MathDB

1

sum of nine consecutive quotients of a_{n}= 1111...111 is a multiple of 9

a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111

( with

2n

digits) with

n > 8

. Let

q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n

the remainder of the division of

a_{i}

by

11

. Prove that the sum of nine consecutive quotients:

s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}

is a multiple of

9

for any

i= 1,2,3, ... , (n-8)

2

1

starting with 4 non intersecting circles from thw vertices of a cyclic quadr.

Let

ABCD

be an inscribed quadrilateral in a circle

c(O,R)

(of circle

O

and radius

R

). With centers the vertices

A,B,C,D

, we consider the circles

C_{A},C_{B},C_{C},C_{D}

respectively, that do not intersect to each other . Circle

C_{A}

intersects the sides of the quadrilateral at points

A_{1} , A_{2}

, circle

C_{B}

intersects the sides of the quadrilateral at points

B_{1} , B_{2}

, circle

C_{C}

at points

C_{1} , C_{2}

and circle

C_{D}

at points

C_{1} , C_{2}

. Prove that the quadrilateral defined by lines

A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}

is cyclic.

4

1

partition a set of 1,2,3,.., n into 3 disjoint sets wit equal sum of elements

Givan the set

S = \{1,2,3,....,n\}

. We want to partition the set

S

into three subsets

A,B,C

disjoint (to each other) with

A\cup B\cup C=S

, such that the sums of their elements

S_{A} S_{B} S_{C}

to be equal .Examine if this is possible when: a)

n=2014

b)

n=2015

c)

n=2018

1

system (x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 and | x - y | = 1

Find all the pairs of real numbers

(x,y)

that are solutions of the system:

(x^{2}+y^{2})^{2}-xy(x+y)^{2}=19

| x - y | = 1

2014 Greece JBMO TST

Subcontests

sum of nine consecutive quotients of a_{n}= 1111...111 is a multiple of 9

starting with 4 non intersecting circles from thw vertices of a cyclic quadr.

partition a set of 1,2,3,.., n into 3 disjoint sets wit equal sum of elements

system (x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 and | x - y | = 1