OliForum Matematico

1) determinare il più piccolo N per cui presi N punti nel piano ne posso prendere 4 formanti un quadrilatero convesso.
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<BR>2) determinare il più piccolo M per cui presi M punti nel piano posso trovare due quadrilateri convessi senza vertici in comune.
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<BR>3) prendiamo nello spazio 7 punti non a 3 a 3 allineati e uniamo tutte le coppie di punti con un segmento. dimostrare che comunque si colorino con 2 colori i segmenti non si possono evitare 2 triangoli monocromatici.
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<BR>4)consideriamo la successione a_n generata con la legge ricorsiva
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<BR> a_n+1 = a_n / a_n-1
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<BR> siano a_0 = 6 e a_1 = 3
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<BR>quale è il millesimo elemento?
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1) 5. 4, ovviamente, non bastano, perchè basta scegliere il quarto punto (D)all\'interno del triangolo formato dai primi 3 (A; B; C). Anche il quinto punto deve essere all\'interno di A , B, C. Prendo A, B e D: il nuovo punto deve essere all\'interno del triangolo formato da questi. Ora considero BDC (si ripete lo stesso ragionamento anche su ADC), il nuovo punto deve essere nel semipiano opposto a BDC rispetto alla retta DC, ma a queste condizioni si verifica facilmente che ADCE è convesso.
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<BR>2) 9. 5 mi bastano per un quadrilatero. Quindi mi \"avanza\" un punto. Ne prendo altri 4 e ripeto il ragionamento di prima.
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<BR>4) 1/3. La sequenza si ripete.

3) da un vertice partono 6 segmenti. Ora: caso a) 3 segmenti di un colore 3 di un altro. I tre punti a cui arrivano i segmenti di un colore non possono essere collegati tra loro da segmenti di quel colore, ma se vengono colorati tutti e tre con l\'altro colore si forma in qualunque caso un triangolo monocromatico. caso b) 4 o più segmenti di un colore. Stesso ragionamento di prima prendendo 2 gruppi di 3 punti a cui arrivano i segmenti del colore prevalente.

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SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <IMG SRC="images/forum/icons/icon_biggrin.gif"> <BR><BR>[ Questo Messaggio è stato Modificato da: duce87 il 05-03-2003 18:11 ]

perchè devi fare queste cose?