MathDB

1

sum x/(1+yz/x) >= M if xy + yz + zx = 1

M

for which it holds that:

\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \ge M,

for all real numbers

x, y, z > 0

that satisfy the equation

xy + yz + zx = 1

.

7

1

p(x -2) = x(y -1) , diophantine

If

p

is a prime number and

x, y

are positive integers, find in terms of

p

, all pairs

(x, y)

that satisfy the equation:

p(x -2) = x(y -1).

If

x+y = 21

, find all triples

(x, y, p)

that satisfy this equation.

8

1

PB = PC , 2 circumcenters related

Let

ABC

be an acute triangle and

D

be the foot of the altiutude from

A

on

BC

,

E

and

F

are the midpoints of

BD

and

DC

respectively.

O

and

Q

are the circumcenters of the triangles

\vartriangle BF

and

\vartriangle ACE

respectively.

P

is the intersection point of

OE

and

QF

, show that

PB = PC

.

6

1

collinear, 2 circles concur on circle - 2015 Cuba 2.6

Let

ABC

be a triangle such that

AB > AC

, with a circumcircle

\omega

. Draw the tangents to

\omega

at

B

and

C

and these intersect at

P

. The perpendicular to

AP

through

A

cuts

BC

at

R

. Let

S

be a point on the segment

PR

such that

PS = PC

. (a) Prove that the lines

CS

and

AR

intersect on

\omega

. (b) Let

M

be the midpoint of

BC

and

Q

be the point of intersection of

CS

and

AR

. Circle

\omega

and the circumcircle of

\vartriangle AMP

intersect at a point

J

(

J \ne A

), prove that

P

,

J

and

Q

are collinear.

5

2

crossroads in a forest

In a certain forest there are at least three crossroads, and for any three crossroads of roads A, B and C there is a road from A to B without passing through C. A deer and a hunter are standing at crossroads of different paths. Is it possible that they can exchange positions without their paths crossing at other points, that are not their initial positions?

min { ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 } < = 1/16

Let

a, b

and

c

be real numbers such that

0 < a, b, c < 1

. Prove that:

\min \ \ \{ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 \} \le \frac{1}{16}.

4

2

A, B subests of 1-100, n \in A => 2n + 2 \in B

Let

A

and

B

be two subsets of

\{1, 2, 3, 4, ..., 100\}

, such that

|A| = |B|

and

A\cap B =\emptyset

. If

n \in A

implies that

2n + 2 \in B

, determine the largest possible value of

|A \cup B|

.

4-digit numbers

Let

A = \overline{abcd}

be a

4

-digit positive integer, such that

a\ge 7

and

a > b >c > d > 0

. Let us consider a positive integer

B = \overline{dcba}

. If all digits of

A+B

are odd, determine all possible values of

A

.

3

2

return walk for blocks traveled an odd number of times

A tourist traveling on a bus from the VIAZUL company arrived at the station in the city of Cienfuegos and immediately set out to take a walking tour of the city. After visiting the Terry theater in this city, decided to return to the station but walking only for blocks traveled an odd number of times in his previous walk. Can he do this on his return regardless. of the initial route?

AB/B 3-digit

Determine the smallest integer of the form

\frac{ \overline{AB}}{B}

.where

A

and

B

are three-digit positive integers and

\overline{AB}

denotes the six-digit number that is form by writing the numbers

A

and

B

consecutively.

2

rhombus, 4 equal incircles - 2015 Cuba 2.2

Let

ABCD

be a convex quadrilateral and let

P

be the intersection of the diagonals

AC

and

BD

. The radii of the circles inscribed in the triangles

\vartriangle ABP

,

\vartriangle BCP

,

\vartriangle CDP

and

\vartriangle DAP

are the same. Prove that

ABCD

is a rhombus,

9 cities and two airline companies

In a certain country there are 9 cities and two airline companies: AeroSol and AeroLuna. Between each pair of cities there are flights from one and only one of them. the two companies. Furthermore, for any triple of cities

X

,

Y

,

Z

σt least one of the flights between them is served by AeroLuna. It is possible to find

4

cities such that all flights between them be served by AeroLuna?

1

2

17 goats, 55 wolves and 6 lions on a magical island

On a magical island there are lions, wolves and goats. Wolves can eat goats while lions can eat both wolves and goats. But if a lion eats a wolf, the lion becomes a goat. Likewise if a wolf eats a goat, the wolf becomes a lion. And if a lion eats a goat, the lion becomes a wolf. Initially on the island there are

17

goats,

55

wolves and

6

lions. If they start eating until they no longer possible to eat more, what is the maximum number of animals that they can stay alive?

f(x) f(y) - f(xy) = x/y + y/x

Let

f

be a function of the positive reals in the positive reals, such that

f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .

(a) Find

f(1)

. (b) Find

f(x)

.

2015 Cuba MO

Subcontests

sum x/(1+yz/x) >= M if xy + yz + zx = 1

p(x -2) = x(y -1) , diophantine

PB = PC , 2 circumcenters related

collinear, 2 circles concur on circle - 2015 Cuba 2.6

crossroads in a forest

min { ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 } < = 1/16

A, B subests of 1-100, n \in A => 2n + 2 \in B

4-digit numbers

return walk for blocks traveled an odd number of times

AB/B 3-digit

rhombus, 4 equal incircles - 2015 Cuba 2.2

9 cities and two airline companies

17 goats, 55 wolves and 6 lions on a magical island

f(x) f(y) - f(xy) = x/y + y/x

2015 Cuba MO

Subcontests

sum x/(1+yz/x) &gt;= M if xy + yz + zx = 1

p(x -2) = x(y -1) , diophantine

PB = PC , 2 circumcenters related

collinear, 2 circles concur on circle - 2015 Cuba 2.6

crossroads in a forest

min { ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 } &lt; = 1/16

A, B subests of 1-100, n \in A =&gt; 2n + 2 \in B

4-digit numbers

return walk for blocks traveled an odd number of times

AB/B 3-digit

rhombus, 4 equal incircles - 2015 Cuba 2.2

9 cities and two airline companies

17 goats, 55 wolves and 6 lions on a magical island

f(x) f(y) - f(xy) = x/y + y/x

sum x/(1+yz/x) >= M if xy + yz + zx = 1

min { ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 } < = 1/16

A, B subests of 1-100, n \in A => 2n + 2 \in B