MathDB
Problems
Contests
National and Regional Contests
Spain Contests
pOMA and PErA mathematical olympiads
2024 PErA
2024 PErA
Part of
pOMA and PErA mathematical olympiads
Subcontests
(6)
P1
1
Hide problems
Standard and easy combo
Let
n
n
n
be a positive integer, and let
[
n
]
=
{
1
,
2
,
…
,
n
}
[n]=\{1,2,\dots,n\}
[
n
]
=
{
1
,
2
,
…
,
n
}
. Find the maximum posible cardinality of a subset
S
S
S
of
[
n
]
[n]
[
n
]
with the property that there aren't any distinct
a
,
b
,
c
∈
S
a,b,c\in S
a
,
b
,
c
∈
S
such that
a
+
b
=
c
a+b=c
a
+
b
=
c
.
P6
1
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Not CRT, not dirichlet
For each positive integer
k
k
k
, define
a
k
a_k
a
k
as the number obtained from adding
k
k
k
zeroes and a
1
1
1
to the right of
2024
2024
2024
, all written in base
10
10
10
. Determine wether there's a
k
k
k
such that
a
k
a_k
a
k
has at least
202
4
2024
2024^{2024}
202
4
2024
distinct prime divisors.
P5
1
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Clean R+ to R+ FE
Find all functions
f
:
R
+
→
R
+
f\colon \mathbb{R}^+ \to \mathbb{R}^+
f
:
R
+
→
R
+
such that
f
(
x
f
(
x
)
+
y
2
)
=
x
2
+
y
f
(
y
)
f(xf(x)+y^2) = x^2+yf(y)
f
(
x
f
(
x
)
+
y
2
)
=
x
2
+
y
f
(
y
)
for any positive reals
x
,
y
x,y
x
,
y
.
P4
1
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Simple angle chase
Let
A
B
C
ABC
A
BC
be a triangle, and let
E
E
E
,
F
F
F
be the feet of the altitudes from
B
B
B
and
C
C
C
to sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. Let
P
P
P
and
Q
Q
Q
be the intersections of
E
F
EF
EF
with the tangents from
B
B
B
and
C
C
C
to
(
A
B
C
)
(ABC)
(
A
BC
)
, respectively. If
M
M
M
is the midpoint of
B
C
BC
BC
, prove that
(
P
Q
M
)
(PQM)
(
PQM
)
is tangent to
B
C
BC
BC
at
M
M
M
.
P3
1
Hide problems
Surprising n-var ineq
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots, x_n
x
1
,
x
2
,
…
,
x
n
be positive real numbers such that
x
1
+
x
2
+
⋯
+
x
n
=
1
x_1+x_2+\cdots + x_n=1
x
1
+
x
2
+
⋯
+
x
n
=
1
. Prove that
∑
i
=
1
n
min
{
x
i
−
1
,
x
i
}
⋅
max
{
x
i
,
x
i
+
1
}
x
i
≤
1
,
\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,
i
=
1
∑
n
x
i
min
{
x
i
−
1
,
x
i
}
⋅
max
{
x
i
,
x
i
+
1
}
≤
1
,
where we denote
x
0
=
x
n
x_0=x_n
x
0
=
x
n
and
x
n
+
1
=
x
1
x_{n+1}=x_1
x
n
+
1
=
x
1
.
P2
1
Hide problems
Locus of pastanaga points
Let
A
B
C
D
ABCD
A
BC
D
be a fixed convex quadrilateral. Say a point
K
K
K
is pastanaga if there's a rectangle
P
Q
R
S
PQRS
PQRS
centered at
K
K
K
such that
A
∈
P
Q
,
B
∈
Q
R
,
C
∈
R
S
,
D
∈
S
P
A\in PQ, B\in QR, C\in RS, D\in SP
A
∈
PQ
,
B
∈
QR
,
C
∈
RS
,
D
∈
SP
. Prove there exists a circle
ω
\omega
ω
depending only on
A
B
C
D
ABCD
A
BC
D
that contains all pastanaga points.