
3
r, r_1, r_2, r_3
r
,
r
1
,
r
2
,
r
3
be radii of these circles (
r
1
,
r
2
,
r
3
<
r
r_1, r_2, r_3 < r
r
1
,
r
2
,
r
3
<
r
). Prove that
r
1
+
r
2
+
r
3
≥
r
r_1+r_2+r_3 \geq r
r
1
+
r
2
+
r
3
≥
r
3
1
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Rook set problem, Combinatorics from IZHO 2021
Let
n
≥
2
n\ge 2
n
≥
2
be an integer. Elwyn is given an
n
×
n
n\times n
n
×
n
table filled with real numbers (each cell of the table contains exactly one number). We define a rook set as a set of
n
n
n
cells of the table situated in
n
n
n
distinct rows as well as in n distinct columns. Assume that, for every rook set, the sum of
n
n
n
numbers in the cells forming the set is nonnegative.\\ \\ By a move, Elwyn chooses a row, a column, and a real number
a
,
a,
a
,
and then he adds
a
a
a
to each number in the chosen row, and subtracts
a
a
a
from each number in the chosen column (thus, the number at the intersection of the chosen row and column does not change). Prove that Elwyn can perform a sequence of moves so that all numbers in the table become nonnegative.
2
1
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interesting hexagon problem from IZHO 2021
In a convex cyclic hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
,
B
C
=
E
F
BC=EF
BC
=
EF
and
C
D
=
A
F
CD=AF
C
D
=
A
F
. Diagonals
A
C
AC
A
C
and
B
F
BF
BF
intersect at point
Q
,
Q,
Q
,
and diagonals
E
C
EC
EC
and
D
F
DF
D
F
intersect at point
P
.
P.
P
.
Points
R
R
R
and
S
S
S
are marked on the segments
D
F
DF
D
F
and
B
F
BF
BF
respectively so that
F
R
=
P
D
FR=PD
FR
=
P
D
and
B
Q
=
F
S
.
BQ=FS.
BQ
=
FS
.
The segments
R
Q
RQ
RQ
and
P
S
PS
PS
intersect at point
T
.
T.
T
.
Prove that the line
T
C
TC
TC
bisects the diagonal
D
B
DB
D
B
.
1
1
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Number theory
Prove that there exists a positive integer
n
n
n
, such that the remainder of
3
n
3^n
3
n
when divided by
2
n
2^n
2
n
is greater than
1
0
2021
10^{2021}
1
0
2021
.