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Contests
National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2016 Costa Rica - Final Round
2016 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(15)
F3
1
Hide problems
f (2016)=? f(ab) = f (a) + f (b), f(10) = 0, f(a) = 0 when unit digit of a is 7
Let
f
:
Z
+
→
Z
+
∪
{
0
}
f: Z^+ \to Z^+ \cup \{0\}
f
:
Z
+
→
Z
+
∪
{
0
}
a function that meets the following conditions: a)
f
(
a
b
)
=
f
(
a
)
+
f
(
b
)
f (a b) = f (a) + f (b)
f
(
ab
)
=
f
(
a
)
+
f
(
b
)
, b)
f
(
a
)
=
0
f (a) = 0
f
(
a
)
=
0
provided that the digits of the unit of
a
a
a
are
7
7
7
, c)
f
(
10
)
=
0
f (10) = 0
f
(
10
)
=
0
. Find
f
(
2016
)
.
f (2016).
f
(
2016
)
.
F2
1
Hide problems
f (x)=1/(\sqrt[3]{x^2 + 6x + 9} + \sqrt[3]{x^2 + 4x + 3} + \sqrt[3]{x^2 + 2x +1}
Sea
f
:
R
+
→
R
f: R^+ \to R
f
:
R
+
→
R
defined as
f
(
x
)
=
1
x
2
+
6
x
+
9
3
+
x
2
+
4
x
+
3
3
+
x
2
+
2
x
+
1
3
f (x) = \frac{1}{\sqrt[3]{x^2 + 6x + 9} + \sqrt[3]{x^2 + 4x + 3} + \sqrt[3]{x^2 + 2x + 1}}
f
(
x
)
=
3
x
2
+
6
x
+
9
+
3
x
2
+
4
x
+
3
+
3
x
2
+
2
x
+
1
1
Calculate
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
.
.
.
+
f
(
2016
)
.
f (1) + f (2) + f (3) + ... + f (2016).
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
...
+
f
(
2016
)
.
F1
1
Hide problems
trinomial inequalities
Let
a
,
b
a, b
a
,
b
and
c
c
c
be real numbers, and let
f
(
x
)
=
a
x
2
+
b
x
+
c
f (x) = ax^2 + bx + c
f
(
x
)
=
a
x
2
+
b
x
+
c
and
g
(
x
)
=
c
x
2
+
b
x
+
a
g (x) = cx^2 + bx + a
g
(
x
)
=
c
x
2
+
b
x
+
a
functions such that
∣
f
(
−
1
)
∣
≤
1
| f (-1) | \le 1
∣
f
(
−
1
)
∣
≤
1
,
∣
f
(
0
)
∣
≤
1
| f (0) | \le 1
∣
f
(
0
)
∣
≤
1
and
∣
f
(
1
)
∣
≤
1
| f (1) | \le 1
∣
f
(
1
)
∣
≤
1
. Show that if
−
1
≤
x
≤
1
-1 \le x \le 1
−
1
≤
x
≤
1
, then
∣
f
(
x
)
∣
≤
5
4
| f (x) | \le \frac54
∣
f
(
x
)
∣
≤
4
5
and
∣
g
(
x
)
∣
≤
2
| g (x) | \le 2
∣
g
(
x
)
∣
≤
2
.
A1
2
Hide problems
sum ...+1/(\sqrt{2015}+\sqrt{2016}) OLCOMA Costa Rica Final 2016 Shortlist A1 D1
Prove that
(
1
1
+
2
+
1
2
+
3
+
1
3
+
4
+
.
.
.
+
1
2015
+
2016
)
2
(
2017
+
24
14
)
=
201
5
2
\left( \frac{1}{\sqrt1+\sqrt2}+\frac{1}{\sqrt2+\sqrt3}+\frac{1}{\sqrt3+\sqrt4}+...+\frac{1}{\sqrt{2015}+\sqrt{2016}}\right)^2(2017+24\sqrt{14})=2015^2
(
1
+
2
1
+
2
+
3
1
+
3
+
4
1
+
...
+
2015
+
2016
1
)
2
(
2017
+
24
14
)
=
201
5
2
4x4 non linear system of equations with cubic radicals
Find all solutions of the system
y
z
4
x
2
3
+
2
w
x
=
0
\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0
3
x
2
y
z
4
+
2
w
x
=
0
x
z
4
y
3
+
5
w
y
=
0
\sqrt[3]{\frac{xz^4}{y}}+5wy=0
3
y
x
z
4
+
5
w
y
=
0
x
y
x
3
+
7
w
z
−
1
/
3
=
0
\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0
3
x
x
y
+
7
w
z
−
1/3
=
0
x
12
+
125
4
y
5
+
343
2
z
4
=
16
x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16
x
12
+
4
125
y
5
+
2
343
z
4
=
16
where
x
,
y
,
z
≥
0
x, y, z \ge 0
x
,
y
,
z
≥
0
and
w
∈
R
w \in R
w
∈
R
I attached the system, in case I have any typos
A2
2
Hide problems
diophantine p (x + y) = xy OLCOMA Costa Rica Final 2016 Shortlist A2 D1
Find all integer solutions of the equation
p
(
x
+
y
)
=
x
y
p (x + y) = xy
p
(
x
+
y
)
=
x
y
, where
p
p
p
is a prime number.
perfect square +1000= other perfect square +1
The initial number of inhabitants of a city of more than
150
150
150
inhabitants is a perfect square. With an increase of
1000
1000
1000
inhabitants it becomes a perfect square plus a unit. After from another increase of
1000
1000
1000
inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.
A3
1
Hide problems
(x+1/x)(y+1/y)>=9 if x,y>0, x+y=1 OLCOMA Costa Rica Final 2016 Shortlist A3 D1
Let
x
x
x
and
y
y
y
be two positive real numbers, such that
x
+
y
=
1
x + y = 1
x
+
y
=
1
. Prove that
(
1
+
1
x
)
(
1
+
1
y
)
≥
9
\left(1 +\frac{1}{x}\right)\left(1 +\frac{1}{y}\right) \ge 9
(
1
+
x
1
)
(
1
+
y
1
)
≥
9
N1
2
Hide problems
x - [ x/2016]= 2016
Find all
x
∈
R
x \in R
x
∈
R
such that
x
−
[
x
2016
]
=
2016
x - \left[ \frac{x}{2016} \right]= 2016
x
−
[
2016
x
]
=
2016
, where
[
k
]
[k]
[
k
]
represents the largest smallest integer or equal to
k
k
k
.
diophantine x^4 + p = 3y^4
Let
p
>
5
p> 5
p
>
5
be a prime such that none of its digits is divisible by
3
3
3
or
7
7
7
. Prove that the equation
x
4
+
p
=
3
y
4
x^4 + p = 3y^4
x
4
+
p
=
3
y
4
does not have integer solutions.
N3
2
Hide problems
diophantine 3 x 2^a + 1 = b^2
Find all nonnegative integers
a
a
a
and
b
b
b
that satisfy the equation
3
⋅
2
a
+
1
=
b
2
.
3 \cdot 2^a + 1 = b^2.
3
⋅
2
a
+
1
=
b
2
.
(n -1)2^{n - 1} + 5 = m^2 + 4m
Find all natural values of
n
n
n
and
m
m
m
, such that
(
n
−
1
)
2
n
−
1
+
5
=
m
2
+
4
m
(n -1)2^{n - 1} + 5 = m^2 + 4m
(
n
−
1
)
2
n
−
1
+
5
=
m
2
+
4
m
.
N2
2
Hide problems
a^4 + 4b^4 is prime
Determine all positive integers
a
a
a
and
b
b
b
for which
a
4
+
4
b
4
a^4 + 4b^4
a
4
+
4
b
4
be a prime number.
x + y + z divides x^2 + y^2 + z^2 when x^3=y^3=z^3 same remainder mod p
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be positive integers and
p
p
p
a prime such that
x
<
y
<
z
<
p
x <y <z <p
x
<
y
<
z
<
p
. Also
x
3
,
y
3
,
z
3
x^3, y^3, z^3
x
3
,
y
3
,
z
3
leave the same remainder when divided by
p
p
p
. Prove that
x
+
y
+
z
x + y + z
x
+
y
+
z
divides
x
2
+
y
2
+
z
2
x^2 + y^2 + z^2
x
2
+
y
2
+
z
2
.
G3
2
Hide problems
collinear wanted, peprpendiculars around a rectangle
Let the
J
H
I
Z
JHIZ
J
H
I
Z
be a rectangle and let
A
A
A
and
C
C
C
be points on the sides
Z
I
ZI
Z
I
and
Z
J
ZJ
Z
J
, respectively. The perpendicular from
A
A
A
on
C
H
CH
C
H
intersects line
H
I
HI
H
I
at point
X
X
X
and perpendicular from
C
C
C
on
A
H
AH
A
H
intersects line
H
J
HJ
H
J
at point
Y
Y
Y
. Show that points
X
,
Y
X, Y
X
,
Y
, and
Z
Z
Z
are collinear.
6 points concyclic wanted, incircle and incenter related
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be acute, with incircle
Γ
\Gamma
Γ
and incenter
I
I
I
.
Γ
\Gamma
Γ
touches sides
A
B
AB
A
B
,
B
C
BC
BC
and
A
C
AC
A
C
at
Z
Z
Z
,
X
X
X
and
Y
Y
Y
, respectively. Let
D
D
D
be the intersection of
X
Z
XZ
XZ
with
C
I
CI
C
I
and
L
L
L
the intersection of
B
I
BI
B
I
with
X
Y
XY
X
Y
. Suppose
D
D
D
and
L
L
L
are outside of
△
A
B
C
\vartriangle ABC
△
A
BC
. Prove that
A
A
A
,
D
D
D
,
Z
Z
Z
,
I
I
I
,
Y
Y
Y
, and
L
L
L
lie on a circle.
G2
2
Hide problems
computational geo right triangle, AF _|_ EI, AI/IF=4/3
Consider
△
A
B
C
\vartriangle ABC
△
A
BC
right at
B
,
F
B, F
B
,
F
a point such that
B
−
F
−
C
B - F - C
B
−
F
−
C
and
A
F
AF
A
F
bisects
∠
B
A
C
\angle BAC
∠
B
A
C
,
I
I
I
a point such that
A
−
I
−
F
A - I - F
A
−
I
−
F
and CI bisect
∠
A
C
B
\angle ACB
∠
A
CB
, and
E
E
E
a point such that
A
−
E
−
C
A- E - C
A
−
E
−
C
and
A
F
⊥
E
I
AF \perp EI
A
F
⊥
E
I
. If
A
B
=
4
AB = 4
A
B
=
4
and
A
I
I
F
=
43
\frac{AI}{IF}={4}{3}
I
F
A
I
=
4
3
, determine
A
E
AE
A
E
. Notation:
A
−
B
−
C
A-B-C
A
−
B
−
C
means than points
A
,
B
,
C
A,B,C
A
,
B
,
C
are collinear in that order i.e.
B
B
B
lies between
A
A
A
and
C
C
C
.
AL + JB = LJ wanted, cyclic ABCD, angle bisectors
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, such that
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
lie on a circle, with
∠
D
A
B
<
∠
A
B
C
\angle DAB < \angle ABC
∠
D
A
B
<
∠
A
BC
. Let
I
I
I
be the intersection of the bisector of
∠
A
B
C
\angle ABC
∠
A
BC
with the bisector of
∠
B
A
D
\angle BAD
∠
B
A
D
. Let
ℓ
\ell
ℓ
be the parallel line to
C
D
CD
C
D
passing through point
I
I
I
. Suppose
ℓ
\ell
ℓ
cuts segments
D
A
DA
D
A
and
B
C
BC
BC
at
L
L
L
and
J
J
J
, respectively. Prove that
A
L
+
J
B
=
L
J
AL + JB = LJ
A
L
+
J
B
=
L
J
.
G1
2
Hide problems
equilateral wanted, DE//AC, AE: BE = 2: 1
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be isosceles with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
ω
\omega
ω
be its circumscribed circle and
O
O
O
its circumcenter. Let
D
D
D
be the second intersection of
C
O
CO
CO
with
ω
\omega
ω
. Take a point
E
E
E
in
A
B
AB
A
B
such that
D
E
∥
A
C
DE \parallel AC
D
E
∥
A
C
and suppose that
A
E
:
B
E
=
2
:
1
AE: BE = 2: 1
A
E
:
BE
=
2
:
1
. Show that
△
A
B
C
\vartriangle ABC
△
A
BC
is equilateral.
collinear wanted, 2 circles and orthocenter related
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be acute with orthocenter
H
H
H
. Let
X
X
X
be a point on
B
C
BC
BC
such that
B
−
X
−
C
B-X-C
B
−
X
−
C
. Let
Γ
\Gamma
Γ
be the circumscribed circle of
△
B
H
X
\vartriangle BHX
△
B
H
X
and
Γ
2
\Gamma_2
Γ
2
be the circumscribed circle of
△
C
H
X
\vartriangle CHX
△
C
H
X
. Let
E
E
E
be the intersection of
A
B
AB
A
B
with
Γ
\Gamma
Γ
, and
D
D
D
be the intersection of
A
C
AC
A
C
with
Γ
2
\Gamma_2
Γ
2
. Let
L
L
L
be the intersection of line
H
D
HD
HD
with
Γ
\Gamma
Γ
and
J
J
J
be the intersection of line
E
H
EH
E
H
with
Γ
2
\Gamma_2
Γ
2
. Prove that points
L
L
L
,
X
X
X
, and
J
J
J
are collinear.Notation:
A
−
B
−
C
A-B-C
A
−
B
−
C
means than points
A
,
B
,
C
A,B,C
A
,
B
,
C
are collinear in that order i.e.
B
B
B
lies between
A
A
A
and
C
C
C
.
LR3
1
Hide problems
sum of squares of 100 terms of arithmetic progression
Consider an arithmetic progression made up of
100
100
100
terms. If the sum of all the terms of the progression is
150
150
150
and the sum of the even terms is
50
50
50
, find the sum of the squares of the
100
100
100
terms of the progression.
LR2
1
Hide problems
2016 participants in the Olcotournament of chess
There are
2016
2016
2016
participants in the Olcotournament of chess. It is known that in any set of four participants, there is one of them who faced the other three. Prove there is at least
2013
2013
2013
participants who faced everyone else.
LR1
1
Hide problems
21 tiles, some white and some black, in a 3 x 7 rectangle
With
21
21
21
tiles, some white and some black, a
3
×
7
3 \times 7
3
×
7
rectangle is formed. Show that there are always four tokens of the same color located at the vertices of a rectangle.