2017 Ecuador Juniors

Part of Ecuador Juniors

Subcontests

(6)

1

CD _|_ EO if AB=AC, O circumcenter, AD=//DB. E centroid ACD

Given an isosceles triangle

ABC

AB = AC

O

be the circumcenter of

ABC

D

the midpoint of

AB

E

the centroid of

ACD

CD \perp EO

1

1-8 at vertices of regular octagon - 2017 Ecuador Juniors (OMEC) L2 p4

Indicate whether it is possible to write the integers

1, 2, 3, 4, 5, 6, 7, 8

at the vertices of an regular octagon such that the sum of the numbers of any

3

consecutive vertices is greater than: a)

11

13

1

x^2 + 3y = 10 , 3 + y = 10/ x - 2017 Ecuador Juniors (OMEC) L2 p2

Find all pairs of real numbers

x, y

that satisfy the following system of equations

\begin{cases} x^2 + 3y = 10 \\ 3 + y = \frac{10}{ x} \end{cases}

1

year monster wakes up - 2017 Ecuador Juniors (OMEC) L2 p1

An ancient Inca legend tells that a monster lives among the mountains that when wakes up, eats everyone who read this issue. After such a task, the monster returns to the mountains and sleeps for a number of years equal to the sum of its digits of the year in which you last woke up. The monster woke up for the first time in the year

234

. a) Would the monster have woken up between the years

2005

2015

? b) Will we be safe in the next

10

1

max of k consecutive integers, coprime with divisors of 8775

Two positive integers are coprime if their greatest common divisor is

1

C

be the set of all divisors of the number

8775

that are greater than

1

k

consecutive positive integers satisfies that each of them is coprime with some element of

C

. Determine the largest possible value of

K

1

p^2- p + 1 is a perfect cube

Find all primes

p

p^2- p + 1

is a perfect cube.