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Problems
Contests
International Contests
Mathematical Excellence Olympiad
2020 IMEO
2020 IMEO
Part of
Mathematical Excellence Olympiad
Subcontests
(6)
Problem 6
1
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Hard geometry with concurrency on OI
Let
O
O
O
,
I
I
I
, and
ω
\omega
ω
be the circumcenter, the incenter, and the incircle of nonequilateral
△
A
B
C
\triangle ABC
△
A
BC
. Let
ω
A
\omega_A
ω
A
be the unique circle tangent to
A
B
AB
A
B
and
A
C
AC
A
C
, such that the common chord of
ω
A
\omega_A
ω
A
and
ω
\omega
ω
passes through the center of
ω
A
\omega_A
ω
A
. Let
O
A
O_A
O
A
be the center of
ω
A
\omega_A
ω
A
. Define
ω
B
,
O
B
,
ω
C
,
O
C
\omega_B, O_B, \omega_C, O_C
ω
B
,
O
B
,
ω
C
,
O
C
similarly. If
ω
\omega
ω
touches
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
D
D
,
E
E
E
,
F
F
F
respectively, prove that the perpendiculars from
D
D
D
,
E
E
E
,
F
F
F
to
O
B
O
C
,
O
C
O
A
,
O
A
O
B
O_BO_C , O_CO_A , O_AO_B
O
B
O
C
,
O
C
O
A
,
O
A
O
B
are concurrent on the line
O
I
OI
O
I
.Pitchayut Saengrungkongka
Problem 5
1
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Primes make difference in Summatory Liouville function
For a positive integer
n
n
n
with prime factorization
n
=
p
1
α
1
p
2
α
2
⋯
p
k
α
k
n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}
n
=
p
1
α
1
p
2
α
2
⋯
p
k
α
k
let's define
λ
(
n
)
=
(
−
1
)
α
1
+
α
2
+
⋯
+
α
k
\lambda(n) = (-1)^{\alpha_1 + \alpha_2 + \dots + \alpha_k}
λ
(
n
)
=
(
−
1
)
α
1
+
α
2
+
⋯
+
α
k
. Define
L
(
n
)
L(n)
L
(
n
)
as sum of
λ
(
x
)
\lambda(x)
λ
(
x
)
over all integers from
1
1
1
to
n
n
n
.Define
K
(
n
)
K(n)
K
(
n
)
as sum of
λ
(
x
)
\lambda(x)
λ
(
x
)
over all composite integers from
1
1
1
to
n
n
n
.For some
N
>
1
N>1
N
>
1
, we know, that for every
2
≤
n
≤
N
2\le n \le N
2
≤
n
≤
N
,
L
(
n
)
≤
0
L(n)\le 0
L
(
n
)
≤
0
.Prove that for this
N
N
N
, for every
2
≤
n
≤
N
2\le n \le N
2
≤
n
≤
N
,
K
(
n
)
≥
0
K(n)\ge 0
K
(
n
)
≥
0
.Mykhailo Shtandenko
Problem 4
1
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Sorting permutation. Who wins?
Anna and Ben are playing with a permutation
p
p
p
of length
2020
2020
2020
, initially
p
i
=
2021
−
i
p_i = 2021 - i
p
i
=
2021
−
i
for
1
≤
i
≤
2020
1\le i \le 2020
1
≤
i
≤
2020
. Anna has power
A
A
A
, and Ben has power
B
B
B
. Players are moving in turns, with Anna moving first.In his turn player with power
P
P
P
can choose any
P
P
P
elements of the permutation and rearrange them in the way he/she wants.Ben wants to sort the permutation, and Anna wants to not let this happen. Determine if Ben can make sure that the permutation will be sorted (of form
p
i
=
i
p_i = i
p
i
=
i
for
1
≤
i
≤
2020
1\le i \le 2020
1
≤
i
≤
2020
) in finitely many turns, ifa)
A
=
1000
,
B
=
1000
A = 1000, B = 1000
A
=
1000
,
B
=
1000
b)
A
=
1000
,
B
=
1001
A = 1000, B = 1001
A
=
1000
,
B
=
1001
c)
A
=
1000
,
B
=
1002
A = 1000, B = 1002
A
=
1000
,
B
=
1002
Anton Trygub
Problem 3
1
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Yet another functional equation
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R^+} \to \mathbb{R^+}
f
:
R
+
→
R
+
such that for all positive real
x
,
y
x, y
x
,
y
holds
x
f
(
x
)
+
y
f
(
y
)
=
(
x
+
y
)
f
(
x
2
+
y
2
x
+
y
)
xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)
x
f
(
x
)
+
y
f
(
y
)
=
(
x
+
y
)
f
(
x
+
y
x
2
+
y
2
)
. Fedir Yudin
Problem 2
1
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Impossible dominoes
You are given an odd number
n
≥
3
n\ge 3
n
≥
3
. For every pair of integers
(
i
,
j
)
(i, j)
(
i
,
j
)
with
1
≤
i
≤
j
≤
n
1\le i \le j \le n
1
≤
i
≤
j
≤
n
there is a domino, with
i
i
i
written on one its end and with
j
j
j
written on another (there are
n
(
n
+
1
)
2
\frac{n(n+1)}{2}
2
n
(
n
+
1
)
domino overall). Amin took this dominos and started to put them in a row so that numbers on the adjacent sides of the dominos are equal. He has put
k
k
k
dominos in this way, got bored and went away. After this Anton came to see this
k
k
k
dominos, and he realized that he can't put all the remaining dominos in this row by the rules. For which smallest value of
k
k
k
is this possible? Oleksii Masalitin
Problem 1
1
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Easy but cute geometry
Let
A
B
C
ABC
A
BC
be a triangle and
A
′
A'
A
′
be the reflection of
A
A
A
about
B
C
BC
BC
. Let
P
P
P
and
Q
Q
Q
be points on
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
P
A
′
=
P
C
PA'=PC
P
A
′
=
PC
and
Q
A
′
=
Q
B
QA'=QB
Q
A
′
=
QB
. Prove that the perpendicular from
A
′
A'
A
′
to
P
Q
PQ
PQ
passes through the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
.Fedir Yudin