COMBINATORICA, probleme diverse

1. A beetle sits on each square of a 9 × 9 board. At a signal each beetle crawlsonto a neighboring square. Then it may happen that several beetles will sit on thesame square and none on others. Find the minimal possible number of free squares.

2. A beetle sits on each square of a 9 × 9 board. At a signal each beetle crawlsdiagonally onto a neighboring square. Then it may happen that several beetleswill sit on the same square and none on others. Find the minimal possible numberof free squares.

3. Each of the unit squares of an n×n square is colored with red, yellow or green.Find the smallest value of n such that, for every possible coloring, there exist a rowand a column with at least three unit squares of the same color (the same color onboth the row and on the column). (ONM 2006)

4. Avand un numar de trei cifre putem schimba acest numar ın doua feluri: i) seia prima cifra (sau ultima cifra) si se insereaza ıntre celelalte doua; ii) se inverseazaordinea cifrelor. Dupa 2013 astfel de schimbari putem obtine numarul 312 din 123 ?

5. Pe un cerc se scriu patru X si cinci 0 ıntr-o ordine oarecare. Intre oricare douasimboluri vecine intercalam un X daca simbolurile sunt identice si intercalam un 0daca sunt diferite. Apoi stergem numerele initiale. Este posibil ca dupa un anumitnumar de asemenea operatii sa ajungem la noua 0-uri?

6. Intr-un tabel m × n se trec numere reale. Avem voie sa schimbam semneletuturor numerelor dintr-un rand sau dintr-o coloana. Aratati ca dupa un numarde asemenea operatii, putem face ca sumele pr fiecare rand si pe fiecare coloana safie nenegative. (All-Russian Olympiad 1961)

7. Prove that in any set of seven different positive integers there are three num-bers such that the greatest common divisor of any two of them leaves the sameremainder when divided by three. (Concursul KoMaL)

8. The forum of this journal on the internet has exactly 6n registered members.Each member sends an e-mail to one other member and receives exactly one mailfrom one other member. Prove that we can select a group of at least 2n and atmost 3n members such that no one in the group sent an e-mail to anyone else.

9. La un concurs de sah participa 2n concurenti. In prima runda se joaca n par-tide, ın runda a doua alte n partide, diferite. Demonstrati ca exista n participanticare n-au jucat ıntre ei dar ca nu pot exista n + 1. (Kvant)

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10. Monsieur et Madame Mathon se partagent un plateau de sept fromages. Ilsdesirent en prendre chacun trois en entier, et partager le septieme de sorte quechacun recoive le meme poids total de fromage. Est-il possible, quels que soientles poids des sept fromages, de choisir les trois fromages de l’un, les trois fromagesde l’autre et comment decouper le septieme fromage de sorte que leur desir soitsatisfait ?

11. Gigel si Costel au o colectie J formata din N borcane goale identice si unnumar foarte mare de monede la dispozitie, cand decid sa joace urmatorul jocıncalcit. Stiind ca ın fiecare borcan ıncap exact 100 de monede, cei doi aleg alter-nativ k monede din gramada, cu 1 ≤ k ≤ 10, apoi (ın cadrul aceleiasi mutari) alegun borcan din J . Castiga cel care umple ultimul borcan din J . Daca Gigel mutaprimul, determinati daca vreunul din jucatori are strategie castigatoare. (CosminPohoata, Mathematical Reflections, 2012)

12. Each diagonal of a regular n-gon A1A2...An is colored blue or black. No threevertices are joined with three diagonals of the same color. Find the largest possiblevalue of n. (test juniori Bulgaria, 2012)

13. Pe o foaie alba de hartie cu patratele coloram 40 de patratele cu rosu.Demonstrati ca putem alege 10 patratele rosii astfel ıncat nicicare doua sa nuaiba punct comun. (KoMaL, 2012)

14. An 8 by 8 board (with 64 1 by 1 squares) is painted white.We are allowedto choose any rectangle consisting of 3 of the 64 squares and paint each of the 3squares in the opposite colour (the white ones black, the black ones white). Isit possible to paint the entire board black by means of such operations? (I.S.Rubanov, Turneul Oraselor, 1990)

15. Pe o tabla 9 × 9 sunt colorate 40 din cele 81 de patratele. Despre o linieorizontala sau verticala se spune ca este buna daca ea contine mai multe patratelecolorate decat necolorate. Care este cel mai mare numar de linii bune (orizontalesi verticale) pe care ıl poate avea tabla? (Olimpiada Kazahstan, 2005)

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