2014 CentroAmerican

Part of CentroAmerican

Subcontests

(3)

2

Maximum value of sum of fractions

a

b

c

d

be real numbers such that no two of them are equal,

\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4

ac=bd

. Find the maximum possible value of

\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.

Funny numbers

A positive integer

n

is funny if for all positive divisors

d

n

d+2

is a prime number. Find all funny numbers with the largest possible number of divisors.

2

Trapezoid and circles

ABCD

be a trapezoid with bases

AB

CD

, inscribed in a circle of center

O

P

be the intersection of the lines

BC

AD

. A circle through

O

P

intersects the segments

BC

AD

at interior points

F

G

, respectively. Show that

BF=DG

Three equilateral triangles

A

B

C

D

are chosen on a line in that order, with

AB

CD

BC

. Equilateral triangles

APB

BCQ

CDR

are constructed so that

P

Q

R

are on the same side with respect to

AD

\angle PQR=120^\circ

\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.

2

Tico numbers

A positive integer is called tico if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

Figures with at most 2014 squares

Using squares of side 1, a stair-like figure is formed in stages following the pattern of the drawing.
For example, the first stage uses 1 square, the second uses 5, etc. Determine the last stage for which the corresponding figure uses less than 2014 squares.
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