B. 4734. Some fields (unit cubes) constituting a cubical lattice of edge 2015 units are infected by an unknown disease. The disease will spread if at least \(\displaystyle t\) fields in some row parallel to any edge of the cube are infected \(\displaystyle (1 \le t \le 2015)\). In that case, every field of that row will become infected in one minute. Én is javaslok egyet: How many fields need to be infected initially in order to
\(\displaystyle a)\) make it possible
\(\displaystyle b)\) be certain
that the infection reaches all fields of the cube?
Proposed by G. Mészáros, Budapest
(6 points)
B. 4738. \(\displaystyle C\) is an arbitrary point of a circle \(\displaystyle k\) of diameter \(\displaystyle AB\), different from \(\displaystyle A\) and \(\displaystyle B\). Drop a perpendicular from \(\displaystyle C\) onto diameter \(\displaystyle AB\). The foot of the perpendicular on line segment \(\displaystyle AB\) is \(\displaystyle D\), and the other intersection with the circle \(\displaystyle k\) is \(\displaystyle E\). The circle of radius \(\displaystyle CD\) centred at \(\displaystyle C\) intersects circle \(\displaystyle k\) at points \(\displaystyle P\) and \(\displaystyle Q\). Let \(\displaystyle M\) denote the intersection of line segments \(\displaystyle CE\) and \(\displaystyle PQ\). Dertermine the value of \(\displaystyle \frac{PM}{PE} + \frac{QM}{QE}\).
Proposed by B. Bíró, Eger
(4 points)
K. 469. A certain paint needs to be diluted in a \(\displaystyle 2 : 1.5\) ratio, that is, 2 litres of water need to be added to 1.5 litres of paint. Violette Palette, the artist first made 9 litres of a mixture, half paint, half water. Then she realized that this was the wrong ratio, and calculated how much more water to add to achieve the correct proportion. However, instead of the amount of water needed, she added an equal amount of paint by mistake. The second time she did not make any mistake, and added the appropriate quantity of water required for the correct ratio. How many litres of mixture did she get eventually?
(6 points)
This problem is for grade 9 students only.
K. 471. Barbara has one 5forint coin (HUF, Hungarian currency), one 10forint coin, one 20forint coin, three 50forint coins and three 100forint coins in her purse. How many different amounts can she pay exactly (that is, without getting back any change)?
(6 points)
This problem is for grade 9 students only.
K. 474. Ann and Bob are playing a word guessing game. Anna thinks of a meaningful Hungarian word of four letters, which Bob is trying to guess. If Bob tries a certain Hungarian word of four letters, Ann will tell him how many of its letters occur in her word, too, and how many of those are in the correct place and how many are in the wrong place. What may have been the word that Ann had in mind? (No knowledge of the Hungarian language is required. Note that O and Ó are different letters in the Hungarian alphabet.)
Bob's guesses 
Number of correct letters in the right position 
Number of correct letters in the wrong position 
RÓKA 
1 
0 
OKOS 
0 
0 
IKRA 
2 
0 
RITA 
1 
1 
DANÓ 
0 
3 

(6 points)
This problem is for grade 9 students only.