Mathematical and Physical Journal
for High Schools
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Problem K. 279. (January 2011)

K. 279. Ann, Belle, Carol and Dora played with a deck of 52 cards. In one game, Dora was dealing out the cards one by one to the players, starting with Ann, followed by Belle, Carol and Dora in this order, when suddenly some of the cards she had not dealt out yet slipped out of her hands and fell on the floor. The girls noticed that the number of cards on the floor was 2/3 of the number of cards Ann had already got, and the number of cards that Carol had got was 2/3 of those in the remaining part of the deck in Dora's hand that she had not dealt out yet. How many cards had Dora dealt out altogether?

(6 pont)

Deadline expired on February 10, 2011.

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

Megoldás. Készítsünk táblázatot a lapok számáról a szerint, hogy mikor estek ki a lapok Dóri kezéből, továbbá használjuk feladat feltételeit is:

mikor Anna Bea Csilla Dóri leesett maradék összesen (52)
Dóri után \(\displaystyle k\) \(\displaystyle k\) \(\displaystyle k\) \(\displaystyle k\) \(\displaystyle \frac 23 k\) \(\displaystyle \frac 32 k\) \(\displaystyle \frac{37}{6} k\)
Csilla után \(\displaystyle k\) \(\displaystyle k\) \(\displaystyle k\) \(\displaystyle k-1\) \(\displaystyle \frac 23 k\) \(\displaystyle \frac 32 k\) \(\displaystyle \frac{37}{6} k-1\)
Bea után \(\displaystyle k\) \(\displaystyle k\) \(\displaystyle k-1\) \(\displaystyle k-1\) \(\displaystyle \frac 23 k\) \(\displaystyle \frac 32 (k-1)\) \(\displaystyle \frac{37}{6} k-\frac 72\)
Anna után \(\displaystyle k\) \(\displaystyle k-1\) \(\displaystyle k-1\) \(\displaystyle k-1\) \(\displaystyle \frac 23 k\) \(\displaystyle \frac 32 (k-1)\) \(\displaystyle \frac{37}{6} k-\frac 92\)

Az utolsó oszlopban szereplő kifejezés értéke 52, ahonnan soronként a \(\displaystyle k=8,432;\ 8,595;\ 9;\ 9,162\) adódik. Mivel a kiosztott lapok száma egész, ezért a harmadik sorban leírt esemény történt: Dóri kiosztott 8 kört, majd Annának és Beának a kilencedik lapot, ezután leejtett a földre 6 darabot, a kezében pedig maradt 12 kártya.

Dóri összesen 34 lapot osztott ki.

Statistics:

132 students sent a solution.
6 points:64 students.
5 points:29 students.
4 points:9 students.
3 points:12 students.
2 points:8 students.
1 point:4 students.
0 point:2 students.
Unfair, not evaluated:4 solutionss.

Problems in Mathematics of KöMaL, January 2011