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Problems
Contests
National and Regional Contests
Spain Contests
pOMA and PErA mathematical olympiads
2023 pOMA
2023 pOMA
Part of
pOMA and PErA mathematical olympiads
Subcontests
(6)
6
1
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Config issues, you say?
Let
Ω
\Omega
Ω
be a circle, and let
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
and
K
K
K
be distinct points on it, in that order, and such that lines
B
C
BC
BC
and
A
D
AD
A
D
are parallel. Let
A
′
≠
A
A'\neq A
A
′
=
A
be a point on line
A
K
AK
A
K
such that
B
A
=
B
A
′
BA=BA'
B
A
=
B
A
′
. Similarly, let
C
′
≠
C
C'\neq C
C
′
=
C
be a point on line
C
K
CK
C
K
such that
D
C
=
D
C
′
DC=DC'
D
C
=
D
C
′
. Prove that segments
A
C
AC
A
C
and
A
′
C
′
A'C'
A
′
C
′
have the same length.
5
1
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$AB>AC$ combogeo
Let
n
≥
2
n\ge 2
n
≥
2
be a positive integer, and let
P
1
P
2
…
P
2
n
P_1P_2\dots P_{2n}
P
1
P
2
…
P
2
n
be a polygon with
2
n
2n
2
n
sides such that no two sides are parallel. Denote
P
2
n
+
1
=
P
1
P_{2n+1}=P_1
P
2
n
+
1
=
P
1
. For some point
P
P
P
and integer
i
∈
{
1
,
2
,
…
,
2
n
}
i\in\{1,2,\ldots,2n\}
i
∈
{
1
,
2
,
…
,
2
n
}
, we say that
i
i
i
is a
P
P
P
-good index if
P
P
i
>
P
P
i
+
1
PP_{i}>PP_{i+1}
P
P
i
>
P
P
i
+
1
, and that
i
i
i
is a
P
P
P
-bad index if
P
P
i
<
P
P
i
+
1
PP_{i}<PP_{i+1}
P
P
i
<
P
P
i
+
1
. Prove that there's a point
P
P
P
such that the number of
P
P
P
-good indices is the same as the number of
P
P
P
-bad indices.
4
1
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Equality algebra
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
be positive real numbers such that
x
1
+
1
x
2
=
x
2
+
1
x
3
=
x
3
+
1
x
4
=
⋯
=
x
n
−
1
+
1
x
n
=
x
n
+
1
x
1
.
x_1+\frac{1}{x_2} = x_2+\frac{1}{x_3} = x_3+\frac{1}{x_4} = \dots = x_{n-1}+\frac{1}{x_n} = x_n+\frac{1}{x_1}.
x
1
+
x
2
1
=
x
2
+
x
3
1
=
x
3
+
x
4
1
=
⋯
=
x
n
−
1
+
x
n
1
=
x
n
+
x
1
1
.
Prove that
x
1
=
x
2
=
x
3
=
⋯
=
x
n
x_1=x_2=x_3=\dots=x_n
x
1
=
x
2
=
x
3
=
⋯
=
x
n
.
3
1
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Two and a half variable diophantine
Find all positive integers
l
l
l
for which the equation
a
3
+
b
3
+
a
b
=
(
l
a
b
+
1
)
(
a
+
b
)
a^3+b^3+ab=(lab+1)(a+b)
a
3
+
b
3
+
ab
=
(
l
ab
+
1
)
(
a
+
b
)
has a solution over positive integers
a
,
b
a,b
a
,
b
.
2
1
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Elementary area geometry
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle, and let
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively be three points on sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
such that
A
E
D
F
AEDF
A
E
D
F
is a cyclic quadrilateral. Let
O
B
O_B
O
B
and
O
C
O_C
O
C
be the circumcenters of
△
B
D
F
\triangle BDF
△
B
D
F
and
△
C
D
E
\triangle CDE
△
C
D
E
, respectively. Finally, let
D
′
D'
D
′
be a point on segment
B
C
BC
BC
such that
B
D
′
=
C
D
BD'=CD
B
D
′
=
C
D
. Prove that
△
B
D
′
O
B
\triangle BD'O_B
△
B
D
′
O
B
and
△
C
D
′
O
C
\triangle CD'O_C
△
C
D
′
O
C
have the same surface.
1
1
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Permutations and apples
Let
n
n
n
be a positive integer. Marc has
2
n
2n
2
n
boxes, and in particular, he has one box filled with
k
k
k
apples for each
k
=
1
,
2
,
3
,
…
,
2
n
k=1,2,3,\ldots,2n
k
=
1
,
2
,
3
,
…
,
2
n
. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than
2
n
+
1
2n+1
2
n
+
1
apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly
2
n
2^n
2
n
distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.