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Problems
Contests
International Contests
International Olympiad of Metropolises
2020 IOM
2020 IOM
Part of
International Olympiad of Metropolises
Subcontests
(6)
5
1
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Alice & Eva take turns filling an empty table with 2^{100} rows and 100 columns
There is an empty table with
2
100
2^{100}
2
100
rows and
100
100
100
columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first). The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies?Proposed by Denis Afrizonov
6
1
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Weird geometry finale
In convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
points
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
,
D
1
D_1
D
1
,
E
1
E_1
E
1
are intersections of pairs of diagonals
(
B
D
,
C
E
)
(BD, CE)
(
B
D
,
CE
)
,
(
C
E
,
D
A
)
(CE, DA)
(
CE
,
D
A
)
,
(
D
A
,
E
B
)
(DA, EB)
(
D
A
,
EB
)
,
(
E
B
,
A
C
)
(EB, AC)
(
EB
,
A
C
)
and
(
A
C
,
B
D
)
(AC, BD)
(
A
C
,
B
D
)
respectively. Prove that if four of quadrilaterals
A
B
1
A
1
B
AB_{1}A_{1}B
A
B
1
A
1
B
,
B
C
1
B
1
C
BC_{1}B_{1}C
B
C
1
B
1
C
,
C
D
1
C
1
D
CD_{1}C_{1}D
C
D
1
C
1
D
,
D
E
1
D
1
E
DE_{1}D_{1}E
D
E
1
D
1
E
and
E
A
1
E
1
A
EA_{1}E_{1}A
E
A
1
E
1
A
are cyclic then the fifth one is also cyclic.
4
1
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Algebra IOM
Given three positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that following holds
a
2
=
b
2
+
b
c
a^2=b^2+bc
a
2
=
b
2
+
b
c
,
b
2
=
c
2
+
a
c
b^2=c^2+ac
b
2
=
c
2
+
a
c
Prove that
1
c
=
1
a
+
1
b
\frac{1}{c}=\frac{1}{a}+\frac{1}{b}
c
1
=
a
1
+
b
1
.
3
1
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every lucky set of values {a_1,a_2,..,a_n} satisfies a_1+a_2+...+a_n >n2^{n-1}
Let
n
>
1
n>1
n
>
1
be a given integer. The Mint issues coins of
n
n
n
different values
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
, where each
a
i
a_i
a
i
is a positive integer (the number of coins of each value is unlimited). A set of values
{
a
1
,
a
2
,
.
.
.
,
a
n
}
\{a_1, a_2,..., a_n\}
{
a
1
,
a
2
,
...
,
a
n
}
is called lucky, if the sum
a
1
+
a
2
+
.
.
.
+
a
n
a_1+ a_2+...+ a_n
a
1
+
a
2
+
...
+
a
n
can be collected in a unique way (namely, by taking one coin of each value). (a) Prove that there exists a lucky set of values
{
a
1
,
a
2
,
.
.
.
,
a
n
}
\{a_1, a_2, ..., a_n\}
{
a
1
,
a
2
,
...
,
a
n
}
with
a
1
+
a
2
+
.
.
.
+
a
n
<
n
⋅
2
n
.
a_1+ a_2+...+ a_n < n \cdot 2^n.
a
1
+
a
2
+
...
+
a
n
<
n
⋅
2
n
.
(b) Prove that every lucky set of values
{
a
1
,
a
2
,
.
.
.
,
a
n
}
\{a_1, a_2,..., a_n\}
{
a
1
,
a
2
,
...
,
a
n
}
satisfies
a
1
+
a
2
+
.
.
.
+
a
n
>
n
⋅
2
n
−
1
.
a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.
a
1
+
a
2
+
...
+
a
n
>
n
⋅
2
n
−
1
.
Proposed by Ilya Bogdanov
2
1
Hide problems
Does there exist a positive integer $n$
Does there exist a positive integer
n
n
n
such that all its digits (in the decimal system) are greather than 5, while all the digits of
n
2
n^2
n
2
are less than 5?
1
1
Hide problems
Prove that $CK=ML$
In a triangle
A
B
C
ABC
A
BC
with a right angle at
C
C
C
, the angle bisector
A
L
AL
A
L
(where
L
L
L
is on segment
B
C
BC
BC
) intersects the altitude
C
H
CH
C
H
at point
K
K
K
. The bisector of angle
B
C
H
BCH
BC
H
intersects segment
A
B
AB
A
B
at point
M
M
M
. Prove that
C
K
=
M
L
CK=ML
C
K
=
M
L