**A. 497.** There is given an acute triangle *ABC* in the plane. Denote by *A*_{1}, *B*_{1} and *C*_{1} the perpendicular feet of an arbitrary interior point *P* of the triangle on the sides *BC*, *CA *and *AB*, respectively. Let the inradii of the triangles *PAC*_{1}, *PC*_{1}*B*, *PBA*_{1}, *PA*_{1}*C*, *PCB*_{1} and *PB*_{1}*A* be _{1},_{2},...,_{6}, respectively. Determine the locus of those points *P* for which

_{1}+_{3}+_{5}=_{2}+_{4}+_{6}.

Proposed by *Viktor Vígh,* Szeged

(5 points)

**A. 499.** Prove that there exist positive constants *c* and *n*_{0} with the following property. If *A* is a finite set of integers, |*A*|=*n*>*n*_{0}, then

|*A*-*A*|-|*A*+*A*|*n*^{2}-*cn*^{8/5}.

*Miklós Schweitzer Memorial Competition,* 2009

(5 points)

**B. 4235.** The sides of a convex quadrilateral \(\displaystyle ABCD\) are divided into \(\displaystyle n\ge 2\) equal parts. On the sides \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CD\), \(\displaystyle DA\), let \(\displaystyle A_k\), \(\displaystyle B_k\), \(\displaystyle C_k\), \(\displaystyle D_k\) denote the \(\displaystyle k\)th points of division, counting from the vertices \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle D\), respectively. For which pairs \(\displaystyle (n,k)\) is it true that the quadrilateral \(\displaystyle ABCD\) is a parallelogram if and only if the quadrilateral \(\displaystyle A_kB_kC_kD_k\) is also a parallelogram?

Suggested by *G. Mészáros,* Kemence

(3 points)

**B. 4236.** Let \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) be the sides of triangle \(\displaystyle ABC\), let \(\displaystyle f_a\), \(\displaystyle f_b\) and \(\displaystyle f_c\) denote the lengths of the interior angle bisectors, and let \(\displaystyle t_a\), \(\displaystyle t_b\) and \(\displaystyle t_c\) be the segments of the interior angle bisectors that lie inside the circumscribed circle. Show that \(\displaystyle a^2b^2c^2=
f_af_bf_ct_at_bt_c\).

(Mathematics Magazine, 1977)

(3 points)

**C. 1018.** A group of 5 little girls and 7 little boys in the kindergarten are playing at weddings. They choose a bride and groom, a registrar, two bridesmaids, one best man for the bride and one for the groom, one witness for the bride and one for the groom. Three of the girls each have a brother in the group, too, and there are no other pairs of brothers or sisters. In how many different ways can the selection be made if a sister and brother cannot play the bride and groom, and the registrar cannot be the brother or sister of either of the newly weds. (Bridesmaids are girls and best men are boys. There is no restriction on the sex of the witnesses.)

(5 points)

**K. 235.** The official currency on the planet Kloppar is the tirof, for which the small change is the dimar, and there is an even smaller change called the nepi. One dimar is worth a whole number of nepis, and a tirof is worth a whole number of dimars. Sangi had dinner in a restaurant on the planet. After dinner he received the following bill: Fried vondar: 7 dimars and 2 nepis; Pickled rotins: 10 dimars and 5 nepis; Vundeg bread: 1 dimar and 6 nepis; Carbonated nestaki drink: 6 dimars and 4 nepis; Krabban cake: 4 dimars; Total: 2 tirofs, 8 dimars and 1 nepi. How many dimars are worth a tirof, and how many nepis are worth a dimar?

(6 points)

This problem is for grade 9 students only.

**K. 237.** Customers at a parking lot are charged 120 forints (HUF, Hungarian currency) an hour for parking between 8 a.m. and 6 p.m. They need to insert at least 30 forints in the parking meter, and the longest possible period covered by a ticket is 3 hours. The meter takes 10, 20, 50 and 100-forint coins. During the coins being inserted, the meter continually displays the end of the parking time covered by the money inserted. The end of the parking time must not extend further than 6:00 p.m. (that is, at 17:15 p.m. the meter does not accept a 100-forint coin any more.)

*a*) In what time interval is it impossible to obtain a valid parking ticket from the meter?

*b*) A customer is standing in front of the meter at 11:28 a.m. He would like to stay till 7 p.m. How many times should he throw money in the meter, when, and how much if he wants to minimise the number of visits to the meter but does not want to park (not even for a single minute) without a valid ticket?

(6 points)

This problem is for grade 9 students only.

**K. 238.** The centre of the regular pentagon *ABCDE* is the point *O*. Each of the points *A*, *B*, *C*, *D*, *E*, *O* is coloured red, yellow, blue or green (each point is given a single colour), so that none of the triangles *OAB*, *OBC*, *OCD*, *ODE*, *OEA* has two vertices of the same colour. In how many different ways is that possible?

(6 points)

This problem is for grade 9 students only.

**K. 239.** Someone bought a bar of chocolate, and then realised that the same chocolate in another shop cost as many percent more, as the price in forints (HUF, Hungarian currency) that he had paid for it. How much did he pay for the chocolate if it cost 75 forints in the other shop?

(6 points)

This problem is for grade 9 students only.

**K. 240.** A company of *T* members went on a trip to the countryside. Exactly half of them went on foot. The distance to the destination was *T* kilometres along the footpath, and they covered it in 1 hour. The other half of the company went by bike along another route. Their speed (not counting periods when they stopped) was *T* less than *T* times the speed of the walkers.

The bikers had to stop on their way for *T* minutes because of a flat tire, and later on they had to stop again, for *T* times as much time, because of a broken chain on another bike. When they continued the journey, a biker with no technical problems so far took the lead of the group for *T* minutesminutes, and then let somebody else to take the lead. Finally, the bikers arrived at the destination *T* minutes sooner than the walkers. How long was the footpath, and how long was the bike route?

(6 points)

This problem is for grade 9 students only.