**A. 449.** In the convex quadrilateral *ABCD*, denote by *r*_{A}, *r*_{B}, *r*_{C} and *r*_{D} the inradii of triangles *BCD*, *CDA*, *DAB* and *ABC*, respectively. Prove that the quadrilateral *ABCD* is cyclic if and only if *r*_{A}+*r*_{C}=*r*_{B}+*r*_{D}.

*Chinese problem*

(5 points)

**K. 165.** Billy and Sophie are making rings out of paper strips and link them together to form chains. Each of them uses a 21×30 cm sheet of paper for making the chain. Billy cuts up the sheet parallel to the 21-cm sides, and a Sophie cuts it parallel to the 30-cm sides. Both of them make strips 1 cm wide, curve them into rings and glue them with 0-cm overlaps. The rings link together to form a chain without any deformation. Which chain is longer? (In calculating the perimeter of a circle, use 3.14 for the value of , and disregard the thickness of the paper.)

(6 points)

This problem is for grade 9 students only.

**K. 166.** Alex is building a pyramid out of congruent cubes, following to the pattern. (He uses no glue, only places the cubes simply next to on top of each other.) Each vertical layer is two cubes taller than the previous one, and forms ascending steps on both sides. When Alex has reached the height of 10 cubes, he goes on decreasing the height of the layers by two cubes in the same fashion. How many cubes does he use altogether?

(6 points)

This problem is for grade 9 students only.

**K. 167.** In honour to the Scottish mathematician *Dudley Langford,* numbers of the following property are called DudLa numbers: Every digit of the number occurs at least twice, and between any pair of identical digits there is as many other digits as the value of the digit in question. For example, 723 121 327 is a DudLa number since there is 1 other digit between two 1's, 2 other digits between two 2's, 3 other digits between two 3's and 7 other digits between two 7's. Find as many 7-digit DudLa numbers as you can.

(6 points)

This problem is for grade 9 students only.

**K. 168.** Consider all possible seven-digit numbers formed by using each of the digits 1, 2, 3, 4, 5, 6, 7 once.

*a*) Show that the difference of any pair of such seven-digit numbers is divisible by 9.

*b*) How many pairs of different numbers can be selected out of them, such that one is a factor of the other?

(6 points)

This problem is for grade 9 students only.