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Contests
National and Regional Contests
Brazil Contests
Mathematicians for Fun Olympiad
2022 OMpD
2022 OMpD
Part of
Mathematicians for Fun Olympiad
Subcontests
(4)
3
2
Hide problems
Person on the chair n says that the n on their left are liars
Let
n
≥
3
n \geq 3
n
≥
3
be a positive integer. In an election debate, we have
n
n
n
seats arranged in a circle and these seats are numbered from
1
1
1
to
n
n
n
, clockwise. In each of these chairs sits a politician, who can be a liar or an honest one. Lying politicians always tell lies, and honest politicians always tell the truth.At one heated moment in the debate, they accused each other of being liars, with the politician in chair
1
1
1
saying that the politician immediately to his left is a liar, the politician in chair
2
2
2
saying that all the
2
2
2
politicians immediately to his left are liars, the politician in the char
3
3
3
saying that all the
3
3
3
politicians immediately to his left are liars, and so on. Note that the politician in chair
n
n
n
accuses all
n
n
n
politicians (including himself) of being liars.For what values of
n
n
n
is this situation possible to happen?
Removing digits 1 and adding N to the number on the blackboard
Let
N
N
N
be a positive integer. Initially, a positive integer
A
A
A
is written on the board. At each step, we can perform one of the following two operations with the number written on the board:(i) Add
N
N
N
to the number written on the board and replace that number with the sum obtained;(ii) If the number on the board is greater than
1
1
1
and has at least one digit
1
1
1
, then we can remove the digit
1
1
1
from that number, and replace the number initially written with this one (with removal of possible leading zeros)For example, if
N
=
63
N = 63
N
=
63
and
A
=
25
A = 25
A
=
25
, we can do the following sequence of operations:
25
→
88
→
151
→
51
→
5
25 \rightarrow 88 \rightarrow 151 \rightarrow 51 \rightarrow 5
25
→
88
→
151
→
51
→
5
And if
N
=
143
N = 143
N
=
143
and
A
=
2
A = 2
A
=
2
, we can do the following sequence of operations:
2
→
145
→
288
→
431
→
574
→
717
→
860
→
1003
→
3
2 \rightarrow 145 \rightarrow 288 \rightarrow 431 \rightarrow 574 \rightarrow 717 \rightarrow 860 \rightarrow 1003 \rightarrow 3
2
→
145
→
288
→
431
→
574
→
717
→
860
→
1003
→
3
For what values of
N
N
N
is it always possible, regardless of the initial value of
A
A
A
on the blackboard, to obtain the number
1
1
1
on the blackboard, through a finite number of operations?
1
2
Hide problems
Squares with same distance from rooks
Consider a chessboard
6
×
6
6 \times 6
6
×
6
, made up of
36
36
36
single squares. We want to place
6
6
6
chess rooks on this board, one rook on each square, so that there are no two rooks on the same row, nor two rooks on the same column. Note that, once the rooks have been placed in this way, we have that, for every square where a rook has not been placed, there is a rook in the same row as it and a rook in the same column as it. We will say that such rooks are in line with this square.For each of those
30
30
30
houses without rooks, color it green if the two rooks aligned with that same house are the same distance from it, and color it yellow otherwise. For example, when we place the
6
6
6
rooks (
T
T
T
) as below, we have:(a) Is it possible to place the rooks so that there are
30
30
30
green squares? (b) Is it possible to place the rooks so that there are
30
30
30
yellow squares? (c) Is it possible to place the rooks so that there are
15
15
15
green and
15
15
15
yellow squares?
Phi function and its conjugate
Given a positive integer
n
≥
2
n \geq 2
n
≥
2
, whose canonical prime factorization is
n
=
p
1
α
1
p
2
α
2
…
p
k
α
k
n = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}
n
=
p
1
α
1
p
2
α
2
…
p
k
α
k
, we define the following functions:
φ
(
n
)
=
n
(
1
−
1
p
1
)
(
1
−
1
p
2
)
…
(
1
−
1
p
k
)
;
φ
‾
(
n
)
=
n
(
1
+
1
p
1
)
(
1
+
1
p
2
)
…
(
1
+
1
p
k
)
\varphi(n) = n\bigg(1 -\frac{1}{p_1}\bigg) \bigg(1 -\frac{1}{p_2}\bigg) \ldots \bigg(1 -\frac {1}{p_k}\bigg) ; \overline{\varphi}(n) = n\bigg(1 +\frac{1}{p_1}\bigg) \bigg(1 +\frac{1}{p_2}\bigg) \ldots \bigg(1 + \frac{1}{p_k}\bigg)
φ
(
n
)
=
n
(
1
−
p
1
1
)
(
1
−
p
2
1
)
…
(
1
−
p
k
1
)
;
φ
(
n
)
=
n
(
1
+
p
1
1
)
(
1
+
p
2
1
)
…
(
1
+
p
k
1
)
Consider all positive integers
n
n
n
such that
φ
‾
(
n
)
\overline{\varphi}(n)
φ
(
n
)
is a multiple of
n
+
φ
(
n
)
n + \varphi(n)
n
+
φ
(
n
)
. (a) Prove that
n
n
n
is even. (b) Determine all positive integers
n
n
n
that satisfy this property.
2
3
Hide problems
Phika sextuplets satisfy a strange inequality
We say that a sextuple of positive real numbers
(
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
)
(a_1, a_2, a_3, b_1, b_2, b_3)
(
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
)
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
h
i
k
a
<
/
s
p
a
n
>
<span class='latex-italic'>phika</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
hika
<
/
s
p
an
>
if
a
1
+
a
2
+
a
3
=
b
1
+
b
2
+
b
3
=
1
a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = 1
a
1
+
a
2
+
a
3
=
b
1
+
b
2
+
b
3
=
1
.(a) Prove that there exists a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
h
i
k
a
<
/
s
p
a
n
>
<span class='latex-italic'>phika</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
hika
<
/
s
p
an
>
sextuple
(
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
)
(a_1, a_2, a_3, b_1, b_2, b_3)
(
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
)
such that:
a
1
(
b
1
+
a
2
)
+
a
2
(
b
2
+
a
3
)
+
a
3
(
b
3
+
a
1
)
>
1
−
1
202
2
2022
a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) > 1 - \frac{1}{2022^{2022}}
a
1
(
b
1
+
a
2
)
+
a
2
(
b
2
+
a
3
)
+
a
3
(
b
3
+
a
1
)
>
1
−
202
2
2022
1
(b) Prove that for every
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
h
i
k
a
<
/
s
p
a
n
>
<span class='latex-italic'>phika</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
p
hika
<
/
s
p
an
>
sextuple
(
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
)
(a_1, a_2, a_3, b_1, b_2, b_3)
(
a
1
,
a
2
,
a
3
,
b
1
,
b
2
,
b
3
)
, we have:
a
1
(
b
1
+
a
2
)
+
a
2
(
b
2
+
a
3
)
+
a
3
(
b
3
+
a
1
)
<
1
a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) < 1
a
1
(
b
1
+
a
2
)
+
a
2
(
b
2
+
a
3
)
+
a
3
(
b
3
+
a
1
)
<
1
<FCE=? <FEC=< CEB$ , < DFC=< CFE in rectangle ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle. The point
E
E
E
lies on side
A
B
‾
\overline{AB}
A
B
and the point
F
F
F
is lies side
A
D
‾
\overline{AD}
A
D
, such that
∠
F
E
C
=
∠
C
E
B
\angle FEC=\angle CEB
∠
FEC
=
∠
CEB
and
∠
D
F
C
=
∠
C
F
E
\angle DFC=\angle CFE
∠
D
FC
=
∠
CFE
. Determine the measure of the angle
∠
F
C
E
\angle FCE
∠
FCE
and the ratio
A
D
/
A
B
AD/AB
A
D
/
A
B
.
Prove that A^p \neq I_p
Let
p
≥
3
p \geq 3
p
≥
3
be a prime number and let
A
A
A
be a matrix of order
p
p
p
with complex entries. Assume that
Tr
(
A
)
=
0
\text{Tr}(A) = 0
Tr
(
A
)
=
0
and
det
(
A
−
I
p
)
≠
0
\det(A - I_p) \neq 0
det
(
A
−
I
p
)
=
0
. Prove that
A
p
≠
I
p
A^p \neq I_p
A
p
=
I
p
.Note:
Tr
(
A
)
\text{Tr}(A)
Tr
(
A
)
is the sum of the main diagonal elements of
A
A
A
and
I
p
I_p
I
p
is the identity matrix of order
p
p
p
.
4
1
Hide problems
PL _|_ ST wanted in cyclic ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral and
M
,
N
M,N
M
,
N
be the midpoints of
A
B
AB
A
B
,
C
D
CD
C
D
respectively. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
L
L
L
. Suppose that the circumcircle of
L
M
N
LMN
L
MN
, with center
T
T
T
, intersects the circumcircle of
A
B
C
D
ABCD
A
BC
D
at two distinct points
X
,
Y
X,Y
X
,
Y
. If the line
M
N
MN
MN
intersects the line
X
Y
XY
X
Y
at
S
S
S
and the line
X
M
XM
XM
intersects the line
Y
N
YN
Y
N
at
P
P
P
, prove that
P
L
PL
P
L
is perpendicular to
S
T
ST
ST
.