Problem K. 819. (September 2024)
K. 819. Kati has written the number \(\displaystyle +1\) ten times on the blackboard. In each move she can change the sign of five numbers on the board. She can repeat this move an arbitrary number of times. Is it possible that after a series of moves she will end up with nine \(\displaystyle +1\)'s and a single \(\displaystyle -1\) on the board? If the answer is yes, find the minimum number of moves needed.
(5 pont)
Deadline expired on October 10, 2024.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. Igen, elérhető, például az alábbi módon:
| \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) |
| \(\displaystyle -1\) | \(\displaystyle -1\) | \(\displaystyle -1\) | \(\displaystyle -1\) | \(\displaystyle -1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) |
| \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle -1\) | \(\displaystyle -1\) | \(\displaystyle -1\) | \(\displaystyle -1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) |
| \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle -1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) | \(\displaystyle +1\) |
Minden lépésben a \(\displaystyle -1\)-ek száma 5-tel, 3-mal vagy 1-gyel változik (nő vagy csökken). Az első lépés után 5 db \(\displaystyle +1\) és 5 db \(\displaystyle -1\) lesz. Innen páros számú lépéssel érhető el, hogy 1 db \(\displaystyle -1\) legyen, hiszen 4-gyel kell csökkenteni a \(\displaystyle -1\)-ek darabszámát. Tehát még legalább két lépés kell, és ez elég is a fentiek szerint, vagyis három lépés a minimum.
Statistics:
175 students sent a solution. 5 points: 53 students. 4 points: 71 students. 3 points: 8 students. 2 points: 5 students. 1 point: 9 students. 0 point: 5 students. Unfair, not evaluated: 1 solutions. Not shown because of missing birth date or parental permission: 23 solutions.
Problems in Mathematics of KöMaL, September 2024