2002 Silk Road

Part of Silk Road

Subcontests

(4)

1

constant sum in all rows and all collums replacing a number, if divided by 2012

In each unit cell of a finite set of cells of an infinite checkered board, an integer is written so that the sum of the numbers in each row, as well as in each column, is divided by

2002

. Prove that every number

\alpha

can be replaced by a certain number

\alpha'

2002

|\alpha-\alpha'| <2002

and the sum of the numbers in all rows, and in all columns will not change.

1

Problem 4(Number Theory)

Observe that the fraction \frac{1}{7}\equal{}0,142857 is a pure periodical decimal with period 6\equal{}7\minus{}1,and in one period one has 142\plus{}857\equal{}999.For n\equal{}1,2,\dots find a sufficient and necessary condition that the fraction \frac{1}{2n\plus{}1} has the same properties as above and find two such fractions other than

\frac{1}{7}

1

Problem 1(Geometry)

\triangle ABC

be a triangle with incircle

\omega(I,r)

and circumcircle

\zeta(O,R)

l_{a}

be the angle bisector of

\angle BAC

.Denote P\equal{}l_{a}\cap\zeta.Let

D

be the point of tangency

\omega

[BC]

.Denote Q\equal{}PD\cap\zeta.Show that PI\equal{}QI if PD\equal{}r.

1

Nice inequality

I tried to search SRMC problems,but i didn't find them(I found only SRMC 2006).Maybe someone know where on this site i could find SRMC problems?I have all SRMC problems,if someone want i could post them, :wink: Here is one of them,this is one nice inequality from first SRMC: Let

n

be an integer with

n>2

and a_{1},a_{2},\dots,a_{n}\in R^{\plus{}}.Given any positive integers

t,k,p

1<t<n

,set m\equal{}k\plus{}p,prove the following inequalities: a) \frac{a_{1}^{p}}{a_{2}^{k}\plus{}a_{3}^{k}\plus{}\dots\plus{}a_{t}^{k}}\plus{}\frac{a_{2}^{p}}{a_{3}^{k}\plus{}a_{4}^{k}\plus{}\dots\plus{}a_{t\plus{}1}^{k}}\plus{}\dots\plus{}\frac{a_{n\minus{}1}^{p}}{a_{n}^{k}\plus{}a_{1}^{k}\plus{}\dots\plus{}a_{t\minus{}2}^{k}}\plus{}\frac{a_{n}^{p}}{a_{1}^{k}\plus{}a_{2}^{k}\plus{}\dots\plus{}a_{t\minus{}1}^{k}}\geq\frac{(a_{1}^{p}\plus{}a_{2}^{p}\dots\plus{}a_{n}^{p})^{2}}{(t\minus{}1) ( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})} b) \frac{a_{2}^{k}\plus{}a_{3}^{k}\dots\plus{}a_{t}^{k}}{a_{1}^{p}}\plus{}\frac{a_{3}^{k}\plus{}a_{4}^{k}\dots\plus{}a_{t\plus{}1}^{k}}{a_{2}^{p}}\plus{}\dots\plus{}\frac{a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{t\minus{}1}^{k}}{a_{n}^{p}}\geq\frac{(t\minus{}1)(a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{n}^{k})^{2}}{( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})} :wink: