2016 Ecuador Juniors

Day 1

1

A natural number of five digits is called Ecuadorian if it satisfies the following conditions: All its digits are different. The digit on the far left is equal to the sum of the other four digits. Example: 91350 is an Ecuadorian number since 9=1+3+5+0, but 54210 is not since 54+2+1+0. Find how many Ecuadorian numbers exist.

2

Prove that there are no positive integers x,y such that: (x+1)2+(x+2)2+...+(x+9)2=y2

3

Let P1P2...P2016 be a cyclic polygon of 2016 sides. Let K be a point inside the polygon and let M be the midpoint of the segment P1000P2000. Knowing that KP1=KP2011=2016 and KM is perpendicular to P1000P2000, find the length of segment KP2016.

Day 2

4

Two sums, each consisting of n addends , are shown below: S=1+2+3+4+... T=100+98+96+94+... . For what value of n is it true that S=T ?

5

In the parallelogram ABCD, a line through C intersects the diagonal BD at E and AB at F. If F is the midpoint of AB and the area of BEC is 100, find the area of the quadrilateral AFED.

6

Determine the number of positive integers N=¯abcd, with a,b,c,d nonzero digits, which satisfy (2a1)(2b1)(2c1)(2d1)=2abcd1.