MathDB
Problems
Contests
National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2023 Costa Rica - Final Round
2023 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(6)
3.6
1
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The number obtained by making the ones digit the left-most digit
Given a positive integer
N
N
N
, define
u
(
N
)
u(N)
u
(
N
)
as the number obtained by making the ones digit the left-most digit of
N
N
N
, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of
N
N
N
until it becomes the first (left-most) digit; for example,
u
(
2023
)
=
3202
u(2023) = 3202
u
(
2023
)
=
3202
. (1) Find a
6
6
6
-digit positive integer
N
N
N
such that
u
(
N
)
N
=
23
35
.
\frac{u(N)}{N} = \frac{23}{35}.
N
u
(
N
)
=
35
23
.
(2) Prove that there is no positive integer
N
N
N
with less than
6
6
6
digits such that
u
(
N
)
N
=
23
35
.
\frac{u(N)}{N} = \frac{23}{35}.
N
u
(
N
)
=
35
23
.
3.5
1
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If t^4 + t^{-4} = 2023, then determine t^3 + t^{-3}
Let
t
t
t
be a positive real number such that
t
4
+
t
−
4
=
2023
t^4 + t^{-4} = 2023
t
4
+
t
−
4
=
2023
. Determine the value of
t
3
+
t
−
3
t^3 + t^{-3}
t
3
+
t
−
3
in the form of
a
b
a\sqrt b
a
b
, where
a
a
a
and
b
b
b
are positive integers.
3.4
1
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Clubs of three people in a class of n students
A teacher wants her
N
N
N
students to know each other, so she creates various clubs of three people, so that each student can participate in several clubs. The clubs are formed in such a way that if
A
A
A
and
B
B
B
are two people, then there is a single club such that
A
A
A
and
B
B
B
are two of its three members. (1) Show that there is no way for the teacher to form the clubs if
N
=
11
N = 11
N
=
11
. (2) Show that the teacher can do it if
N
=
9
N = 9
N
=
9
.
3.3
1
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Diagonals AE, BG, CK of 14-gon ABCD...KLMN are concurrent
Let
A
B
C
D
…
K
L
M
N
ABCD \dots KLMN
A
BC
D
…
K
L
MN
be a regular polygon with
14
14
14
sides. Show that the diagonals
A
E
AE
A
E
,
B
G
BG
BG
, and
C
K
CK
C
K
are concurrent.
3.2
1
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LCM(a, b) = 2^r * 3^s
Find all ordered pairs of positive integers
(
r
,
s
)
(r, s)
(
r
,
s
)
for which there are exactly
35
35
35
ordered pairs of positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that the least common multiple of
a
a
a
and
b
b
b
is
2
r
⋅
3
s
2^r \cdot 3^s
2
r
⋅
3
s
.
3.1
1
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f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2 and f(0)=f(1)=1
Let
Z
≥
0
\mathbb Z^{\geq 0}
Z
≥
0
be the set of all non-negative integers. Consider a function
f
:
Z
≥
0
→
Z
≥
0
f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}
f
:
Z
≥
0
→
Z
≥
0
such that
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
and
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
, and that for any integer
n
≥
1
n \geq 1
n
≥
1
, we have
f
(
n
+
1
)
f
(
n
−
1
)
=
n
f
(
n
)
f
(
n
−
1
)
+
(
f
(
n
)
)
2
.
f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.
f
(
n
+
1
)
f
(
n
−
1
)
=
n
f
(
n
)
f
(
n
−
1
)
+
(
f
(
n
)
)
2
.
Determine the value of
f
(
2023
)
/
f
(
2022
)
f(2023)/f(2022)
f
(
2023
)
/
f
(
2022
)
.