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<P>[<A href=3D"http://utenti.quipo.it/base5/index.htm"><STRONG>BASE=20
Cinque</STRONG> - <EM>Appunti di Matematica ricreativa</EM></A>]</P>
<H1 style=3D"TEXT-ALIGN: center">Olimpiadi matematiche per=20
ragazzi<BR>Aritmetica</H1>
<P style=3D"TEXT-ALIGN: center"><STRONG=20
style=3D"MARGIN-LEFT: auto; MARGIN-RIGHT: auto">Esercizi tratti dalle =
gare=20
nazionali e internazionali di matematica</STRONG><BR=20
style=3D"MARGIN-LEFT: auto; MARGIN-RIGHT: auto"><STRONG=20
style=3D"MARGIN-LEFT: auto; MARGIN-RIGHT: auto">per ragazzi della scuola =

elementare e media</STRONG></P>
<P>Non ho scelto i pi=F9 difficili, ma quelli che danno spunti per fare =
buona=20
matematica con tutti.</P>
<HR>

<H1><B>Numeri</B></H1>
<P><B>Le buche nel giardino</B></P>
<P>Nel giardino di Matteo vi sono 8 buche. Il giardiniere copre met=E0 =
di ciascuna=20
buca con delle pietre. Quante sono le buche in quel giardino dopo il =
lavoro del=20
giardiniere?</P>
<P>(Kangourou, 2005)</P>
<P>[8, perch=E9 una buca coperta a met=E0 =E8 ancora una buca]</P>
<P><B>La portata del ponte</B></P>
<P>Due segnali stradali si trovano all=92imboccatura di un ponte. Essi =
indicano la=20
massima larghezza, 325 cm, ed il massimo peso, 4300 kg, consentiti per =
un=20
veicolo che debba attraversare quel ponte.<BR>Quale dei seguenti =
automezzi =E8=20
autorizzato ad attraversare quel ponte?</P>
<P>A) Uno largo 315 cm e pesante 4307 kg.</P>
<P>B) Uno largo 330 cm e pesante 4250 kg.</P>
<P>C) Uno largo 325 cm e pesante 4400 kg.</P>
<P>D) Uno largo 322 cm e pesante 4298 kg.</P>
<P>E) Nessuno dei precedenti.</P>
<P>(Kangourou, 2005)</P>
<P>[D) Uno largo 322 cm e pesante 4298 kg]</P>
<P><B>Il numero misterioso</B></P>
<P>Tra i 5 numeri elencati qui sotto, quello che ho scelto =E8 pari; =
tutte le=20
cifre che lo formano sono diverse fra loro; la cifra delle centinaia =E8 =
doppia di=20
quella delle unit=E0; la cifra delle decine =E8 maggiore della cifra =
delle=20
centinaia.</P>
<P>Che numero ho scelto?</P>
<P>A) 1246</P>
<P>B) 3874</P>
<P>C) 4683</P>
<P>D) 4894</P>
<P>E) 8462</P>
<P>(Kangourou, 2005)</P>
<P>[E) 8462]</P>
<P><B>Il numero pi=F9 grande</B></P>
<P>Osserva la figura: sei numeri sono scritti su altrettanti =
foglietti.</P>
<P><IMG alt=3D"" =
src=3D"http://utenti.quipo.it/base5/scuola/olimparit34.jpg"></P>
<P>Accostando tutti i sei foglietti puoi formare dei numeri di dieci =
cifre, ad=20
esempio 6841309572.</P>
<P>Qual =E8 il pi=F9 piccolo?</P>
<P>(Kangourou, 2005)</P>
<P>[2309415687]</P>
<P><B>Il 7 eliminato!</B></P>
<P>Carla ha scritto i numeri da 1 a 120 nelle caselle di un nastro.</P>
<P><IMG alt=3D"" =
src=3D"http://utenti.quipo.it/base5/scuola/olimparit35.jpg"></P>
<P>Il fratellino si =E8 divertito a cancellare tutte le cifre =AB 7 =
=BB.</P>
<P>Ecco una parte del nastro con i =AB 7 =BB cancellati :</P>
<P>Sul nastro dei numeri da 1 a 120 dove i =AB 7 =BB sono stati =
cancellati,</P>
<P>- quante caselle vuote ci sono ?</P>
<P>- quante caselle con un numero di una sola cifra ?</P>
<P>Spiegate come avete trovato le vostre risposte.</P>
<P>(Rally Matematico Transalpino, 2005)</P>
<P>[2 caselle vuote (7, 77), 25 caselle da una cifra (i numeri da 1 a 9 =
escluso=20
il 7 e i numeri 17, 27, 37, 47, 57, 67, 70, 71, 72, 73, 74, 75, 76, 78, =
79, 87,=20
97)]</P>
<P><B>Ieri, oggi, domani</B></P>
<P>Ieri Milena mi ha detto: =93Dopodomani sar=E0 il 13 dicembre=94. Oggi =
mi chiedo che=20
giorno del mese sar=E0 domani. Sai rispondere?</P>
<P>(Pristem 2004)</P>
<P>[13 dicembre]</P>
<P><B>Il giorno del mio compleanno</B></P>
<P>Un giorno dopo il mio compleanno, quest'anno, sarebbe stato corretto =
dire=20
"Dopo domani sar=E0 un gioved=EC". In che giorno della settimana ho =
compiuto gli=20
anni quest'anno?</P>
<P>[Luned=EC]</P>
<P>(Kangourou, 2003)</P>
<P><B>Mesi di 30 giorni</B></P>
<P>Quanti sono, in un anno solare, i mesi che contengono il giorno =
30?</P>
<P>(Kangourou, 2005)</P>
<P>[11, tutti tranne febbraio]</P>
<P><B>La targa dell'auto</B></P>
<P>La polizia cerca l=92auto di un ladro.</P>
<P>- un primo testimone ha osservato che il numero della targa =E8 =
formato da=20
cinque cifre, tutte differenti,</P>
<P>- un secondo testimone ricorda che la prima cifra =E8 9,</P>
<P>- un terzo testimone ha notato che l=92ultima cifra =E8 8,</P>
<P>- un quarto testimone, che ha 22 anni, ricorda che la somma delle =
cinque=20
cifre della targa =E8 uguale alla sua et=E0.</P>
<P>Quale pu=F2 essere il numero della targa dell=92auto che la polizia =
cerca?=20
Scrivete tutte le possibilit=E0 e spiegate come le avete trovate.</P>
<P>(Rally Matematico Transalpino, 2005)</P>
<P>[9 014 8, 9 041 8 , 9 104 8 , 9 140 8 , 9 401 8 , 9 410 8 , 9 023 8 , =
9 032 8=20
, 9 203 8 , 9 230 8 , 9 302 8 , 9 320 8]</P>
<HR>

<H1><B>Quattro operazioni</B></H1>
<P><B>Il naso di Pinocchio</B></P>
<P>Il naso di Pinocchio =E8 lungo 5 centimetri. Quando Pinocchio dice =
una bugia la=20
Fata dai capelli turchini glielo fa allungare di 3 centimetri, ma quando =

Pinocchio d=E0 una risposta sincera la Fata glielo fa accorciare di 2=20
centimetri.</P>
<P>Alla fine della giornata Pinocchio ha il naso lungo 20 centimetri e =
ha detto=20
7 bugie.</P>
<P>Quante risposte sincere ha dato Pinocchio alla Fata nel corso della=20
giornata?</P>
<P>Spiegate come avete fatto a trovare la risposta.</P>
<P>(Rally Matematico Transalpino, 1999)</P>
<P>[3]</P>
<P><B>Le maglie del Rally</B></P>
<P>Le maglie della squadra di calcio del RMT (Rally Matematico =
Transalpino) sono=20
state lavate e sono stese ad asciugare, appese una accanto =
all=92altra.</P>
<P><IMG height=3D97 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit02.gif" =
width=3D603></P>
<P>Un tifoso della squadra, senza scrupoli, durante la notte ruba =
quattro=20
maglie.</P>
<P>Le maglie che restano non sono state spostate. Al mattino, esse =
formano due=20
gruppi separati. C'=E8 dello spazio vuoto tra i due gruppi.</P>
<P>L'allenatore della squadra nota che, in ogni gruppo, la somma dei =
numeri=20
delle maglie =E8 la stessa.</P>
<P>Quali sono le maglie che sono state rubate?</P>
<P>(Rally Matematico Transalpino, 2000)</P>
<P>[6, 9, 10, 11]</P>
<P><B>Che belle colonne!</B></P>
<P><IMG alt=3D"" =
src=3D"http://utenti.quipo.it/base5/scuola/olimparit36.jpg"></P>
<P>Scrivete un numero in ogni casella rispettando le seguenti =
consegne:</P>
<P>- Utilizzate soltanto i numeri 1, 2, 3, 4, 5 ma tutte le volte che=20
volete.</P>
<P>- In ogni riga, tutti i numeri sono diversi.</P>
<P>- In ogni colonna, tutti i numeri sono diversi.</P>
<P>- In ciascuna colonna il numero scritto nel triangolo =E8 la somma =
degli altri=20
tre numeri.</P>
<P>Completate le colonne e spiegate il vostro ragionamento.</P>
<P>(Rally Matematico Transalpino, 2005)</P>
<P>[Le tre righe di numeri sono: 3 - 2 - 4 - 5 - 1, 5 - 1 - 3 - 4 - 2, 1 =
- 4 - 5=20
- 2 - 3]</P>
<P><B>Triangoli numerici</B></P>
<P><IMG alt=3D"" =
src=3D"http://utenti.quipo.it/base5/scuola/olimparit37.jpg"></P>
<P>In ognuno dei nove dischetti devono risultare scritti nove numeri =
diversi da=20
1 a 9. Il 6 e l=928 sono gi=E0 inseriti; disponete gli altri sette =
numeri in modo=20
che siano tra loro diversi e che la somma dei numeri scritti nei tre =
dischi di=20
ogni triangolo piccolo sia 15.</P>
<P>(Matematica senza frontiere, 2006)</P>
<P><B>L'et=E0 dei nonni</B></P>
<P>- Dimmi, Camilla, quanti anni hanno i tuoi nonni?</P>
<P>- Posso dirti che se addiziono le loro et=E0, trovo 132.</P>
<P>- Dammi ancora una informazione in pi=F9.</P>
<P>- Mio nonno ha 6 anni in pi=F9 di mia nonna.</P>
<P>- E vivono insieme da molto tempo?</P>
<P>- Si sono sposati esattamente 42 anni fa.</P>
<P>Quanti anni avevano i nonni di Camilla il giorno del loro =
matrimonio?</P>
<P>(Rally Matematico Transalpino, 2004)</P>
<P>[21, 27]</P>
<P><B>I numeri in cerchio</B></P>
<P>Inserite in ciascuna delle sette "regioni chiuse" formate da questi =
tre=20
cerchi uno dei numeri 1, 2, 3, 4, 5, 6, 7.</P>
<P><IMG height=3D203 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit03.gif" =
width=3D218></P>
<P>Cercate una disposizione in cui la somma dei numeri all'interno di =
ogni=20
cerchio sia la stessa e <B>la pi=F9 grande possibile</B>.</P>
<P>Cercate anche una disposizione in cui la somma dei numeri all'interno =
di ogni=20
cerchio sia sempre la stessa, ma <B>la pi=F9 piccola possibile</B>.</P>
<P>Mostrate le vostre soluzioni e spiegate il vostro ragionamento.</P>
<P>(Rally Matematico Transalpino, 2004)</P>
<P><B>Griglie di fiammiferi</B></P>
<P>Per costruire la prima figura ci sono voluti 12 fiammiferi.</P>
<P><IMG height=3D60 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit04.gif" =
width=3D58></P>
<P>Per la seconda =E8 stato necessario usare qualche fiammifero in =
pi=F9!</P>
<P><IMG height=3D58 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit05.gif" =
width=3D83></P>
<P>E per la terza, ancora altri fiammiferi!</P>
<P><IMG height=3D58 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit06.gif" =
width=3D110></P>
<P>Continuando a costruire figure nello stesso modo, quanti fiammiferi =
saranno=20
necessari per la costruzione della centesima figura? Giustificate la =
vostra=20
risposta.</P>
<P>(Rally Matematico Transalpino, 2005)</P>
<P>[12 + 99 x 5 =3D 507]</P>
<P><B>Sport in Svezia</B></P>
<P>In un negozio di articoli sportivi di Stoccolma le combinazioni =
di</P>
<P>oggetti illustrate hanno questi prezzi:</P>
<P><IMG height=3D81 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit07.gif" =
width=3D440></P>
<P>Puoi acquistare anche ogni oggetto singolarmente. Quante corone</P>
<P>costa uno di quei palloni da calcio?</P>
<P>(Kangourou, 2006)</P>
<P>[200 corone]</P>
<P><B>Matteo e il suo orologio</B><BR>Matteo si diverte a calcolare la =
somma=20
delle cifre che legge sul proprio</P>
<P>orologio digitale (per esempio, se l'orologio mostra 21:17, allora =
Matteo=20
trova 11).</P>
<P>Qual =E8 la massima somma che pu=F2 trovare?</P>
<P>(Kangourou, 2003)</P>
<P>[24]</P>
<P><B>Una somma particolare</B></P>
<P>In questa somma ogni cifra (esclusa una) =E8 stata sostituita con una =
lettera;=20
a lettere uguali corrispondono cifre uguali e viceversa</P>
<P><B>A A B +</B></P>
<P><B>A A A +</B></P>
<P><B>A A C =3D</B></P>
<P><B>-------</B></P>
<P><B>1 C C C</B></P>
<P>Quanto valgono A, B, e C?</P>
<P>(Pristem, 2001)</P>
<P>[ABC =3D 370, 463, 556, 649]</P>
<P><B>Addizione misteriosa</B></P>
<P>A simboli uguali corrispondono cifre uguali, a simboli diversi =
corrispondono=20
cifre diverse.</P>
<P>Nell'addizione visualizzata, che cifra corrisponde al quadrato?</P>
<P><IMG height=3D112 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit08.gif" =
width=3D114></P>
<P>(Kangourou, 2004)</P>
<P>[quadrato =3D 6, cerchio =3D 9, triangolo =3D 1]</P>
<P><B>Biglie o caramelle?</B></P>
<P>Giuliano ha sette oggetti (biglie o caramelle) nella tasca sinistra e =
otto=20
nella tasca destra. Quando prende un oggetto dalla tasca sinistra, se si =
tratta=20
di una caramella la mangia; se si tratta di una biglia, la mette nella =
tasca=20
destra. Alla fine la tasca sinistra =E8 vuota e la tasca destra contiene =
dodici=20
oggetti.</P>
<P>Quante caramelle Giuliano aveva nella sua tasca sinistra all'inizio =
?</P>
<P>(Pristem)</P>
<P>[3]</P>
<P><B>La famiglia di Pietro</B></P>
<P>Pietro ha esattamente un fratello e sua sorella Caterina ha tanti =
fratelli=20
quante sorelle. Quanti sono i figli (maschi e femmine) nella famiglia di =

Pietro?<BR>(Kangourou, 2005)</P>
<P>[5, tre sorelle e due fratelli]</P>
<P><B>La famiglia di Enrico</B></P>
<P>Enrico ha tre sorelle e cinque fratelli. Sua sorella Sabina ha "S" =
sorelle e=20
"F"</P>
<P>fratelli. Quanto vale il prodotto di S per F?</P>
<P>(Kangourou, 2005)</P>
<P>[12]</P>
<P><B>Il drago rosso e il drago verde</B></P>
<P>Se il drago rosso avesse 6 teste in pi=F9 del drago verde, essi =
avrebbero in=20
totale 34 teste. Ma il drago rosso ha 6 teste meno del drago verde.</P>
<P>Quante teste ha il drago rosso?</P>
<P>(Kangourou, 2001)</P>
<P>[8]</P>
<P><B>Le botti</B></P>
<P>Un vignaiolo possiede 15 botti:</P>
<P>- 5 botti vuote;</P>
<P>- 5 botti piene per met=E0;</P>
<P>- 5 botti piene.</P>
<P>Vuole suddividere le botti fra i suoi tre figli senza fare alcun =
travaso, ma=20
in modo che ognuno riceva lo stesso numero di botti e la stessa =
quantit=E0 di=20
vino.</P>
<P>Come pu=F2 fare?</P>
<P><IMG height=3D138 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit09.gif" =
width=3D227></P>
<P>(Matematica senza frontiere, 2005)</P>
<P>[Ecco un esempio di soluzione, ma ne esistono altri:</P>
<P><IMG height=3D152 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit10.gif" =
width=3D240>]</P>
<P><B>Le palline da ping-pong</B></P>
<P>Una cassa contiene 4000 palline da ping-pong.</P>
<P>Alle ore 10, un ragazzo ha iniziato a prelevare le palline, una per =
una, a=20
velocit=E0 costante. Dopo 6 ore, nella cassa sono rimaste 3520 palline. =
Se il=20
ragazzo continuasse con lo stesso ritmo, a che ora si troverebbero 2000 =
palline=20
nella scatola?</P>
<P>(Olimpiada Brasileira de Matematica, 2006)</P>
<P>[Alle 11 del giorno seguente]</P>
<P><B>Il tempo nel pianeta POT</B></P>
<P>Nel pianeta POT il numero delle ore del giorno =E8 uguale al numero =
dei giorni=20
della settimana, che =E8 uguale al numero delle settimane del mese, che =
=E8 uguale=20
al numero dei mesi dell'anno. Sapendo che su POT l'anno ha 4096 ore, =
quante=20
settimane ha un mese?</P>
<P>(Olimpiada Brasileira de Matematica, 2006)</P>
<P>[8]</P>
<P><B>La gita scolastica</B></P>
<P>Gli alunni di una scuola partecipano ad una gita utilizzando due =
autobus.</P>
<P>Al momento della partenza, 57 alunni entrano nel primo autobus e =
soltanto 31=20
entrano nel secondo autobus.</P>
<P>Quanti ragazzi devono passare dal primo autobus al secondo affinch=E9 =
i due=20
autobus trasportino lo stesso numero di alunni?</P>
<P>(Olimpiada Brasileira de Matematica, 2004)</P>
<P>[13]</P>
<P><B>Una corsa in taxi</B></P>
<P>Una corsa in taxi costa 2,50 euro per la chiamata pi=F9 10 centesimi =
per ogni=20
100 metri percorsi.</P>
<P>Ho soltanto 10 eureo nel portafoglio. Perci=F2 posso pagare una corsa =
in taxi=20
lunga... quanto?</P>
<P>(Olimpiada Brasileira de Matematica, 2004)</P>
<P>[7,5 km]</P>
<P><B>Fabbricanti di braccialetti</B></P>
<P>Un artigiano inizia a lavorare alle ore 8 e produce 6 braccialetti =
ogni 20=20
minuti.</P>
<P>Il suo aiutante inizia a lavorare un'ora dopo e produce 8 =
braccialetti ogni=20
mezz'ora.</P>
<P>L'artigiano smette di lavorare a mezzogiorno ma avverte il suo =
aiutante:=20
"Dovrai continuare a lavorare finch=E9 non avrai prodotto tanti =
braccialetti=20
quanti ne ho fatti io!"</P>
<P>A che ora l'aiutante potr=E0 andare a riposarsi?</P>
<P>(Olimpiada Brasileira de Matematica, 2004)</P>
<P>[13:30]</P>
<P><B>Il costo della benzina</B></P>
<P>Un litro di benzina costa 1,50 euro. L'auto di Enrico percorre 25 km =
con tre=20
litri di benzina.</P>
<P>Quanto costa la benzina necessaria per percorrere 600 km?</P>
<P>(Olimpiada Brasileira de Matematica, 2000)</P>
<P>[54]</P>
<P><B>Produzione di refrigerante</B></P>
<P>Tutta la produzione mensile di bombole di refrigerante di una certa =
fabbrica=20
=E8 stata venduta in tre lotti. Il primo lotto comprendeva la met=E0 =
della=20
produzione, il secondo lotto ne comprendeva i 2/5 e il terzo lotto era =
formato=20
da 2500 bombole.</P>
<P>Quante bombole ha prodotto la fabbrica in quel mese?</P>
<P>(Olimpiada Brasileira de Matematica, 2002)</P>
<P>[25000]</P>
<HR>

<H1><B>Misure (lunghezza, peso, tempo, capacit=E0)</B></H1>
<P><B>Le fermate dell'autobus</B></P>
<P>Nove fermate di autobus consecutive sono situate lungo una strada in =
modo che=20
la distanza tra due fermate adiacenti sia sempre la stessa. La distanza =
fra la=20
prima e la terza fermata =E8 600 metri.</P>
<P>Quanti metri dista la prima fermata dalla nona?</P>
<P>(Kangourou, 2004)</P>
<P>[2400 m]</P>
<P><B>Mesi di 30 giorni</B></P>
<P>Quanti sono, in un anno solare, i mesi che contengono il giorno =
30?</P>
<P>(Kangourou, 2005)</P>
<P>[11, tutti tranne febbraio]</P>
<P><B>Ieri, oggi, domani</B></P>
<P>Ieri Milena mi ha detto: =93Dopodomani sar=E0 il 13 dicembre=94. Oggi =
mi chiedo che=20
giorno del mese sar=E0 domani. Sai rispondere?</P>
<P>(Pristem 2004)</P>
<P>[13 dicembre]</P>
<P><B>Il giorno del mio compleanno</B></P>
<P>Un giorno dopo il mio compleanno, quest'anno, sarebbe stato corretto =
dire=20
"Dopo domani sar=E0 un gioved=EC". In che giorno della settimana ho =
compiuto gli=20
anni quest'anno?</P>
<P>[Luned=EC]</P>
<P>(Kangourou, 2003)</P>
<P><B>Dopo 2003 minuti</B></P>
<P>Quale data sar=E0 2003 minuti dopo le 20.03 del 20-03-2003?</P>
<P>(Kangourou, 2003)</P>
<P>[B) 22-03-2003]</P>
<P><B>Il peso delle scatole</B></P>
<P>Alcune casse cubiche, tutte uguali, sono state impilate come vedete =
nella=20
figura qui sotto.</P>
<P>Se ciascuna cassa pesa 25 kg, quanto pesa tutta la pila?</P>
<P>(Olimpiada Brasileira de Matematica, 2002)</P>
<P><IMG height=3D163 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit11.gif" =
width=3D124></P>
<P>[14 x 25 =3D 350 kg]</P>
<HR>

<H1><B>Espressioni e sequenze di operazioni</B></H1>
<P><B>I tre conigli</B></P>
<P>Tre conigli mangiano le verdure del mio orto!</P>
<P>Il coniglio bianco mangia ogni sera una carota.</P>
<P>Il coniglio marrone mangia ogni sera una rapa o, se non ce ne sono =
pi=F9, tre=20
carote.</P>
<P>Il coniglio nero mangia ogni sera un cavolo o, se non ce ne sono =
pi=F9, tre=20
rape, o se non ce ne sono pi=F9, 5 carote.</P>
<P>Questa mattina, ho raccolto una parte degli ortaggi. Ho lasciato per =
i=20
conigli: 45 carote, 21 rape, 5 cavoli. Per quanti giorni possono =
nutrirsi tutti=20
e tre i conigli?</P>
<P>(Rally Matematico Transalpino, 2005)</P>
<P>[13]</P>
<P><B>Castelli di carta</B></P>
<P>Andrea si diverte a costruire castelli con le carte da gioco.</P>
<P>Ha costruito questi due castelli: il primo ha due piani ed =E8 fatto =
con 7=20
carte; il secondo ha tre piani ed =E8 fatto con 15 carte.</P>
<P><IMG height=3D91 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit12.gif" =
width=3D168></P>
<P>Per costruire un castello di 25 piani, quante carte dovrebbe =
utilizzare=20
Andrea?</P>
<P>(Rally Matematico Transalpino, 2006)</P>
<P>[950]</P>
<P><B>Una sequenza di operazioni</B></P>
<P>La figura illustra una sequenza di operazioni che d=E0 come risultato =
50. Da=20
quale numero siamo partiti?</P>
<P><IMG height=3D87 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit13.gif" =
width=3D519></P>
<P>(Kangourou, 2004)</P>
<P>[42]</P>
<P><B>Il numero di partenza</B></P>
<P>Ho in mente un numero: sottraggo da esso 203, quindi aggiungo al =
risultato=20
2003 ottenendo cos=EC 20003.</P>
<P>Da quale numero sono partito?</P>
<P>Kangourou, 2003</P>
<P>[18203]</P>
<P><B>Divisione di salamini</B></P>
<P>Ciascuno dei 5 salamini, sul tavolo, viene affettato con 5 tagli. =
Cinque=20
amici ricevono lo stesso numero di fette.</P>
<P>Quante fette ricever=E0 ogni ragazzo?</P>
<P>(Pristem 2002, Cat. C1)</P>
<P>[6]</P>
<P><B>I soldatini di Pierino</B></P>
<P>Pierino dispone di stampi per fabbricare soldatini di piombo. Fa =
fondere=20
alcuni lingottini di piombo e osserva che produce un soldatino per ogni=20
lingottino, inoltre, con gli avanzi del piombo fuso, riesce a costruire =
un altro=20
soldatino ogni 4.</P>
<P>Quanti soldatini fabbrica con 24 lingottini?</P>
<P>(Gran Premio di Matematica Applicata, 2006)</P>
<P>[32]</P>
<P><B>Il videogioco portentoso</B></P>
<P>Silvio =E8 un appassionato di videopoker e slot-machine e tutti i =
giorni perde=20
sistematicamente le 15 monete da un euro che porta con s=E9.</P>
<P>Una sera prova una nuova macchina la =93magic-video-pack=94: =
introducendo nel=20
videogioco una cifra fissa, essa raddoppia il capitale che Silvio ha in=20
tasca!</P>
<P>Fa una prova, introduce alcune monete e effettivamente sente che gli =
euro che=20
aveva in tasca raddoppiano.</P>
<P>Riprova e le monete raddoppiano ancora.</P>
<P>Decide allora di continuare, ma al quarto tentativo si accorge di =
aver perso=20
tutto come al solito!</P>
<P>Quante monete da un euro devono essere introdotte per giocare alla=20
=93magic-video-pack=94?</P>
<P>(Gran Premio di Matematica Applicata, 2005)</P>
<P>[8]</P>
<P><B>La calcolatrice di Sofia</B></P>
<P>La calcolatrice di Sofia =E8 alquanto strana. Essa possiede due tasti =

speciali:<BR>- un tasto D che raddoppia il numero scritto sul =
visore;</P>
<P>- un tasto E che elimina la cifra delle unit=E0 del numero scritto =
sul=20
visore.</P>
<P>Per esempio, se sul visore =E8 scritto 123 e noi premiamo D, =
otterremo 246. Se=20
poi premiamo E otterremo 24.</P>
<P>Supponiamo che sul visore sia scritto 1999.</P>
<P>Premendo in successione i tasti: D, E, D, E, quale numero si =
ottiene?</P>
<P><I>(Olimpiada Brasileira de Matematica, 1999)</I></P>
<P>[79]</P>
<HR>

<H1><B>Multipli e divisori</B></H1>
<P><B>Il gioco del BUM!</B></P>
<P>Si sa che la "tabellina del 7" =E8 alquanto ostica!</P>
<P>Per insegnarla, un maestro elementare fa con i suoi alunni il =
seguente gioco:=20
li dispone a cerchio e ognuno, a turno, deve dire un numero della =
sequenza=20
naturale, senza per=F2 nominare i multipli di 7, sostituendoli con la =
parola=20
"BUM"; la stessa cosa deve essere fatta se il numero finisce per 7. Chi =
sbaglia=20
=E8 eliminato e l'ultimo alunno rimasto vince il gioco. Per esempio =
nella sequenza=20
da 1 a 20 gli alunni gli alunni gridano: "1, 2, 3, 4, 5, 6, BUM, 8, 9, =
10, 11,=20
12, 13, BUM, 15, 16, BUM, 18, 19, 20".</P>
<P>Quanti BUM vengono detti in una sequenza da 1 a 200?</P>
<P>(Gran Premio di Matematica Applicata, 2004)</P>
<P>[45]</P>
<P><B>Saltando, saltando</B></P>
<P>Una rana, un canguro e una lepre saltellano sulla =ABpista dei =
numeri=BB:</P>
<P><IMG height=3D89 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit14.gif" =
width=3D502></P>
<P>Partono tutti dalla casella 0.</P>
<P>La rana fa sempre salti da tre caselle (quindi con il primo salto =
arriva=20
sulla casella 3), il canguro fa sempre salti da sei caselle e la lepre =
fa sempre=20
salti da quattro caselle.</P>
<P>Con l=92ultimo salto ogni animale arriva sulla casella finale del =
percorso.</P>
<P>Ciascun animale lascia le proprie impronte sulla casella su cui =
poggia le=20
zampe.</P>
<P>Terminato il gioco, ci sono 9 caselle contenenti ciascuna impronte di =
tutti e=20
tre gli animali.</P>
<P>Indicate qual =E8 il numero della casella finale della pista.</P>
<P>(Rally Matematico Transalpino, 2002)</P>
<P>[45]</P>
<P><B>Prodotti in riga</B></P>
<P>Disponete i dieci numeri da 1 a 10 nei cerchi di questa figura in =
modo tale=20
che il prodotto di tre numeri allineati sia il numero indicato alla fine =
della=20
riga.</P>
<P><IMG height=3D289 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit15.gif" =
width=3D273></P>
<P>Calcolate i due prodotti mancanti.</P>
<P>Quanti sono i modi di disporre questi dieci numeri?</P>
<P>Rally Matematico Transalpino, 2002</P>
<P><IMG height=3D203 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit16.gif" =
width=3D187></P>
<P><B>Il prezzo dei palloni</B></P>
<P>Simonetta vuole acquistare dei palloni da basket, tutti uguali fra =
loro. Se=20
comprasse cinque palloni, le rimarrebbero 10 euro nel portafoglio. Se ne =

comprasse sette, dovrebbe chiedere un prestito di 22 euro.</P>
<P>Quanti euro costa un pallone da basket?</P>
<P>(Kangourou, 2003)</P>
<P>[16 euro]</P>
<P><B>Nel paese di Piovepoco</B></P>
<P>Nel paese PIOVEPOCO manca l'acqua. Due amiche, Laura e Paola, vanno a =

prendere l'acqua con un secchio alla fontana ACQUACHIARA.</P>
<P>I loro due secchi insieme contengono 26 litri. Con l'acqua contenuta =
nel=20
secchio di Laura si pu=F2 riempire per tre volte il secchio di Paola e =
restano=20
ancora 2 litri d'acqua nel secchio di Laura.</P>
<P>Quanti litri contiene il secchio di Paola? E quello di Laura?</P>
<P>(Rally Matematico Transalpino, 2004)</P>
<P>[6, 20]</P>
<P><B>I nipotini di nonna Alice</B></P>
<P>=C8 sabato e nonna Alice aspetta i suoi nipotini.</P>
<P>Ha preparato 3 dolcetti al cioccolato per ciascuno di loro.</P>
<P>Ma..sorpresa! I nipotini arrivano con due amichetti.</P>
<P>Per non farli litigare nonna Alice mangia un dolcetto e cos=EC ne =
pu=F2 dare 2 ad=20
ogni bambino.</P>
<P>Quanti sono i nipotini di nonna Alice?</P>
<P>(Rally Matematico Transalpino, 2004)</P>
<P>[5]</P>
<P><B>Sulla pista di pattinaggio</B></P>
<P>Nikita e Sasha corrono sulla pista dello stadio. Nikita impiega 3 =
minuti per=20
ogni giro, mentre Sasha impiega 4 minuti per ogni giro. Partono allo =
stesso=20
momento.</P>
<P>Dopo quanti minuti essi passeranno ancora la linea di partenza =
insieme?</P>
<P>(Kangourou, 2001)</P>
<P>[12]</P>
<P><B>L=92emiro prolifico</B></P>
<P>L=92emiro Abdul Azir divenne aveva pi=F9 di 39 figli, inclusi molti =
gemelli.</P>
<P>Lo storico Ahmed Aab afferma in uno dei suoi scritti che tutti i =
figli=20
dell=92emiro erano gemelli doppi, tranne 39, tutti erano gemelli tripli, =
tranne 39=20
e tutti erano gemelli quadrupli, tranne 39.</P>
<P>Quanti erano i figli dell=92emiro Abdul Azir?</P>
<P>(Olimpiada Brasileira de Matematica, 2000)</P>
<P><I>Suggerimento: dal testo si capisce che num. gemelli doppi =3D num. =
gemelli=20
tripli =3D num. gemelli quadrupli; inoltre tale numero deve essere =
divisibile sia=20
per 2 sia per 3 sia per 4.</I></P>
<P>[12]</P>
<HR>

<H1><B>Ordinamenti, sequenze</B></H1>
<P><B>Le scimmiette colorate</B></P>
<P>Anna ha cinque scimmiette di peluche: una blu, una rossa, una gialla, =
una=20
marrone e una verde. Le dispone in fila, una accanto all=92altra, su un =
ripiano=20
della sua libreria.</P>
<P>Anna vede che:</P>
<P>- la scimmia gialla =E8 a destra della verde e a sinistra della =
marrone</P>
<P>- ci sono tre scimmiette a sinistra della rossa</P>
<P>- la scimmia blu non =E8 agli estremi della fila.</P>
<P>Colorate le cinque scimmiette sul ripiano.</P>
<P><IMG height=3D98 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit01.gif" =
width=3D377></P>
<P>Se trovate un'altra possibilit=E0 colorate il disegno qui sotto</P>
<P><IMG height=3D98 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit01.gif" =
width=3D377></P>
<P>Le due soluzioni possibili: V =96 B =96 G =96 R =96 M e V =96 G =96 B =
=96 R =96 M</P>
<P>(Rally Matematico Transalpino, 2005, Cat. 3, 4)</P>
<P><B>Tre cassette e tre oggetti</B></P>
<P>Su un tavolo ci sono tre scatole e tre oggetti, una moneta, un =
fermaglio e=20
una gomma.</P>
<P>Ogni oggetto si trova in una scatola diversa.</P>
<P>Si sa che:</P>
<P>- la scatola verde =E8 a sinistra della scatola blu;</P>
<P>- la moneta =E8 a sinistra della gomma;</P>
<P>- la scatola rossa =E8 a destra del fermaglio;</P>
<P>- la gomma =E8 a destra della scatola rossa.</P>
<P>Disegna le tre scatole con i rispettivi contenuti.</P>
<P>(Olimpiada Brasileira de Matematica, 2004)</P>
<P><I>Suggerimento: considera dapprima le ultime due informazioni. Ti =
permettono=20
di disegnare il seguente schema.</I></P>
<TABLE cellSpacing=3D0 cellPadding=3D5 border=3D0>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter></P>
      <P align=3Dcenter><IMG height=3D39 alt=3D""=20
      src=3D"http://utenti.quipo.it/base5/scuola/olimparit17.gif" =
width=3D70></P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter></P>
      <P align=3Dcenter><IMG height=3D49 alt=3D""=20
      src=3D"http://utenti.quipo.it/base5/scuola/olimparit18.gif" =
width=3D79></P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><IMG height=3D63 alt=3D""=20
      src=3D"http://utenti.quipo.it/base5/scuola/olimparit19.gif"=20
  width=3D82></P></TD></TR></TBODY></TABLE>
<P>[verde-fermaglio, rossa-moneta, blu-gomma]</P>
<P><B>Numeri in cerchio</B></P>
<P>Scrivete i numeri da 1 a 10 nei cerchi a fianco, in ordine crescente =
in senso=20
anti-orario.</P>
<P>Dopo sottraete 1 ai numeri dispari e aggiungete 1 ai numeri pari.</P>
<P>Dopo aver fatto queste operazioni scegliete 3 cerchi consecutivi in =
modo che=20
la somma dei numeri in essi contenuti sia la pi=F9 grande possibile.</P>
<P>Qual =E8 tale somma?</P>
<P><IMG height=3D200 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit20.gif" =
width=3D205></P>
<P>(Olimpiada Brasileira de Matematica, 2003)</P>
<P><B>L'ultimo vagone</B></P>
<P>Qual =E8 il numero dell'ultimo vagone del trenino del kangourou?</P>
<P><IMG height=3D54 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit21.gif" =
width=3D335></P>
<P>(Kangourou 2001)</P>
<P>[66]</P>
<P><B>Una sequenza autodescrittiva</B></P>
<P>La sequenza di cifre "22", (due 2) descrive se stessa perch=E9 =E8 =
formata=20
esattamente da due "2".</P>
<P>Analogamente la sequenza "31 12 33 15" descrive se stessa perch=E9 =
=E8 formata=20
esattamente da tre "1", un "2", tre "3", un "5".</P>
<P>Quale delle seguenti sequenze non descrive se stessa?</P>
<P>A) 21 32 23 16</P>
<P>B) 31 12 33 18</P>
<P>C) 31 22 33 17 19</P>
<P>D) 21 32 33 24 15</P>
<P>E) 41 32 23 24 15 16 18</P>
<P>(Olimpiada Brasileira de Matematica, 2003)</P>
<P>[D]</P>
<HR>

<H1><B>Frazioni</B></H1>
<P><B>Una frazione di giorno</B></P>
<P>Quanti minuti dura la met=E0 di un terzo di un quarto di un =
giorno?</P>
<P>Kangourou, 2005</P>
<P>[60]</P>
<P><B>L'aumento del quadrato</B></P>
<P>Quale numero aumenta del 500% quando se ne fa il quadrato?</P>
<P>Kangourou, 2006</P>
<P>[6]</P>
<P><B>La bottiglietta</B></P>
<P>Una bottiglietta da 1/3 di litro =E8 piena per 3/4. Quanti centilitri =
di=20
liquido conterr=E0 dopo averne versato in un bicchiere 20 =
centilitri?</P>
<P>Kangourou, 2006</P>
<P>[5]</P>
<HR>

<H1><B>Rapporti, percentuali e proporzioni</B></H1>
<P><B>Tessera con sconto</B></P>
<P>In un bar un caff=E8 costa 80 =80 cent. La tessera per 15 caff=E8 10 =
=80. Qual =E8 lo=20
sconto applicato?</P>
<P>(Gran premio di Matematica Applicata, 2002)</P>
<P>[circa il 16,7%]</P>
<P><B>Tre per due</B></P>
<P>Un supermercato applica l'offerta 3 x 2 (compri 3 paghi 2). Qual =E8 =
lo sconto=20
applicato?</P>
<P>(Gran premio di Matematica Applicata, 2002)</P>
<P>[circa il 33%]</P>
<P><B>Le macchine difettose</B></P>
<P>Un'azienda meccanica produce cuscinetti a sfera utilizzando due =
diverse=20
macchine.</P>
<P>La prima =E8 in grado di fornire 10.000 pezzi in 4 ore, ma il 6% di =
questi sono=20
difettosi.</P>
<P>La seconda 6.000 pezzi in 2 ore con il 10% di pezzi difettosi.</P>
<P>In un giorno (8 ore) quanti sono i pezzi difettosi prodotti dalle due =

macchine?</P>
<P>(Gran premio di Matematica Applicata, 2003)</P>
<P>[3600 =3D 8,18%]</P>
<P><B>Il vitalizio dei vincitori</B></P>
<P>Anche a Mathemandia si fanno Gare Matematiche e lo scorso anno hanno =
vinto=20
ex-equo due studenti: Teo Rema e Assi Omar. Il premio consiste in un =
vitalizio=20
annuo di 1.000 =80 .</P>
<P>Il vitalizio di Teo viene pagato a fine anno e poi si incrementa di =
20 =80=20
all'anno, mentre quello di Omar viene pagato a fine semestre (500 =80) e =
poi si=20
incrementa di 5 =80 semestrali. Allora ogni anno (scegli l'affermazione=20
esatta):</P>
<P>A) Teo incassa 10 =80 pi=F9 di Omar</P>
<P>B) Omar incassa 10 =80 pi=F9 di Teo</P>
<P>C) Teo incassa 5 =80 pi=F9 di Omar</P>
<P>D) Omar incassa 5 =80 pi=F9 di Teo</P>
<P>(Gran premio di Matematica Applicata, 2003)</P>
<P>[D]</P>
<P><B>Il gattile</B></P>
<P>In un gattile vi sono 160 gatti; il 20% dei gatti =E8 nero e il 25% =
dei gatti=20
ha gli occhi gialli. Quale di queste affermazioni =E8 sicuramente =
falsa:</P>
<P>A) 30 gatti sono neri con gli occhi gialli</P>
<P>B) 35 gatti hanno gli occhi gialli e sono neri</P>
<P>C) nessun gatto nero ha gli occhi gialli</P>
<P>D) tutti i gatti neri hanno gli occhi gialli</P>
<P>(Gran premio di Matematica Applicata, 2005)</P>
<P>[B]</P>
<P><B>L'investimento triennale</B></P>
<P>Tizio deve investire 50.000 euro in 3 anni e ha 4 opportunit=E0:</P>
<P>- l'investimento A frutta il 2% all'anno per 3 anni</P>
<P>- l'investimento B frutta l'1,1% il primo anno, il 2% il secondo e il =
3% il=20
terzo</P>
<P>- l'investimento C frutta il 3,1% il primo anno, il 2% il secondo e =
l=921% il=20
terzo</P>
<P>- l'investimento D frutta 3.100 euro complessivi nei tre anni.</P>
<P>Sapendo che i rendimenti sono netti e supponendo che il tasso di =
inflazione=20
sia lo stesso nei quattro anni, determinate quale investimento =E8 pi=F9 =

conveniente.</P>
<P>(Gran premio di Matematica Applicata, 2006)</P>
<P>[B]</P>
<P><B>L'investimento di Gianni</B></P>
<P>Gianni vuole incrementare la somma C regalatagli dalla nonna.</P>
<P>Investendo al 5% annuo (interesse composto), dopo due anni quanto =
avr=E0=20
complessivamente?</P>
<P>(Gran premio di Matematica Applicata, 2003)</P>
<P>[1,1025 C]</P>
<P><B>Succo di frutta</B></P>
<P>Pierino ha versato in un recipiente 3 l d'acqua e 1 l di succo =
composto dal=20
20% di polpa e dall'80% d'acqua. Dopo aver mescolato bene, quale =
percentuale del=20
volume finale =E8 polpa?</P>
<P>(Olimpiada Brasileira de Matematica, 2005)</P>
<P>[5%]</P>
<P><B>Mele disidratate</B></P>
<P>Una mela contiene circa il 90% d'acqua e il 10% di materia solida. Un =

produttore mette 100 kg di mele a disidratare (seccare) fino a quando =
l'acqua =E8=20
il 60% del peso totale.</P>
<P>Quanti litri di acqua sono evaporati? (ricorda che un litro d'acqua =
pesa un=20
chilogrammo)</P>
<P>(Olimpiada Brasileira de Matematica, 2001)</P>
<P><I>Suggerimento: il peso della materia solida rimane costante e alla =
fine=20
rappresenta il 40% del peso totale.</I></P>
<P>[75]</P>
<HR>

<H1><B>Tabelle e grafici</B></H1>
<P><B>Il colore dei capelli</B></P>
<P>Un'indagine sul colore dei capelli di 1200 persone ha dato i =
risultati=20
riassunti nel grafico seguente.</P>
<P><IMG height=3D205 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit22.gif" =
width=3D219></P>
<P>Quante persone hanno i capelli neri?</P>
<P>(Olimpiada Brasileira de Matematica, 2004)</P>
<P>[360]</P>
<P><B>Analisi del fatturato</B></P>
<P>Il grafico qui sotto rappresenta il fatturato mensile di due imprese, =
A e B=20
nel secondo semestre del 2001.</P>
<P><IMG height=3D269 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit23.gif" =
width=3D399></P>
<P></P>
<P>Quale delle seguenti affermazioni =E8 esatta?</P>
<P>A) C'=E8 un mese in cui il fatturato dell'impresa A =E8 doppi del =
fatturato=20
dell'impresa B.</P>
<P>B) Nel mese di luglio la differenza fra i fatturati =E8 maggiore che =
negli=20
altri mesi.</P>
<P>C) L'impresa B fu quella che ebbe il maggior calo di fatturato in due =
mesi=20
successivi.</P>
<P>D) Nel semestre considerato, il fatturato totale di A fu maggiore di =
quello=20
di B.</P>
<P>E) La differenza fra i fatturati totali delle due imprese nel =
semestre supera=20
i 20 milioni di euro.</P>
<P>(Olimpiada Brasileira de Matematica, 2002)</P>
<P>[D]</P>
<P><B>Imbottigliamento del latte</B></P>
<P>Un allevatore di mucche imbottiglia ogni giorno tutto il latte =
prodotto dalla=20
sua fattoria.</P>
<P>Il latte appena munto =E8 convogliato in un barile di forma =
cilindrica e da qui=20
viene travasato nelle bottiglie, come vedi nella figura qui sotto.</P>
<P><IMG height=3D173 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit24.gif" =
width=3D244></P>
<P>La tabella seguente riporta la quantit=E0 di bottiglie riempite e il=20
corrispondente livello del latte nel barile.</P>
<TABLE cellSpacing=3D1 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter>Numero di bottiglie riempite</TD>
    <TD vAlign=3Dcenter>0</TD>
    <TD vAlign=3Dcenter>200</TD>
    <TD vAlign=3Dcenter>400</TD>
    <TD vAlign=3Dcenter>600</TD>
    <TD vAlign=3Dcenter>?......</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>Livello del latte nel barile (cm)</TD>
    <TD vAlign=3Dcenter>210</TD>
    <TD vAlign=3Dcenter>170</TD>
    <TD vAlign=3Dcenter>130</TD>
    <TD vAlign=3Dcenter>90</TD>
    <TD vAlign=3Dcenter>0</TD></TR></TBODY></TABLE>
<P>Quante bottiglie saranno riempite quando il barile sar=E0 vuoto?</P>
<P>Scrivi il dato mancante nella tabella.</P>
<P>(Olimpiada Brasileira de Matematica, 2002)</P>
<P>[1050]</P>
<P><B>Consumo d'acqua in una famiglia</B></P>
<P>Nella tabella seguente =E8 riportato il consumo d'acqua di una =
famiglia nei=20
primi cinque mesi del 2003.</P>
<TABLE cellSpacing=3D1 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter><B>Mesi</B></TD>
    <TD vAlign=3Dcenter><B>Consumo (m<SUP>3</SUP>)</B></TD></TR>
  <TR>
    <TD vAlign=3Dcenter>Gennaio</TD>
    <TD vAlign=3Dcenter>12,5</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>Febbraio</TD>
    <TD vAlign=3Dcenter>13,8</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>Marzo</TD>
    <TD vAlign=3Dcenter>13,7</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>Aprile</TD>
    <TD vAlign=3Dcenter>11,4</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>Maggio</TD>
    <TD vAlign=3Dcenter>12,1</TD></TR></TBODY></TABLE>
<P>Qual =E8 stato il consumo mensile medio di questa famiglia nel =
periodo=20
considerato?</P>
<P>(Olimpiada Brasileira de Matematica, 2003)</P>
<P>[12,7 m<SUP>3</SUP>]</P>
<HR>

<H1><B>Combinatoria (labirinti, combinazioni etc)</B></H1>
<P><B>Sudoku</B></P>
<P><IMG height=3D217 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit25.gif" =
width=3D217></P>
<P>Dovete sistemare in ogni casella vuota di questa tabella una di =
queste=20
quattro lettere: una A o una B o una C o una D, rispettando le regole=20
seguenti:</P>
<P>Ci devono essere le quattro lettere, differenti</P>
<P>- in ogni riga</P>
<P>- in ogni colonna</P>
<P>- in ognuno dei quattro quadrati di quattro caselle (bianchi o =
grigi)</P>
<P>Spiegate come avete fatto a riempire le caselle.</P>
<P>(Kangourou 2006)</P>
<P><IMG height=3D155 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit26.gif" =
width=3D155></P>
<P><B>Il labirinto</B></P>
<P><IMG height=3D117 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit27.gif" =
width=3D180></P>
<P>Trova un percorso per attraversare (solo in orizzontale o in =
verticale)=20
questo labirinto, sapendo che:</P>
<P>- non puoi passare pi=F9 volte per una stessa casella;</P>
<P>- la somma dei numeri delle caselle attraversate deve essere uguale a =
13.</P>
<P>(Pristem 2004)</P>
<P>[3, 2, 0, 2, 1, 4, 0, 1]</P>
<P><B>Aiutiamo Braccio di Ferro!</B></P>
<P>Dobbiamo soccorrere =93Braccio di Ferro=94 inviandogli immediatamente =
degli=20
spinaci. Ne abbiamo 8 scatolette, ma una =E8 stata avvelenata da Brutus. =
Le=20
scatole sono esattamente uguali, ma, per fortuna sappiamo che quella =
avvelenata=20
pesa poco di pi=F9 delle altre. Disponendo di una bilancia a bracci =
uguali, quante=20
pesate sono necessarie, come minimo, per individuare con certezza la =
scatoletta=20
avvelenata?</P>
<P>(Gran premio di Matematica Applicata, 2006)</P>
<P>[Bastano due pesate. Mettendo sui piatti della bilancia tre =
scatolette per=20
parte, se la bilancia =E8 in equilibrio la scatola avvelenata =E8 una =
delle altre=20
due, che si pu=F2 individuare con una seconda pesata; se la bilancia non =
=E8 in=20
equilibrio, quella avvelenata appartiene alla terna pi=F9 pesante, =
quindi pesando=20
due delle tre scatolette, si individua quella avvelenata (infatti =E8 la =
pi=F9=20
pesante, oppure se sono in equilibrio quella non pesata)]</P>
<P><B>Ordinare le carte</B></P>
<P>Cinque carte, numerate da 1 a 5, sono allineate sul tavolo =
nell=92ordine 5, 1,=20
4, 3, 2.</P>
<P><IMG alt=3D"" =
src=3D"http://utenti.quipo.it/base5/scuola/olimparit38.gif"></P>
<P>Devi mettere le carte nell=92ordine 1, 2, 3, 4, 5 eseguendo solo =
mosse del=20
seguente tipo: ogni mossa consiste nello scambiare fra loro due carte. =
Qual =E8 il=20
minimo numero di mosse che ti consente di realizzare l=92operazione?</P>
<P>(Kangourou, 2005)</P>
<P>[3]</P>
<P><B>Tricolorazione</B></P>
<P><IMG height=3D148 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit28.gif" =
width=3D167></P>
<P>In questa griglia triangolare, si devono colorare certi triangoli =
rispettando=20
le condizioni seguenti:</P>
<P>- un triangolo con un numero non pu=F2 essere colorato</P>
<P>- il numero scritto in un triangolo indica il numero dei triangoli =
vicini da=20
colorare</P>
<P>Esempio:</P>
<P><IMG height=3D64 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit29.gif" =
width=3D295></P>
<P>(Pristem 2002)</P>
<P>[<IMG height=3D133 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit30.gif" =
width=3D152>]</P>
<P><B>La sfida della lettura</B></P>
<P>Dodici allievi devono leggere quattro libri: "La galassia =
dimenticata", "Il=20
piccolo fantasma", "L=92elefante verde", "La giungla". In classe ci sono =
due copie=20
di ciascun libro e il maestro d=E0 in prestito un libro alla settimana. =
Quante=20
settimane come minimo servono perch=E9 ogni allievo abbia letto tutti e =
quattro i=20
libri?</P>
<P>(Matematica senza frontiere, 2005)</P>
<P>[6]</P>
<P><B>Il pannello luminoso</B></P>
<P>Un pannello luminoso =E8 composto da 30 quadri (disposti come in =
figura) che=20
possono essere accesi o spenti. I numeri scritti in ogni quadro indicano =
il=20
numero di quadri accesi che lo circondano (nel senso che hanno in comune =
un lato=20
o anche solo un vertice).</P>
<P>Sul foglio risposte, ricopiate i numeri nei quadri accesi.</P>
<TABLE cellSpacing=3D1 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>0</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>1</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>2</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>2</P></TD></TR>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>1</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>2</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>4</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD></TR>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>2</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>2</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>5</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD></TR>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>5</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>6</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>4</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>4</P></TD></TR>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>2</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>4</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>4</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD></TR>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>2</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>4</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter>3</P></TD></TR></TBODY></TABLE>
<P>(Pristem 2004)</P>
<P><I>Suggerimento: questo esercizio assomiglia al noto gioco "Campo =
minato"=20
fornito con il sistema operativo Windows.</I></P>
<TABLE cellSpacing=3D1 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>2</B></P></TD>
    <TD vAlign=3Dcenter></TD></TR>
  <TR>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>4</B></P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>3</B></P></TD>
    <TD vAlign=3Dcenter></TD></TR>
  <TR>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>2</B></P></TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>3</B></P></TD>
    <TD vAlign=3Dcenter></TD></TR>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>3</B></P></TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>4</B></P></TD>
    <TD vAlign=3Dcenter></TD></TR>
  <TR>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>2</B></P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>3</B></P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>4</B></P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>4</B></P></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>3</B></P></TD></TR>
  <TR>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>
      <P align=3Dcenter><B>3</B></P></TD>
    <TD vAlign=3Dcenter></TD></TR></TBODY></TABLE>
<P><B>Nove per lato</B></P>
<P>In un contenitore come quello mostrato in figura, Andrea dispone le =
proprie=20
biglie in modo che in ogni scompartimento ce ne siano 3, cio=E8 9 per =
lato.</P>
<P>Giocando, Andrea perde 4 delle sue biglie, ma riesce ugualmente a =
distribuire=20
le rimanenti nel contenitore, in modo che:</P>
<P>- nessuno scompartimento rimanga vuoto;</P>
<P>- negli scompartimenti centrali di ogni lato ci sia lo stesso numero =
di=20
biglie;</P>
<P>- su ogni lato ci siano ancora 9 biglie.</P>
<P>In quali modi Andrea pu=F2 aver disposto le sue biglie?</P>
<P>Dopo qualche giorno Andrea ritrova le 4 biglie che aveva perso e ne =
riceve 4=20
nuove in regalo.</P>
<P>Quali possibilit=E0 ha ancora Andrea di sistemarle in modo che siano =
sempre=20
rispettate le precedenti condizioni?</P>
<P><IMG height=3D214 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit31.gif" =
width=3D477></P>
<P>(Rally Matematico Transalpino, 1998)</P>
<P>[Con 20 biglie:</P>
<TABLE cellSpacing=3D0 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter>4</TD>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>4</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>1</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>4</TD>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>4</TD></TR></TBODY></TABLE>
<P></P>
<TABLE cellSpacing=3D0 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter>5</TD>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>3</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>1</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>3</TD>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>5</TD></TR></TBODY></TABLE>
<P></P>
<TABLE cellSpacing=3D0 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter>7</TD>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>1</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>1</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>7</TD></TR></TBODY></TABLE>
<P>Con 28 biglie:</P>
<TABLE cellSpacing=3D0 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter>2</TD>
    <TD vAlign=3Dcenter>5</TD>
    <TD vAlign=3Dcenter>2</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>5</TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>5</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>2</TD>
    <TD vAlign=3Dcenter>5</TD>
    <TD vAlign=3Dcenter>2</TD></TR></TBODY></TABLE>
<P></P>
<TABLE cellSpacing=3D0 cellPadding=3D5 border=3D1>
  <TBODY>
  <TR>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>5</TD>
    <TD vAlign=3Dcenter>3</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>5</TD>
    <TD vAlign=3Dcenter></TD>
    <TD vAlign=3Dcenter>5</TD></TR>
  <TR>
    <TD vAlign=3Dcenter>1</TD>
    <TD vAlign=3Dcenter>5</TD>
    <TD vAlign=3Dcenter>3</TD></TR></TBODY></TABLE>
<HR>

<H1><B>Ottimizzazione e strategie particolari</B></H1>
<P><B>Lucchetti e monete</B></P>
<P>In un baule ci sono 5 valigie, ciascuna delle quali contiene 3 =
cassettine, in=20
ognuna delle quali ci sono 10 monete d'oro. Baule, valigie e cassettine =
sono=20
tutti chiusi con lucchetti.</P>
<P>Qual =E8 il minimo numero di lucchetti che ti basta aprire per avere =
50 monete=20
d'oro?<BR>(Kangourou, 2005)</P>
<P>[8 lucchetti (un baule, due valige e cinque cassettine)]</P>
<P><B>Il fabbro veloce</B></P>
<P>Un fabbro ha 10 pezzi di catena di 3 anelli ciascuno, come mostrato =
nella=20
figura qui sotto.</P>
<P><IMG height=3D96 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit32.gif" =
width=3D279></P>
<P>Egli vuole costruire una unica catena di 30 anelli.</P>
<P>Per aprire e poi saldare un anello il fabbro impiega 5 minuti.</P>
<P>Quanti minuti impiegher=E0 come minimo per costruire la catena?</P>
<P>(Olimpiada Brasileira de Matematica, 2001)</P>
<P><B>I salti del canguro Jumpy</B></P>
<P>Quando il canguro Jumpy si spinge con la gamba sinistra salta 2 =
metri; quando=20
si spinge con la destra salta 4 metri; infine quando si spinge con =
entrambe le=20
gambe, salta 7 metri. Qual =E8 il minimo numero di salti sufficienti a =
Jumpy per=20
coprire una distanza di 997 metri esatti?</P>
<P>(Kangourou, 2006)</P>
<P>[142 + 1 =3D 143]</P>
<HR>

<H1><B>Probabilit=E0 e statistica</B></H1>
<P><B>I barattoli di caramelle</B></P>
<P>Nonna Matilde mette in un barattolo 6 caramelle all=92arancia e 10 al =

limone.</P>
<P>In un secondo barattolo mette 8 caramelle all=92arancia e 14 al =
limone.</P>
<P>Le caramelle hanno la stessa forma e sono incartate nello stesso =
modo.</P>
<P>La nonna sa che a Giulio non piacciono le caramelle al</P>
<P>limone e quindi gli dice: "Puoi prendere una caramella. Ti lascio =
scegliere=20
il barattolo nel quale puoi infilare la mano, senza guardare =
dentro."</P>
<P><IMG height=3D196 alt=3D""=20
src=3D"http://utenti.quipo.it/base5/scuola/olimparit33.gif" =
width=3D240></P>
<P>Giulio ci pensa un po=92 e sceglie infine il barattolo che, secondo =
lui, gli=20
offre pi=F9 possibilit=E0 di prendere una caramella all=92arancia.</P>
<P>Al posto di Giulio quale barattolo sceglieresti? [il primo]</P>
<P>(Rally Matematico Transalpino, 2006)</P>
<P>[Il primo]</P>
<P><B>La moneta falsa</B></P>
<P>Un sacchetto contiene 5 monete d'oro identiche, ma una =E8 falsa e ha =
un peso=20
leggermente diverso da quelle vere. Non sappiamo se tale peso =E8 =
maggiore o=20
minore, ma sappiamo che le vere hanno esattamente lo stesso peso. Usando =
una=20
bilancia di precisione a bracci, qual =E8 la probabilit=E0 di =
individuare due monete=20
vere con una sola pesata?</P>
<P>A) Il 50%</P>
<P>B) Il 64%</P>
<P>C) L' 80%</P>
<P>D) Il 100%</P>
<P>(Gran premio di Matematica Applicata, 2003)</P>
<P>[E' certo che individuiamo (almeno) due monete vere con una sola =
pesata:=20
infatti, mettendo una moneta su ogni piatto della bilancia, se si ha=20
l'equilibrio esse sono vere, mentre, se non sono in equilibrio, una =E8 =
falsa e=20
quindi le altre tre sono vere! La risposta =E8 quindi D.]</P>
<P><B>I cacciatori fallibili</B></P>
<P>Due cacciatori vedono una lepre e sparano simultaneamente. Sappiamo =
che i due=20
cacciatori colpiscono la preda una volta su quattro, qual =E8 la =
probabilit=E0 che=20
almeno uno colpisca la lepre ?</P>
<P>A) 40%</P>
<P>B) 43,75%</P>
<P>C) 50%</P>
<P>D) 56,25%</P>
<P>Commento:</P>
<P>(Gran premio di Matematica Applicata, 2003)</P>
<P>[La probabilit=E0 che uno colpisca la lepre =E8 1/4 =3D 25%, che la =
colpiscano=20
entrambi =E8 1/16 =3D 6,25%; quindi che almeno uno la colpisca =E8 25% + =
25% =96 6,25% =3D=20
43,75%]</P>
<P><EM>Data creazione: settembre 2006</EM></P>
<P><EM>Ultimo aggiornamento settembre 2006:</EM></P>
<P>xhtml 1.1</P>
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<P style=3D"TEXT-ALIGN: center">Sito Web realizzato da =
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	BORDER-RIGHT: #808080 1px solid; PADDING-RIGHT: 3px; BORDER-TOP: =
#808080 1px solid; PADDING-LEFT: 3px; PADDING-BOTTOM: 3px; BORDER-LEFT: =
#808080 1px solid; PADDING-TOP: 3px; BORDER-BOTTOM: #808080 1px solid
}
UL {
	PADDING-RIGHT: 0em; MARGIN-TOP: 0.5em; PADDING-LEFT: 0em; =
MARGIN-BOTTOM: 0.5em; PADDING-BOTTOM: 0em; MARGIN-LEFT: 0em; =
PADDING-TOP: 0em
}
OL {
	PADDING-RIGHT: 0em; MARGIN-TOP: 0.5em; PADDING-LEFT: 0em; =
MARGIN-BOTTOM: 0.5em; PADDING-BOTTOM: 0em; MARGIN-LEFT: 0em; =
PADDING-TOP: 0em
}
LI {
	PADDING-RIGHT: 0em; MARGIN-TOP: 0.5em; PADDING-LEFT: 0em; =
MARGIN-BOTTOM: 0.5em; PADDING-BOTTOM: 0em; MARGIN-LEFT: 1.5em; =
PADDING-TOP: 0em; TEXT-ALIGN: left
}
DL {
	PADDING-RIGHT: 0em; MARGIN-TOP: 0.5em; PADDING-LEFT: 0em; =
MARGIN-BOTTOM: 0.5em; PADDING-BOTTOM: 0em; MARGIN-LEFT: 0em; =
PADDING-TOP: 0em
}
DT {
	PADDING-RIGHT: 0em; MARGIN-TOP: 0.3em; PADDING-LEFT: 0em; FONT-WEIGHT: =
bold; MARGIN-BOTTOM: 0.3em; PADDING-BOTTOM: 0em; PADDING-TOP: 0em
}
DD {
	PADDING-RIGHT: 0em; MARGIN-TOP: 0.3em; PADDING-LEFT: 0em; =
MARGIN-BOTTOM: 0.3em; PADDING-BOTTOM: 0em; MARGIN-LEFT: 0.5em; =
PADDING-TOP: 0em
}
DL {
	MARGIN-TOP: 0.3em; MARGIN-BOTTOM: 0.3em
}
STRONG {
	FONT-WEIGHT: bold
}
EM {
	FONT-STYLE: italic
}
CODE {
	FONT-FAMILY: Courier New, Courier, monospace
}
INS {
	BACKGROUND-COLOR: yellow; TEXT-DECORATION: underline
}
DEL {
	TEXT-DECORATION: line-through
}
A[href] {
	COLOR: blue; TEXT-DECORATION: underline
}
HR {
	COLOR: #999999; HEIGHT: 1px
}

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