MathDB
Problems
Contests
National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2020 Costa Rica - Final Round
2020 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(6)
4
1
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10x - 6 h(x) = 4 h(2020/x)
Consider the function
h
h
h
, defined for all positive real numbers, such that:
10
x
−
6
h
(
x
)
=
4
h
(
2020
x
)
10x -6h(x) = 4h \left(\frac{2020}{x}\right)
10
x
−
6
h
(
x
)
=
4
h
(
x
2020
)
for all
x
>
0
x > 0
x
>
0
. Find
h
(
x
)
h(x)
h
(
x
)
and the value of
h
(
4
)
h(4)
h
(
4
)
.
5
1
Hide problems
(1 + tan(1^o))(1 + tan(2^o))...(1 + tan(45^o)).
Determine the value of the expression
(
1
+
tan
(
1
o
)
)
(
1
+
tan
(
2
o
)
)
.
.
.
(
1
+
tan
(
4
5
o
)
)
.
(1 +\tan(1^o))(1 + \tan(2^o))...(1 + \tan(45^o)).
(
1
+
tan
(
1
o
))
(
1
+
tan
(
2
o
))
...
(
1
+
tan
(
4
5
o
))
.
6
1
Hide problems
10 persons around a circular table with 22 vases on it
10
10
10
persons sit around a circular table and on the table there are
22
22
22
vases. Two persons can see each other if and only if there are no vases aligned with them. Prove that there are at least two people who can see each other.
3
1
Hide problems
sum x/(x +\sqrt{(x + y)(x + z)) )<=1
Let
x
,
y
,
z
∈
R
+
x, y, z \in R^+
x
,
y
,
z
∈
R
+
. Prove that
x
x
+
(
x
+
y
)
(
x
+
z
)
+
y
y
+
(
y
+
z
)
(
y
+
x
)
+
z
z
+
(
x
+
z
)
(
z
+
y
)
≤
1
\frac{x}{x +\sqrt{(x + y)(x + z)}}+\frac{y}{y +\sqrt{(y + z)(y + x)}}+\frac{z}{z +\sqrt{(x + z)(z + y)}} \le 1
x
+
(
x
+
y
)
(
x
+
z
)
x
+
y
+
(
y
+
z
)
(
y
+
x
)
y
+
z
+
(
x
+
z
)
(
z
+
y
)
z
≤
1
2
1
Hide problems
R =2r/(r+1) , a square and two incircles
Consider a square
A
B
C
D
ABCD
A
BC
D
. Let
M
M
M
be the midpoint of segment
A
B
AB
A
B
,
Γ
1
\Gamma_1
Γ
1
be the circle tangent to
A
D
‾
\overline{AD}
A
D
,
A
M
‾
\overline{AM}
A
M
and
M
C
‾
\overline{MC}
MC
with radius
r
>
0
r > 0
r
>
0
and let
Γ
2
\Gamma_2
Γ
2
be the circle tangent to
A
D
‾
\overline{AD}
A
D
,
D
C
‾
\overline{DC}
D
C
and
M
C
‾
\overline{MC}
MC
with radius
R
>
0
R > 0
R
>
0
. Prove that
R
=
2
r
r
+
1
R =\frac{2r}{r+1}
R
=
r
+
1
2
r
.
1
1
Hide problems
4-digit number = cube of the sum of its digits
Find all the
4
4
4
-digit natural numbers, written in base
10
10
10
, that are equal to the cube of the sum of its digits.