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Problems
Contests
National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2015 Costa Rica - Final Round
2015 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(20)
N4
1
Hide problems
c^2/(a + b),b^2/(a + c),a^2/(c + b) integers, a + c, b + c, a + b coprime
Show that there are no triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that a)
a
+
c
,
b
+
c
,
a
+
b
a + c, b + c, a + b
a
+
c
,
b
+
c
,
a
+
b
do not have common multiples in pairs. b)
c
2
a
+
b
,
b
2
a
+
c
,
a
2
c
+
b
\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}
a
+
b
c
2
,
a
+
c
b
2
,
c
+
b
a
2
are integer numbers.
N3
1
Hide problems
ab-1 | a^2 + 1 OLCOMA Costa Rica 2015 SL N3 day2
Find all the pairs
a
,
b
∈
N
a,b \in N
a
,
b
∈
N
such that
a
b
−
1
∣
a
2
+
1
ab-1 |a^2 + 1
ab
−
1∣
a
2
+
1
.
N1
1
Hide problems
diophantine n^2 = 2^n OLCOMA Costa Rica 2015 SL N1 day2
Find all the values of
n
∈
N
n \in N
n
∈
N
such that
n
2
=
2
n
n^2 = 2^n
n
2
=
2
n
.
G5
1
Hide problems
PQ // AB wanted, equal arcs AF=BF, cyclic ABCD
Let
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
be points that lie on the same circle . Let
F
F
F
be such that the arc
A
F
AF
A
F
is congruent with the arc
B
F
BF
BF
. Let
P
P
P
be the intersection point of the segments
D
F
DF
D
F
and
A
C
AC
A
C
. Let
Q
Q
Q
be intersection point of the
C
F
CF
CF
and
B
D
BD
B
D
segments. Prove that
P
Q
∥
A
B
PQ \parallel AB
PQ
∥
A
B
.
G4
1
Hide problems
AN = ID wanted, incenter and incircle of right triangle
Consider
△
A
B
C
\vartriangle ABC
△
A
BC
, right at
B
B
B
, let
I
I
I
be its incenter and
F
,
D
,
E
F,D,E
F
,
D
,
E
the points where the circle inscribed on sides AB,
B
C
BC
BC
and
A
C
AC
A
C
, respectively. If
M
M
M
is the intersection point of
C
I
CI
C
I
and
E
F
EF
EF
, and
N
N
N
is the intersection point of
D
M
DM
D
M
and
A
B
AB
A
B
. Prove that
A
N
=
I
D
AN = ID
A
N
=
I
D
.
A3
1
Hide problems
x =\sqrt{b-\sqrt{b+x}}, b>=1 OLCOMA Costa Rica 2015 SL A3 day2
Knowing that
b
b
b
is a real constant such that
b
≥
1
b\ge 1
b
≥
1
, determine the sum of the real solutions of the equation
x
=
b
−
b
+
x
x =\sqrt{b-\sqrt{b+x}}
x
=
b
−
b
+
x
A2
1
Hide problems
x+y<0 if x^2/y^2+y^2/x^2+x/y+y/x=0 OLCOMA Costa Rica 2015 SL A2 day2
Determine, if they exist, the real values of
x
x
x
and
y
y
y
that satisfy that
x
2
y
2
+
y
2
x
2
+
x
y
+
y
x
=
0
\frac{x^2}{y^2} +\frac{y^2}{x^2} +\frac{x}{y}+\frac{y}{x} = 0
y
2
x
2
+
x
2
y
2
+
y
x
+
x
y
=
0
such that
x
+
y
<
0.
x + y <0.
x
+
y
<
0.
LR4
1
Hide problems
3 conditions for endpoints on lattice points, (a,b) if a+b=multiple of 3 or ...
Let
P
=
{
(
a
,
b
)
/
a
,
b
∈
{
1
,
2
,
.
.
.
,
n
}
,
n
∈
N
}
P =\{(a, b) / a, b \in \{1, 2, ..., n\}, n \in N\}
P
=
{(
a
,
b
)
/
a
,
b
∈
{
1
,
2
,
...
,
n
}
,
n
∈
N
}
be a set of point of the Cartesian plane and draw horizontal, vertical, or diagonal segments, of length
1
1
1
or
2
\sqrt 2
2
, so that both ends of the segment are in
P
P
P
and do not intersect each other. Furthermore, for each point
(
a
,
b
)
(a, b)
(
a
,
b
)
it is true that i) if
a
+
b
a + b
a
+
b
is a multiple of
3
3
3
, then it is an endpoint of exactly
3
3
3
segments. ii) if
a
+
b
a + b
a
+
b
is an even not multiple of
3
3
3
, then it is an endpoint of exactly
2
2
2
segments. iii) if
a
+
b
a + b
a
+
b
is an odd not multiple of
3
3
3
, then it is endpoint of exactly
1
1
1
segment. a) Check that with
n
=
6
n = 6
n
=
6
it is possible to satisfy all the conditions. b) Show that with
n
=
2015
n = 2015
n
=
2015
it is not possible to satisfy all the conditions.
LR3
1
Hide problems
2 player card game with 2n cards
Ana & Bruno decide to play a game with the following rules.: a) Ana has cards
1
,
3
,
5
,
7
,
.
.
.
,
2
n
−
1
1, 3, 5,7,..., 2n-1
1
,
3
,
5
,
7
,
...
,
2
n
−
1
b) Bruno has cards
2
,
4
,
6
,
8
,
.
.
.
,
2
n
2, 4,6, 8,...,2n
2
,
4
,
6
,
8
,
...
,
2
n
During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after
n
n
n
turns. Determine the highest number of points Ana can earn, and how she manages to do this.
LR2
1
Hide problems
numbers 1-9 in a 3x4 table
In the rectangle in the figure, we are going to write
12
12
12
numbers from
1
1
1
to
9
9
9
, so that the sum of the four numbers written in each line is the same and the sum of the three is also equal numbers in each column. Six numbers have already been written. Determine the sum of the numbers of each row and every column. https://cdn.artofproblemsolving.com/attachments/7/f/3db9ded1e703c5392f258e1608a1800760d78c.png
F1
1
Hide problems
f (f (f (k))) = k if f (n) =n + 3 for n odd , f(n) = n/2 for n rvrn
A function
f
f
f
defined on integers such that
f
(
n
)
=
n
+
3
f (n) =n + 3
f
(
n
)
=
n
+
3
if
n
n
n
is odd
f
(
n
)
=
n
2
f (n) = \frac{n}{2}
f
(
n
)
=
2
n
if
n
n
n
is evenIf
k
k
k
is an odd integer, determine the values for which
f
(
f
(
f
(
k
)
)
)
=
k
f (f (f (k))) = k
f
(
f
(
f
(
k
)))
=
k
.
F2
1
Hide problems
f (f (x) f (y)) = xy and f (k) \ne k for k=0,1,-1
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
such that
f
(
f
(
x
)
f
(
y
)
)
=
x
y
f (f (x) f (y)) = xy
f
(
f
(
x
)
f
(
y
))
=
x
y
and there is no
k
∈
R
−
{
0
,
1
,
−
1
}
k \in R -\{0,1,-1\}
k
∈
R
−
{
0
,
1
,
−
1
}
such that
f
(
k
)
=
k
f (k) = k
f
(
k
)
=
k
.
G3
1
Hide problems
sequence of triangles of from angle trisectors. // sides wanted in triangles
Let
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
and
l
1
,
m
1
,
n
1
l_1, m_1, n_1
l
1
,
m
1
,
n
1
be the trisectors closest to
A
1
B
1
A_1B_1
A
1
B
1
,
B
1
C
1
B_1C_1
B
1
C
1
,
C
1
A
1
C_1A_1
C
1
A
1
of the angles
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
respectively. Let
A
2
=
l
1
∩
n
1
A_2 = l_1 \cap n_1
A
2
=
l
1
∩
n
1
,
B
2
=
m
1
∩
l
1
B_2 = m_1 \cap l_1
B
2
=
m
1
∩
l
1
,
C
2
=
n
1
∩
m
1
C_2 = n_1 \cap m_1
C
2
=
n
1
∩
m
1
. So on we create triangles
△
A
n
B
n
C
n
\vartriangle A_nB_nC_n
△
A
n
B
n
C
n
. If
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
is equilateral prove that exists
n
∈
N
n \in N
n
∈
N
, such that all the sides of
△
A
n
B
n
C
n
\vartriangle A_nB_nC_n
△
A
n
B
n
C
n
are parallel to the sides of
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
.
6
2
Hide problems
NK // ML wanted, trapezoid and 2 circles with diameters legs related
Given the trapezoid
A
B
C
D
ABCD
A
BC
D
with the
B
C
∥
A
D
BC\parallel AD
BC
∥
A
D
, let
C
1
C_1
C
1
and
C
2
C_2
C
2
be circles with diameters
A
B
AB
A
B
and
C
D
CD
C
D
respectively. Let
M
M
M
and
N
N
N
be the intersection points of
C
1
C_1
C
1
with
A
C
AC
A
C
and
B
D
BD
B
D
respectively. Let
K
K
K
and
L
L
L
be the intersection points of
C
2
C_2
C
2
with
A
C
AC
A
C
and
B
D
BD
B
D
respectively. Given
M
≠
A
M\ne A
M
=
A
,
N
≠
B
N\ne B
N
=
B
,
K
≠
C
K\ne C
K
=
C
,
L
≠
D
L\ne D
L
=
D
. Prove that
N
K
∥
M
L
NK \parallel ML
N
K
∥
M
L
.
reflection of BC wrt PQ is tangent to (APQ), <POB = <ABC , <QOC = <ACB
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with circumcenter
O
O
O
. Let
P
P
P
and
Q
Q
Q
be internal points on the sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively such that
∠
P
O
B
=
∠
A
B
C
\angle POB = \angle ABC
∠
POB
=
∠
A
BC
and
∠
Q
O
C
=
∠
A
C
B
\angle QOC = \angle ACB
∠
QOC
=
∠
A
CB
. Show that the reflection of line
B
C
BC
BC
over line
P
Q
PQ
PQ
is tangent to the circumcircle of triangle
△
A
P
Q
\vartriangle APQ
△
A
PQ
.
1
2
Hide problems
<BAC=? DE = BC, lines altitudes intersect circumcircle
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be such that
∠
B
A
C
\angle BAC
∠
B
A
C
is acute. The line perpendicular on side
A
B
AB
A
B
from
C
C
C
and the line perpendicular on
A
C
AC
A
C
from
B
B
B
intersect the circumscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
at
D
D
D
and
E
E
E
respectively. If
D
E
=
B
C
DE = BC
D
E
=
BC
, calculate
∠
B
A
C
\angle BAC
∠
B
A
C
.
cyclic wanted, ABCD with _|_ diagonals related, reflections of intersections
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral whose diagonals are perpendicular, and let
S
S
S
be the intersection of those diagonals. Let
K
,
L
,
M
K, L, M
K
,
L
,
M
and
N
N
N
be the reflections of
S
S
S
on the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
and
D
A
DA
D
A
respectively.
B
N
BN
BN
cuts the circumcircle of
△
S
K
N
\vartriangle SKN
△
S
K
N
at
E
E
E
and
B
M
BM
BM
cuts the circumcircle of
△
S
L
M
\vartriangle SLM
△
S
L
M
at
F
F
F
. Prove that the quadrilateral
E
F
L
K
EFLK
EF
L
K
is cyclic.
5
2
Hide problems
1/(a^3 + 3b)+1/(b^3 + 3a)<=1/2 OLCOMA Costa Rica 2015 p5 day2
Let
a
,
b
∈
R
+
a,b \in R^+
a
,
b
∈
R
+
with
a
b
=
1
ab = 1
ab
=
1
, prove that
1
a
3
+
3
b
+
1
b
3
+
3
a
≤
1
2
.
\frac{1}{a^3 + 3b}+\frac{1}{b^3 + 3a}\le \frac12.
a
3
+
3
b
1
+
b
3
+
3
a
1
≤
2
1
.
f(a) + f(b)<= f (a + b) <= f(a) + f(b) + 1 if kf(n) <= f (kn) <= kf(n)+ k- 1
Let
f
:
N
+
→
N
+
f: N^+ \to N^+
f
:
N
+
→
N
+
be a function that satisfies that
k
f
(
n
)
≤
f
(
k
n
)
≤
k
f
(
n
)
+
k
−
1
,
∀
k
,
n
∈
N
+
kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+
k
f
(
n
)
≤
f
(
kn
)
≤
k
f
(
n
)
+
k
−
1
,
∀
k
,
n
∈
N
+
Prove that
f
(
a
)
+
f
(
b
)
≤
f
(
a
+
b
)
≤
f
(
a
)
+
f
(
b
)
+
1
,
∀
a
,
b
∈
N
+
f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+
f
(
a
)
+
f
(
b
)
≤
f
(
a
+
b
)
≤
f
(
a
)
+
f
(
b
)
+
1
,
∀
a
,
b
∈
N
+
4
2
Hide problems
diophantine p^3 + pm + 2zm = m^2 + pz + z^2
Find all triples
(
p
,
M
,
z
)
(p,M, z)
(
p
,
M
,
z
)
of integers, where
p
p
p
is prime,
m
m
m
is positive and
z
z
z
is negative, that satisfy the equation
p
3
+
p
m
+
2
z
m
=
m
2
+
p
z
+
z
2
p^3 + pm + 2zm = m^2 + pz + z^2
p
3
+
p
m
+
2
z
m
=
m
2
+
p
z
+
z
2
(y+z-x)^2 /4x , (z+x-y)^2 /4y, (x+y-z)^2 / 4z are all integers
Find all triples of integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
not zero and relative primes in pairs such that
(
y
+
z
−
x
)
2
4
x
\frac{(y+z-x)^2}{4x}
4
x
(
y
+
z
−
x
)
2
,
(
z
+
x
−
y
)
2
4
y
\frac{(z+x-y)^2}{4y}
4
y
(
z
+
x
−
y
)
2
and
(
x
+
y
−
z
)
2
4
z
\frac{(x+y-z)^2}{4z}
4
z
(
x
+
y
−
z
)
2
are all integers.
3
2
Hide problems
functional with fractional part , {f(x)} sin^2 x+{x} cos(f(x))cosx=f(x)=f(f(x))
Indicate (justifying your answer) if there exists a function
f
:
R
→
R
f: R \to R
f
:
R
→
R
such that for all
x
∈
R
x \in R
x
∈
R
fulfills thati)
{
f
(
x
)
)
}
sin
2
x
+
{
x
}
c
o
s
(
f
(
x
)
)
c
o
s
x
=
f
(
x
)
\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)
{
f
(
x
))}
sin
2
x
+
{
x
}
cos
(
f
(
x
))
cos
x
=
f
(
x
)
ii)
f
(
f
(
x
)
)
=
f
(
x
)
f (f(x)) = f(x)
f
(
f
(
x
))
=
f
(
x
)
where
{
m
}
\{m\}
{
m
}
denotes the fractional part of
m
m
m
. That is,
{
2.657
}
=
0.657
\{2.657\} = 0.657
{
2.657
}
=
0.657
, and
{
−
1.75
}
=
0.25
\{-1.75\} = 0.25
{
−
1.75
}
=
0.25
.
partition a set X of n people into two sets Y and Z, A knows B ...
In a set
X
X
X
of n people, some know each other and others do not, where the relationship to know is symmetric; that is, if
A
A
A
knows
B
B
B
. then
B
B
B
knows
A
A
A
. On the other hand, given any
4
4
4
people:
A
,
B
,
C
A, B, C
A
,
B
,
C
and
D
D
D
: if
A
A
A
knows
B
B
B
,
B
B
B
knows
C
C
C
and
C
C
C
knows
D
D
D
, then it happens at least one of the following three:
A
A
A
knows
C
,
B
C, B
C
,
B
knows
D
D
D
or
A
A
A
knows
D
D
D
. Prove that
X
X
X
can be partition into two sets
Y
Y
Y
and
Z
Z
Z
so that all elements of
Y
Y
Y
know all those of
Z
Z
Z
or no element in
Y
Y
Y
knows any in
Z
Z
Z
.
2
2
Hide problems
9^2(13^2 + 14^2 + 15^2) black coins, i x j coins in a_{ij} in a 27x27 board
In a video game, there is a board divided into squares, with
27
27
27
rows and
27
27
27
columns. The squares are painted alternately in black, gray and white as follows:
∙
\bullet
∙
in the first row, the first square is black, the next is gray, the next is white, the next is black, and so on;
∙
\bullet
∙
in the second row, the first is white, the next is black, the next is gray, the next is white, and so on;
∙
\bullet
∙
in the third row, the order would be gray-white-black-gray and so on;
∙
\bullet
∙
the fourth row is painted the same as the first, the fifth the same as the second,
∙
\bullet
∙
the sixth the same as the third, and so on. In the box in row
i
i
i
and column
j
j
j
, there are
i
j
ij
ij
coins. For example, in the box in row
15
15
15
and column
20
20
20
there are
(
15
)
(
20
)
=
300
(15) (20) = 300
(
15
)
(
20
)
=
300
coins. Verify that in total there are, in the black squares,
9
2
(
1
3
2
+
1
4
2
+
1
5
2
)
9^2 (13^2 + 14^2 + 15^2)
9
2
(
1
3
2
+
1
4
2
+
1
5
2
)
coins.
NT, p_1 + p_2 + ...+ p_n = c^2_{n-1}
A positive natural number
n
n
n
is said to be comico if its prime factorization is
n
=
p
1
p
2
.
.
.
p
k
n = p_1p_2...p_k
n
=
p
1
p
2
...
p
k
, with
k
≥
3
k\ge 3
k
≥
3
, and also the primes
p
1
,
.
.
.
,
p
k
p_1,..., p_k
p
1
,
...
,
p
k
they fulfill that
p
1
+
p
2
=
c
1
2
p_1 + p_2 = c^2_1
p
1
+
p
2
=
c
1
2
p
1
+
p
2
+
p
3
=
c
2
2
p_1 + p_2 + p_3 = c^2_2
p
1
+
p
2
+
p
3
=
c
2
2
.
.
.
...
...
p
1
+
p
2
+
.
.
.
+
p
n
=
c
n
−
1
2
p_1 + p_2 + ...+ p_n = c^2_{n-1}
p
1
+
p
2
+
...
+
p
n
=
c
n
−
1
2
where
c
1
,
c
2
,
.
.
.
,
c
n
−
1
c_1, c_2, ..., c_{n-1}
c
1
,
c
2
,
...
,
c
n
−
1
are positive integers where
c
1
c_1
c
1
is not divisible by
7
7
7
. Find all comico numbers less than
10
,
000
10,000
10
,
000
.
G1
1
Hide problems
MMO 063 Moscow MO 1940 AD + BD = DC in an inscribed equilateral
Points
A
,
B
,
C
A, B, C
A
,
B
,
C
are vertices of an equilateral triangle inscribed in a circle. Point
D
D
D
lies on the shorter arc \overarc {AB} . Prove that
A
D
+
B
D
=
D
C
AD + BD = DC
A
D
+
B
D
=
D
C
.