**A. 599.** Two parabolas, and have the same focus point. The directrix of meets at points *A* and *B*. The directrix of meets at *C* and *D*. Show that the points *A*, *B*, *C* and *D* are concyclic.

Proposed by: *Gábor Holló, *Budapest

(5 points)

**B. 4574.** Solve the simultaneous equations

*ax*+*by*+*cz*=*d*,

*a*^{2}*x*+*b*^{2}*y*+*c*^{2}*z*=*d*^{2},

where *a*, *b*, *c*, *d* are distinct real parameters.

Matlap, Kolozsvár, 2013

(4 points)

**B. 4575.** According to the tradition of the ancient tribe of Head Counters, every year is designated either as lucky or as sinister. For example, 2013 is a lucky year since the first 2013 positive integers can be partitioned into at least two classes such that each class contains the same number of elements and the sum of the elements is also equal in each class. If such a partition does not exist, the year is said to be sinister. Which years are sinister?

Suggested by *T. Káspári,* Paks

(6 points)

**C. 1195.** Let *a*, *b* and *c*, respectively, be the lengths of the face diagonals *AF*, *FC *and *CA* of a cuboid *ABCDEFGH*. Let , and , respectively, denote the angles enclosed by the face diagonals with the diagonal *BH* of the cuboid. Prove that *a*^{.}cos -*b*^{.}cos +*c*^{.}cos =0 if the length of edge *AB* is between the lengths of the edges *BC* and *BF*.

(5 points)

This problem is for grade 11 - 12 students only.

**K. 392.** An underground supporting cable is to be fixed to a pole that will be entirely under ground, even the top of the pole will be buried. When the pit dug for the pole was half done, the workers lowered the pole into the pit. They made the following observation: When the whole pit is dug out, the top of the pole will be exactly twice as much below ground level as it is sticking out of the pit now. Given that the height of the pole is two metres, find the depth of the pit when it is completely dug out.

(6 points)

This problem is for grade 9 students only.

**K. 393.** Consider a product whose factors are all 7. Is it possible that the result is a 45-digit number in which the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 occur 9, 8, 7, 6, 5, 4, 3, 2, 1 times, respectively?

(6 points)

This problem is for grade 9 students only.