**A. 503.** In space, there are given some vectors *u*_{1},*u*_{2},...,*u*_{n} and *v* such that |*u*_{1}|1, ..., |*u*_{n}|1 and |*v*|1, and *u*_{1}+...+*u*_{n}=0. Show that

|*u*_{1}-*v*|+...+|*u*_{n}-*v*|*n*.

(5 points)

**A. 504.** Prove that for arbitrary integers 0<*r*<*k*<*t* there exists a positive integer *N*(*r*,*k*,*t*) which satisfies the following property: whenever *G* is an *r*-uniform hypergraph with at least *N*(*r*,*k*,*t*) vertices such that there is at least one hyperedge on any *k* vertices, then *G *contains a complete subgraph with *t* vertices. (A *hypergraph* is a graph whose edges are arbitrary subsets of the vertices. The graph is called *r-uniform* if all edges contain exactly *r* vertices. An *r*-uniform hypergraph is *complete* if any *r* of its vertices form an edge.)

(5 points)

**A. 505.** In a cyclic quadrilateral *ABCD*, the points *O*_{1} and *O*_{2} are the incenters of triangles *ABC* and *ABD*, respectively. The line *O*_{1}*O*_{2} meets *BC* and *AD* at *E* and *F*, respectively.

(a) Show that there exists a circle *k* which touches the lines *BC* and *AD* at *E* and *F*, respectively.

(b) Prove that *k* also touches the circumcirlce of *ABCD*.

Proposed by: *János Nagy,* Budapest)

(5 points)

**K. 247.** During the first three years of her life, Sleeping Beauty slept an average of 14 hours a day. Then, until she was 16 years old she slept 8 hours a day, and then from her 16th birthday onwards, when she wounded her finger with the reel, she slept 24 hours a day for 100 years. A filbert mouse hibernates for 5 months of the year (from 1 November to 31 March), and in the rest of the year it spends a daily average of 8 hours awake (during the nighttime hours). After how many years of sleep should the prince have woken up Sleeping Beauty so that her daily average of sleep from her birth to her waking up equals that of a filbert mouse? (For simplicity, ignore leap years, that is, assume that February is always 28 days long.)

(6 points)

This problem is for grade 9 students only.

**K. 248.** The *diagram* shows a net of a cube. In how many different ways is it possible to colour two of the six squares red, and the other four with four different colours: white, green, yellow and blue, so that the resulting cube does not have two adjacent faces of the same colour?

(6 points)

This problem is for grade 9 students only.

**K. 249.** 5, 10, 20, 50, 100 and 200-forint coins (HUF, Hungarian currency) are collected in a huge piggy bank. At the moment, there are 18 200 forints in it. Before the last coin was added, the number of various types of coins was inversely proportional to the value of the coins. How many 200-forint coins are there in the piggy bank now?

(6 points)

This problem is for grade 9 students only.

**K. 251.** Extend each diagonal of a square of side in one direction by the length of the side.

*a*) How long is the line segment connecting the new endpoints of the extensions?

*b*) Show that there is a vertex of the square that forms an isosceles triangle with the new endpoints of the extensions.

(6 points)

This problem is for grade 9 students only.