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2020 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(2)
1
1
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sum of numbers in cells numberes 2^k, inside 600x600 sqaure, 4 types of tiles
Square
600
×
600
600\times 600
600
×
600
is divided into figures of four types, shown in figure. In the figures of the two types, shown on the left, in painted black, the cells recorded number
2
k
2^k
2
k
, where
k
k
k
is the number of the column, where is this cell (columns numbered from left to right by numbers from
1
1
1
to
600
600
600
). Prove that the sum of all recorded numbers are divisible by
9
9
9
. https://cdn.artofproblemsolving.com/attachments/2/5/811a0d1a6bea0e15a3271f62f35b9f4848a172.png
3
1
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Cool geometry but easy)
Altitudes
A
H
1
AH1
A
H
1
and
B
H
2
BH2
B
H
2
of acute triangle
A
B
C
ABC
A
BC
intersect at
H
H
H
. Let
w
1
w1
w
1
be the circle that goes through
H
2
H2
H
2
and touches the line
B
C
BC
BC
at
H
1
H1
H
1
, and let
w
2
w2
w
2
be the circle that goes through
H
1
H1
H
1
and touches the line
A
C
AC
A
C
at
H
2
H2
H
2
. Prove, that the intersection point of two other tangent lines
B
X
BX
BX
and
A
Y
AY
A
Y
(
X
X
X
and
Y
Y
Y
are different from
H
1
H1
H
1
and
H
2
H2
H
2
) to circles
w
1
w1
w
1
and
w
2
w2
w
2
respectively, lies on the circumcircle of triangle
H
X
Y
HXY
H
X
Y
. Proposed by Danilo Khilko