Mathematical and Physical Journal
for High Schools
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Problem C. 1235. (May 2014)

C. 1235. In Flora's flower garden, tulips are grown for mothers' day. One of the flowerbeds has 53 rows with 38 flowers in each row. When Flora inspected all flowers from the first one to the last one, row by row, she observed that every other tulip had coloured streaks in it, every 19th had a broken petal, and every 53rd was not fully open yet. She also discovered that the sum of the numbers of the positions of the perfect tulips (with no coloured streaks or broken petals, fully open) was equal to nineteen times her income in forints (HUF, Hungarian currency). For how many forints did she sell a dozen of perfect tulips?

Suggested by Á. Meszlényi, Budapest

(5 pont)

Deadline expired on June 10, 2014.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. A kérdéses sorszámok összege:

\(\displaystyle (1+3+...+2013)-(53+3\cdot53+5\cdot53+...+37\cdot53)-(19+3\cdot19+...+105\cdot19)+19\cdot53=S.\)

Magyarázat: Csak a páratlan sorszámúak lehetnek tökéletes szépségűek. Viszont minden 19. és minden 53. tulipán sem tökéletes, így ezeket (az olyan páratlan számokat, amik 19, illetve 53 többszörösei) nem számolhatjuk bele az összegbe, ezért kivonjuk őket. Ám a \(\displaystyle 19\cdot53\)-at már kétszer is kivontuk az összegből, így ezt egyszer hozzáadjuk.

Az összeg kiszámolva:

\(\displaystyle S=\frac{1+2013}{2}\cdot1007-53\cdot\frac{1+37}{2}\cdot19-19\cdot\frac{1+105}{2}\cdot53+19\cdot53=942552.\)

A bevétel ezek szerint \(\displaystyle 942552/19=49608\).

A tökéletes virágok száma pedig \(\displaystyle 1007-19-53+1=936\), ami \(\displaystyle 936/12=78\) tucatot jelent. Tehát a tulipánok tucatját \(\displaystyle 49608/78=636\) Ft-ért árulták.


86 students sent a solution.
5 points:66 students.
4 points:12 students.
3 points:3 students.
2 points:3 students.
1 point:1 student.
Unfair, not evaluated:1 solution.

Problems in Mathematics of KöMaL, May 2014