▪ Kazakhstan National Olympiad 2013

 Grade 9 

Day 1

1. On the board written numbers from 1 to 25 . Bob can pick any three of them say and replace by . Prove that last number on the board can not be . 

2. Prove that for all natural there exists such that 

.

 Day 2 

1. Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that 

and . 

Prove that and are concurrent.

2. a) Does there exist for any rational number some rational numbers such that 

and

a) Does there exist for any rational number some rational numbers such that 

and .

 Grade 10 

 Day 1 

1. On the board written numbers from 1 to 25 . Bob can pick any three of them say and replace by . Prove that last number on the board can not be . 

2. Let for natural numbers and any natural we have that

divides

  . 

Prove that then . 

3. Let be cyclic quadrilateral. And and intersect at , and at .Let and are points on and such that . And , are intersections of with diagonals.Prove that circumcircles of triangles and are tangent at fixed point. 

 Day 2 

1. Find maximum value of 

 

when are reals in . 

2. Let in triangle incircle touches sides at respectively. Let 

.

Prove that if is intersection of incircle and then

3. How many non-intersecting pairs of paths we have from (0,0) to (n,n) so that path can move two ways:top or right? 

 Grade 11 

 Day 1 

1. Find all triples of positive integer such that

  and .

2. Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that 

and . 

Prove that and are concurrent. 

3. Consider the following sequence:

  . 

Prove that

 Day 2 

1. Find maximum value of 

when are reals in . 

2. Let in triangle incircle touches sides at respectively. Let

  .

Prove that if is intersection of incircle and then .

3. How many non-intersecting pairs of paths we have from (0,0) to (n,n) so that path can move two ways:top or right? 

Πηγή: artofproblemsolving