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CMLIKaJlOBa E. B.
C21 .D:onOJIHHTeJILHlde rJIaBld no MaTeMaTHKe JIH yqHXe.H
5 KJIaeea. CII6: CMMO IIpeee, 2009. - 48 e.
IIoco6He HBJIHeTCH ,lJ;OnOJIHeHHeM K yqe6HHKY MaTeMaTHKH ,lJ;JIH
aID;HXCH 5 KJIaCca. OHO CO,lJ;epmHT BHenporpaMMHLIH TeopeTHe-
CKHH MaTepH8.JI H 3aH no CJIe,lJ;YIOID;HM TeMaM:
1. CHcTeMLI CHCJIeHHH.
2. 3JIeMeHTbI TeopHH MHomeCTB.
3. 3JIeMeHTbI TeopHH rpacpoB.
4. KOM6HHaTopHKa.
5. IIpHHHn HpHXJIe.
BecL MaTepH8.JI H3JIOmeH B ,lJ;OcTynHoM H HHTepeCHOM BH,lJ;e. IIo-
co6He MomHO HCnOJIL30BaTL B KJIaCCHOH H BHeKJIaCCHOH pa6oTe, Ha
3aHHTHHX MaTeMaTHecKoro KpymKa, ,lJ;JIH caMOCTOHTeJILHOrO H3Y-
eHHH MaTeMaTHKH.
ISBN 5-7704-0055-2
@ CMIdKaJlOBa E. B., 2001 r., 2005 r. (HcnpaB.)
@ ryJIhKOBCKHH H. H., o4>opMJIeHHe 06JIOmKH,
2001 r.
@ CMMO llpecc, 2001 r.
Pe.n;aKTopld K. 4>.-M. H. 30Auna H. K., K. 4>.-M. H. qepICacosa T. X.
Xy.n;omecTBeHHIdH pe.n;aKTOp COAosbesa H. 4.
.D;HpeKTop H3.n;aTeJIhCTBa M op03,Osa H. c.
Ma.n;aTeJIbCTBO «CMMO llpecc»,
CaHKT-lleTep6ypr,YJI. Ce.n;OBa,.n;. 97,K. 3,JIHT.A
TenI<I>aKC (812) 595-94-76, TeJI. (911) 290-90-26 (MTC), (961) 811-17-49 (BHJurliH),
e-mail: smiopress@mail.ru, http://www.smio.ru
llpHHHMaeM aaKaabI «KHHI'a-IIOI'roID)
llo.n;IIHcaHO B IIeqaTb 5 MapTa 2009 r. lleqaTb o<pceTHaSl.
YCJI.-IIeq. JI. 0,8. <I>opMaT 84x 108 1 /32. THpam 2000. 3aKaa '30 .
OTIIeqaTaHO B OAO «lleTpOu;eHTp» Oll llymKHHcKaSl THIIOrpa<pHSI
r. llymKHH, YJI. Cpe.n;HSlSl, .n;. 3/8.

nPEAHCAOBHE
.D:aHHoe noe06Me .HBJI.HeTe.H OnOJIHeHHeM K yqe6HHKY Ma-
TeMaTHKH. OHO eOepmHT BHenpOrpaMMHLIH TeOpeTHqeeKHH
MaTepRaJI H 3aaqH no eJIeyk>HM TeMaM:
1. CHeTeMLI eqHeJIeHH.H.
2. MHomeeTBa.
3. rpacpLI.
4. KOM6RHaTopHKa.
5. IIpHHHn .D:HpHXJIe.
.D:onOJIHHTeJILHlde rJIaBLI MomHO HenOJIL30BaTL Ha 3aH.H-
TH.HX MaTeMaTHqeeKOrO KpymKa, a TaKme JI.H HHHBHyaJIL-
HOH pa60TLI e yqaHMHe.H. .D:onOJIHHTeJILHLIe rJIaBId nOMorYT
ytlHTeJIIO MaTeMaTHKH B peaJIR3aHH OeHOBHLIX npHHHnOB
pa3BRBalOero 06yqeHH.H: .
. 06yqeHHe Ha BLIeOKOM ypOBHe TPYHoeTeH;
. Beya.H pOJIL TeOpeTHqeeKHX 3HaHHH;
. npOBHmeHHe Bnepe 6LIeTpldM TeMnOM;
. e03HaTeJILHOe yqaeTHe IlIKOJILHHKOB B yqe6HOM
npoeeee .
Bee 3MaqH nOe06H.H 6ldJIH anp06HpoBaHId Ha ypOKax Ma-
TeMaTHKH B 5-6 KJIaeeaX rHMHa3HH N2 52 IIpHMopeKoro paH-
OHa CaHKT-IIeTep6ypra B 1997-2005 rr. Ko BeeM 3aaqaM
eeTL OTBeTId, eTpaHH1d e KOTOpldMH MorYT 6LITL OTpe3aHLI
no yeMOTpeHHIO yqHTeJIH.
ABTOp BLIpWKaeT HeKpeHHlO1O npH3HaTeJILHOeTb oeHTY Ka-
cpepLI MeTOHKH o6yqeHR.H MaTeMaTHKe prIIY HM. A. 11. rep-
eHa T. A. Topy6apoBoH 3a p.H eHHLIX 38.MeqaHHH H onOJI-
HeHRH, KaH. cpH3.-MaT. HaYK, o. BITY T. x. qepKaeoBoH,
a TaKme C. A. OeMexHHY H IOpHIO CMLIKaJIOBY 3a nOMOL B
nO)J;rOTOBKe pYKonReH.
3

rna6a I
CHCTEMbICHCAEHHH
CucmeMa C1£ucnenuJl, - 3TO CHCTeMa 3anHCH 'tIHCeJI.
B HacToHee BpeMH nO'tITH Bce HapobI nOJIb3YIOTCH
oecJl,mu 1£nou n03ulfuonnou cucmeMOu C1£ucnenuJl,.
B 3TOM CHCTeMe Bce 'tIHCJIa 3aIIHCbIBaIOTCH C IIOMObIO
eCHTHH<PP: 0,1,2,3,4,5,6,7,8,9. qHCJIO 10-ocHo-
BaHHe CHCTeMbI C'tIHCJIeHHH. C"tIeT HeT eCHTKaMH, COT-
HHMH, TbICH'tIaMH H T. .
1 eCHTOK = 1 0 eHHH;
1 COTHH = 10 eCHTKOB;
1 TbICH'tIa = 10 COTeH.
EHHHa Kamoro CJIe,ZJy1Oero pa3pa B 10 pa3 60JIb-
IIle eHHHbI npebIyero pa3pHa. 3Ha'tIeHHe KamoM
H<PPbI 3aBHCHT OT ee MeCTa (II03HHH) B 3anHCH 'tIHCJIa.
HanpHMep, 'tIHCJIO, COCTOHee H3 BYX COTeH, Tpex
eCHTKOB H 'tIeTbIpex eHHH, MbI 3anHCbIBaeM TaK:
234 = 2 . 10 . 10 + 3 . 10 + 4.
llCn0JIb30Bamre tlliCJIa 10 KaK OCHOBamrH CHCTeMbI CtfilC-
JIeHHH CBH3aHO C TeM, 'tITO Y JIIOeM Ha PYKax 10 narrbeB,
KOTopbIe yo6Hee Bcero 6bIJIO HCnOJIb30BaTb npH C'tIeTe.
COBpeMeHHaH eCHTH'tIHaH n03HHOHH8JI CHCTeMa C'tIHC-
JIeHHH B03HHKJIa B VI BeKe B llHHH. llHHHcKaH CHCTeMa
C"tIHCJIeHHH npHIIlJIa CHa'tIarra B apa6cKHe CTpaHbI, 3aTeM B
XVI BeKe B 3anHYIO EBpony H B XVII BeKe B POCCHIO.
OCHoBaHHeM CHCTeMbI C'tIHCJIeHHH MomeT 6bITb JII060e
'tIHCJIO, 60JIbIIlee eHHHbI. PaHbIIle HeKOTopbIe HapobI
npHMeHHJIH CHCTeMbI C'tIHCJIeHHH C OCHOBaHHHMH: 2, 5,
8, 12, 16, 20, 40, 60 H p.
HeeCHTH'tIHbIe CHCTeMbI C'tIHCJIeHHH COXpaHHJIHCb o
CHX nop. TaR, HanpHMep, MbI eJIHM ro Ha 12 MeCHeB,
CYTRH - Ha 24 'tIaca, 1 'tIac - Ha 60 MHHYT, 1 MHHYTY - Ha
60 ceRYH.
4

CWCTel\t1bl C"fI>tCl\eHHR
IIpH paCqeTaX Ha SBM IlIHpOKO rrpHMeHHIOTCH CHCTe-
MbI CqHCJIeHHH C OCHOBaHHHMH 2, 8 H 16.
QTo6bI pa3JIHqHTb qHCJIa, 3arrHcaHHbIe B pa3HbIX CHC-
TeMax CqHCJIeHHH, BHH3Y crrpaBa YKa3bIBaIOT OCHOBaHHe
CHCTeMbI CqHCJIeHHH. eCHTHqHOe OCHOBaHHe 06bIqHO He
YKa3bIBaeTCH, rrOCKOJIbKY Bce MbI rrOJIb3yeMcH eCHTHq-
HOM CHCTeMOM CqHCJIeHHH.
TaK, HarrpHMep: 5 = 101 2 = 123 = 114 = 105.
ABOHIIHaA CHCTeMa CIIHCAeHHA
B BOHqHOM CHCTeMe CqHCJIeHHH TOJIbKO Be H<PPbI 0
H 1. C HX rrOMOIIJ;bIO MomHO 3arrHcaTb JII060e qHCJIO. 3a
OCHOBaHHe CHCTeMbI CqHCJIeHHH rrpHHHMaeTCH qHCJIO 2.
CqeT BeeTCH BoMKaMH. QHCJIO 2 CqHTaeTCH eHHHeM
2-ro pa3pHa H 3arrHCbIBaeTCH B BHe 10, qHTaeTCH
«OHH-HYJIb». KajKaH eHHHa CJIeYIOIIJ;ero pa3pHa B
2 pa3a 60JIbllIe rrpebIYIIJ;eM, T. e. 3TH eHHHbI COCTaB-
JIHIOT rrOCJIeOBaTeJIbHOCTb qHCeJI 2, 4, 8, 16, 32, ...
1=123=112 5=101 2 7=1112 9=10012
2 = 102 4 = 1002 6 = 110 2 8 = 1000 2 10 = 1010 2
· Ha OeCJ£mU1£uou.8 080U1£UYlO
JIH Toro qTo6bI qHCJIO, 3arrHCaHHOe B eCHTHqHOM CHC-
TeMe CqHCJIeHHH, 3anHcaTb B BOHqHOH CHCTeMe CqHCJIeHHH,
ero eJIHT rrOCJIeOBaTeJIbHO Ha 2 H 3anHCbIBaIOT rrOJIyqaIO-
IIJ;HeCH OCTaTKH 0 H 1 B rropHKe OT rrOCJIeHerO K rrepBOMY,
rrepBaH H<ppa qHCJIa - rrOCJIeHee qaCTHOe, HanpHMep:
43 : 2 = 21 (OCT. 1);
21 : 2 = 10 (OCT. 1);
10 : 2 = 5 (OCT. 0);
5 : 2 = 2 (OCT. 1);
2 : 2 = 1 (OCT. 0).
Om6em: 43 = 101011 2 .
IlpoBepbTe caMOCTOHTeJIbHO 3arrHCb: 35 = 1000112.
5

rAasa I
. Ha 080Ul£nOU 8 OecJI,mul£nyJO
QTo6bI rrepeBeCTH 'tIHCJIO 1010112 B eCHTH'tIHYIO CHC-
TeMY C'tIHCJIeHHH, Hao rrpecTaBHTb 3TO 'tIHCJIO B cJIey-
lOeM BHe:
1 . 25 + 0 . 2 4 + 1 · 2 3 + 0 . 2 2 + 1 . 2 1 + 1 . 2° =
= 32 + 0 + 8 + 0 + 2 + 1 = 43.
Om6em: 1010112 = 43.
IIpoBepbTe caMOCTOHTeJIbHO 3arrHCb: 11101 2 = 29.
. Cl£o3ICenue
B BOH'tIHOH CHCTeMe C'tIHCJIeHHH:
0+ 0 = 0; 0 + 1 = 1; 1 + 0 = 1; 1 + 1 = 10.
Ta6JIHa
CJIOmeHHJI:
+ 0 1
0 0 1
1 1 10
IIpHMep CJIOmeHHH B BOH'tIHOH CHCTeMe C'tIHCJIeHHH:
1010112 + 10101 2 = 10000002.
. Y Mn03ICenue
B BOH'tIHOH CHCTeMe C'tIHCJIeHHH:
o . 0 = 0; 0 . 1 = 0; 1 · 0 = 0; 1 · 1 = 1.
Ta6JIHa
YMHomeHHJI:
x 0 1
0 0 0
1 0 1
IIpHMep YMHomeHHH B BOH'tIHOH CHCTeMe C'tIHCJIeHHH:
1011 2 . 101 2 = 1101112.
nATepHIIHaJi CHCTeMa CIIHCAeHHJI
B ITHTepH'tIHOH CHCTeMe C'tIHCJIeHHH HCrrOJIb3YIOTCH
HcppbI 0, 1,2, 3 H 4. OCHoBaHHe CHCTeMbI C'tIHCJIeHHH-
'tIHCJIO 5. C'tIeT BeeTCH ITHTepKaMH. QHCJIO 5 C'tIHTaeTCH
6

CMCTeMbl C"IMCAeHMR
eAHHHeH 2-ro paapHAa H aanHChIBaeTCH B BHAe 10. Ka-
mAaH eAHHHa CJIeAYIOero paapHAa paBHa nHTH eAH-
HHaM npeAhIAyero paapHAa:
1 = 15 3 = 35 5 = 10 5 7 = 125
2=25 4=45 6=11 5 8=13 5
. Ha OeCJI,mU1f,nOU 8 nJl,mepU1f,nyJO
IIepeBoA qHCeJI B nHTepHqHYIO CHCTeMY CqHCJIeHHH Ha
AeCHTHqHOH npOHCxOAHT TaK me, KaK H B ABOHqHyJO,
TOJIhKO AeJIeHHeM Ha 5:
42 : 5 = 8 (OCT. 2);
8 : 5 = 1 (OCT. 3).
Om8em: 42 = 1325.
. Ha nJl,mepU1f,nOU 8 oeCJI,mU1f,nyJO
QTo6hI nepeBeCTH qHCJIO 234215 B AeCHTHqHYIO CHCTe-
MY CqHCJIeHHH, HaAO npeACTaBHTh 3TO qHCJIO B BHAe:
2 · 54 + 3 · 53 + 4 · 52 + 2 . 51 + 1 . 5° =
= 1250 + 375 + 100 + 10 + 1 = 1736.
Om8em: 23421 5 = 1 736.
. CJl,03ICenue
B nHTepHqHOH CHCTeMe CqHCJIeHHH:
1+1=2; 2+2=4; 3+3=11; 4+4=13.
1+2=3; 2+3=10; 3+4=12;
1+3=4; 2+4=11;
1+4=10;
9 = 145
10 = 205
Ta6JIHa
CJIOmeHHJI:
+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 10
2 2 3 4 10 11
3 3 4 10 11 12
4 4 10 11 12 13
7

rAasa I
IIpHMep CJIOmeHHSI B nSlTepHqHOH CHCTeMe CqHCJIe-
HHSI: 13045 + 3345 = 21435.
. Y Mn03ICenue
B nHTepHqHOH CHCTeMe CqHCJIeHHSI:
1 .1= 1;
1 · 2 = 2;
1 . 3 = 3;
1 . 4 = 4;
Ta6JIHa
YMHomeHHJI:
2 · 2 = 4;
2.3=11;
2 . 4 = 13.
,
3 . 3 = 14;
3 · 4 = 22.
,
4.4=31.
x 1 2 3 4
1 1 2. 3 4
2 2 4 11 13
3 3 11 14 22
4 4 13 22 31
IIpHMep YMHomeHHSI B nSlTepHqHOH CHCTeMe CqHCJIe-
HHH: 213 5 · 35 = 11445.
BOCbMepHqHaA CHCTeMa CqHCAeHHA
B BOCbMepHqHOH CHCTeMe CqHCJIeHHSI HCnOJIbayroT-
CSI H<PPbI 0, 1, 2, 3, 4, 5, 6, 7. OCHOBaHHe CHCTeMbI
CqHCJIeHHSI - qHCJIO 8. CqeT Be,l);eTCSI BOCbMepKaMH.
qHCJIO 8 CqHTaeTCSI e.n;HHHeH 2-ro paapSl.n;a H aanH-
CbIBaeTCSI B BH.n;e 10. Kam.n;aSl e.n;HHHa CJIe,ll;yroero
paapH.n;a paBHa BOCbMH e.n;HHHaM npe.n;bI.n;yero paa-
pSl.n;a.
1 = 18
2 = 28
3 = 3 8
4 =48
8
5 = 58
6= 68
7 = 7 8
8 = 108
9 = 118
10 = 128

. Ha OeCJ£mUl£nOU 8 80CbMepul£nylO
IIepeBo qHCeJI H3 eCHTHqHOH CHCTeMbI CqHCJIeHHH B
BOCbMepHqHYlO npOHCXOHT TaK me KaK H B npebIY-
He, HO eJIeHHeM Ha 8.
158 : 8 = 19 (OCT. 6);
19 : 8 = 2 (OCT. 3).
Om6em: 158 = 2368.
· Ha 8ocbMepul£nou l! OeCJ£mUl£nylO
QTo6hI nepeBeCTH qHCJIO 123 8 B eCHTHqHYlO cHcTeMY
CqHCJIeHHH, Hao npecTaBHTb ero B BHe:
210
1.8 +2. 8 +3.8 =64+16+3=83.
Om6em: 1233 = 83.
. Cl£o3ICenue
Ta6JIHa CJIOmeHHJI:
+ 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 10
2 2 3 4 5 6 7 10 11
.
3 3 4 5 6 7 10 11 12
4 4 5 6 7 10 11 12 13
5 5 6 7 10 11 12 13 14
6 6 7 10 11 12 13 14 15
7 7 10 11 12 13 14 15 16
IIpHMep CJIOmeHHH B BOCbMepHqHOH CHCTeMe CqHCJIe-
HHH: 12348 + 65448 = 100008.
9

rABSB I
. Y Mn03ICenUe
Ta6JIHa YMHOmeHHJI:
x 1 2 -3 4 5 6 7
1 1 2 3 4 5 6 7
2 2 4 6 10 12 14 16
3 3 6 11 14 17 22 25
4 4 10 14 20 24 30 34
5 5 12 17 24 31 36 43
6 6 14 22 30 36 44 52
7 7 16 25 34 43 52 61
IIpHMep YMHOmeHHH B BOCbMepHqHOH CHCTeMe CqHC-
JIeHHH: 378 . 528 = 24268.
38A8qll
1. 3aIIHillHTe HaTypaJIbHbIe qHCJIa OT 1 ,l);O 10 B TPOHq-
HOH CHCTeMe CqHCJIeHHH.
2. B KaKoH CHCTeMe CqHCJIeHHH qHCJIa 3aIIHCbIBaJOTCH:
a) 23 KaK 212;
6) 33 KaK 53;
B) 42 KaK 52?
3. KaKoe H3 qHCeJI 60JIbille:
a) 3 HJIH 38;
6) 145 HJIH 148;
B) 1112 HJIH 1118?
4. HaIIHillHTe HaH60JIbillee TpeX3HaqHOe qHCJIO B qeTBe-
PHqHOH CHCTeMe CqHCJIeHHH.
10

CMCTeMbl CMCJ\eHMR
5. HanHIIIHTe HaHMeHbIIIee 'tIeTbIpeX3Ha'tlHOe 'tIHCJIO B
,
CeMepH'tIHOH CHCTee C'tIHCJIeHHH.
6. HanHIIIHTe HaH60JIbIIIee nHTH3Ha'tlHOe 'tIHCJIO, B KO-
TOpOM Bce HcppbI pa3JIH'tIHbI B nHTepH'tIHOH CHCTeMe
C'tIHCJIeHHH.
7. HanHIIIHTe HaHMeHbIIIee IIIeCTH3Ha'tlHOe 'tIHCJIO, B
KOTOpOM Bce HcppbI pa3JIH'tIHbI B IIIeCTepH'tIHOH CHC-
TeMe C'tIHCJIeHHH.
8. 3anHIIIHTe:
a) 64 B BOH'tIHOH;
6) 321 B TpOH'tIHOH;
B) 2456 B nHTepH'tIHOH;
r) 3074 B BOCbMepH'tIHOH CHCTeMe C'tIHCJIeHHH.
9. 3anHIIIHTe 'tIHCJIO 130 B CHCTeMax C'tIHCJIeHHH C OCHO-
BaHHHMH OT 2 o 9.
10. 3anHIIIHTe B eCHTH'tIHOH CHCTeMe C'tIHCJIeHHH 'tIHCJIa:
a) 1001012; 6) 120013; B) 4032 5 ; r) 5048.
11. ECJIH 4 .4= 20, TO 'tIeMY paBHO npOH3BeeHHe 5 . 5
B TOH me CHCTeMe C'tIHCJIeHHH?
12. CocTaBbTe Ta6JIHY CJIOmeHHH OH03Ha'tlHbIX 'tIHCeJI
B CHCTeMe C'tIHCJIeHHH C OCHOBaHHeM 6.
13. IIpHYMaHTe npHMep Ha CJIOmeHHe BYX 'tIeTbIpex-
3Ha'tlHbIX 'tIHCeJI B 'tIeTBepH'tIHOH CHCTeMe C'tIHCJIeHHH
H BbInOJIHHTe CJIOmeHHe.
14. CocTaBbTe Ta6JIHY YMHomeHHH OH03Ha'tlHbIX 'tIH-
ceJI B CHCTeMe C'tIHCJIeHHH C OCHOBaHHeM 7.
15. B KaKOH CHCTeMe C'tIHCJIeHHH:
a)2+3=12;
6) 5 + 2 = 11;
B) 2 · 2 = 10;
r) 3 . 3 = 14?
11

rJ\aaif I
16. Bo CKOJIbKO pa3 YBeJIHqHTCH qHCJIO 325 6 , eCJIH npH-
nHcaTb cnpaBa:
a) OHH HYJIb;
6) Ba HYJIH?
17. Bo CKOJIbKO pa3 YMeHbillHTCH qHCJIO 212000 3 , eCJIH
oT6poCHTb:
a) OHH HYJIb;
6) TPH HYJIH?
18. YCTaHoBHTe, B KaKOM CHCTeMe CqHCJIeHHH BblnOJIHH-
u u
JIOCb Kamoe H3 CJIeYIOHX eHCTBHH:
a) 23 + 14 = 42;
6) 71 - 36 = 33;
B) 14 . 2 = 30;
r) 55 : 4 = 13.
19. qHCJIO, 3anHcaHHoe B eCHTHqHOM CHCTeMe CqHCJIe-
HHH, OKaHqHBaeTCH Ha 5. ByeT JIH 3TO qHCJIO KpaT-
HO 5, eCJIH ero 3anHcaTb B CHCTeMe CqHCJIeHHH C oc-
HOBaHHeM 8?
20. XOpOillO H3BeCTHbI npH3HaKH eJIHMOCTH HaTypaJIb-
HbIX qHCeJI Ha 2, 5, 10 B eCHTHqHOM CHCTeMe CqHC-
JIeHHH. BYYT JIH BepHbI 3TH npH3HaKH B HHbIX CHC-
TeMax CqHCJIeHHH, HanpHMep, C OCHOBaHHeM 6?
21. 3annillHTe B TPOHqHOM CHCTeMe CqHCJIeHHH ro CBO-
ero pomeHHH.
22. B KaKoM CHCTeMe CqHCJIeHHH Y KBapaTa 100 CTO-
pOH? CKOJIbKO CTOpOH B 3TOM CHCTeMe CqHCJIeHHH Y
TpeyrOJIbHHKa?
23. PacillHCPPYMTe BblCKa3bIBaHHe: «MHe 1111 JIeT, H H
yqYCb BIll KJIaCce».
24. YqHTeJIb Ha Bonpoc, MHoro JIH Y Hero B KJIaCCe eHH-
KOB, OTBeTHJI: «B KJIaCCe 100 eTeM, H3 HHX 25 MaJIb-
qHKOB H 31 eBOqKa». B KaKOM CHCTeMe CqHCJIeHHH
yqHTeJIb arr oTBeT?
12

CHCTeMbl C4WCI\eHHR
25. «fl OKOH'tIHJI KYpC YHHBepCHTeTa 44 JIeT OT poy.
CnycTH ro, 1 OO-JIeTHHM MOJIObIM 'tIeJIOBeKOM H
meHHJICH Ha 34-JIeTHeH eByrnKe. He3Ha'tlHTeJIbHaH
pa3HHa B B03paCTe - Bcero 11 JIeT - cnoco6cTBoBa-
JIa TOMY, 'tITO MbI mHJIH 06HMH HHTepecaMH H Meq-
TaMH. CnycTH HeCKOJIbKO JIeT y MeHH 6bIJIa yme ce-
MbH H3 1 0 eTeH». B KaKOH CHCTeMe C'tIHCJIeHHH aH
paccKa3?
26. Heo6LIKHoBeHHaJl eBOqKa
EH 6bIJIO TbICH'tIa CTO JIeT,
OHa B CTO nepBbIH KJIaCC xo.n;HJIa,
B nopTcpeJIe no CTO KHHr HOCHJIa -
Bce 3TO npaB.n;a, a He 6pe.
Kora, nbIJIH eCHTKOM Hor,
OHa rnaraJIa no .n;opore,
3a HeH Bcera 6emaJI eHOK
C OHHM XBOCTOM, 3aTO CTOHorHH.
OHa JIOBHJIa Kam.n;bIH 3BYK
CBOHMH .n;eCHTbIO yrnaMH,
II eCHTb 3arOpeJIbIX PYK
I10pTcpeJIb H nOBOOK epmaJIH.
II eCHTb TeMHO-CHHHX rJIa3
PaCCMaTpHBaJIH MHp npHBbI'tIHO...
Ho CTaHeT Bce COBceM 06bI'tIHbIM,
Kora nOHMeTe Harn paCCKa3.
A. H. CmapUttK06
B KaROH CHCTeMe C'tIHCJIeHHH HanHCaHO CTHXOTBopeHHe?

rJl,a8a II
MHO>KECTBA
Mno3ICecm8o - OHO H3 OCHOBHbIX nOHHTHH MaTeMaTH-
KH. Ero CMbICJI BbIpamaeTcH CJIOBaMH: COBOKynHOCTb, co6-
paHHe, KJIaCC, Ha6op, KOMaHa H T. . STOT CMbICJI nOHC-
HHeTCH MHOrO'tlHCJIeHHbIMH npHMepaMH. TaR, MomHO ro-
BOpHTb 0 MHomeCTBe Bcex yqHXCH 5- ro KJIaCCa, 0 MHO-
meCTBe Bcex mHTeJIeH CaHKT-IIeTep6ypra, 0 MHomeCTBe
Bcex HaTYPaJIbHbIX 'tIHCeJI, 0 MHomeCTBe KopHeH aHHoro
ypaBHeHHH. OCHOBaTeJIb TeopHH MHomeCTB HeMeKHH Ma-
TeMaTHK reopr KaHTOp (1845-1918) TaK onpeeJIHJI MHO-
meCTBO - «MHoroe, MbICJIHMOe KaR eHHoe, eJIoe».
MHomeCTBa 0603Ha'tlaIOTCH nponHCHbIMH 6YKBaMH
JIaTHHCKoro aJIcpaBHTa A, B, C, ...
o npeMeTax, COCTaBJIHIOrn;HX MHomeCTBO, rOBopHT,
'tITO OHH npunaone3ICam 3TOMY MHomeCTBY HJIH HBJIHIOTCH
ero 3neMenmaMU. MHomecTBa, 3JIeMeHTaMH KOTOpbIX HB-
JIHIOTCH'tIHCJIa, Ha3bIBaJOTCH l£ucn08b£MU MHomeCTBaMH.
MHomecTBo MomeT 6bITb 3aaHO nepe'tlHCJIeHHeM
Bcex ero 3JIeMeHTOB B npOH3BOJIbHOM nopHKe. TaKoe
MHomeCTBO Ha3bIBaIOT Konel£nb£M. MbI 6yeM paCCMaT-
pHBaTb TOJIbKO KOHe'tlHbIe MHomeCTBa.
3anHcb A = {I, 3, 5} 03Ha'tlaeT, 'tITO 'tIHCJIa 1, 3, 5 HBJIH-
IOTCH 3JIeMeI-ITaMH MHomeCTBa A. STO MomHO 3anHcaTb
HHa'tle, HanpHMep, A = {3, 1, 5}.
B 3anHCH MHomeCTBa C nOMOrn;bIO cpHrYPHbIx CKo6oK
TOtJHO YKa3aHO, KaKHe npeMeTbI HBJIHIOTCH 3JIeMeHTa-
MH MHomeCTBa H KaKHe He HBJIHIOTCH. qHCJIO 3 - 3JIe-
MeHT MHomeCTBa A. IIHIlIYT 3 EA. qHCJIO 4 He HBJISIeTCH
3JIeMeHTOM A. IIHIlIYT 4eA. B MHomeCTBe A coepmHT-
CH TPH 3JIeMeHTa. 8TO 0603Ha'tlaIOT TaK: IAI = 3.
MomHo COCTaBHTb MHomeCTBO C JII06bIM KOJIH'tIeCT-
BOM 3JIeMeHTOB: 1000, 25, 2, 1 H ame o.
14

MHO>KeCTBa
MHomeCTBO, B KOTOpOM HYJIb 3JIeMeHTOB, Ha3bIBaIOT
nycmb£M.
llycmoe MU03ICeCm80 0603HatlaIOT 0, 101 = o.
IIycTb MHomeCTBO A = {I, 3, 5}.
113 HeKOTOpbIX 3JIeMeHTOB MHomeCTBa A MomHO co-
CTaBHTb HOBoe MHomeCTBO, HanpHMep, B = {3, 5}.
KambIH 3JIeMeHT MHOmeCTBa A
B npHHB.)];JIemHT MHomecTBY A.
B TaKOM CJIYtlae rOBopHT, tlTO
B eCTb nooMuo3ICecm8o A H nH-
mYT B c A (pHC. 1).
Ec7m MHOmecTBa COCTOHT H3 o-
HHX H Tex me 3JIeMeHTOB, TO OHH BcA
Ha3bIBaJOTCH pa8ub£Mu. HanpH- PHe. 1
Mep: C = {2, 4}, D = {4, 2}, C = D.
qacTo MHomeCTBa H306pamaIOT KpyraMH, 3TH KpyrH
06bItlHO Ha3bIBaIOT 1(,pyzaMu :JuJI,epa» no HMeHH BeJIH-
tlaHmerO MaTeMaTHKa JIeoHapa 8HJIepa (1707-1783).
JI. 8HJIep pOHJICH H BbIpOC B IIIBeHapHH, a pa60T8.JI B
OCHOBHOM B POCCHH H B repMaHHH. 3a CBOIO mH3Hb
JI. 8HJIep HanHCaJI 60JIee 850 HaHbIX pa6oT. B OHOH
H3 HHX nOHBHJIHCb KpyrH, KOTopbIe OtleHb nOXOHT
JIH Toro, tlTo6bI 06JIertlHTb HamH pa3MbImJIeHHH..
Onepaqllil HIlA MHO)l(eCTBaMIl
HB.)]; MHomeCTBaMH, KaK H Ha tlHCJIaMH, npOH3BOHT
onepaHH. PaCCMOTpHM HeKOTopbIe H3 HHX: nepece1f,e-
uue, oo""eouueuue H pa3uocmb.
. Hepece1f,euue
B03bMeM MHomeCTBO X, COCTOHee H3 6YKB a, 6, 6, Z,
iJ, H MHomeCTBO Y, CocTOHee H3 6YKB Z, iJ, e, JIC:
X = {a, 6, 6, Z, iJ}, Y = {z, iJ, e, JIC}.
8TH MHomeCTBa HMeIOT oOUfue 3J1,eMeUmb£ z H iJ.
MHomeCTBa X H Y Ha3bIBaIOTCH nepece1(,aJOUfuMucB
15

rA8S8 II
x
y
A
x
XnY
PHe. 2
AnX=0
PHe. 3
MHomeCTBaMH. MHomecTBo 06m;HX 3JIeMeHTOB X H Y Ha-
3bIBaIOT nepeCel£eJl,UeM MJI,O:HCeCm8 X H Y H 0603HaqaIOT
C nOMOm;bIO 3HaKa (1: X n Y = {z, iJ} (pHC. 2).
IIycTb MHomeCTBoA = {I, 3, 5}. MHomecTBaAHXHe
HMeIOT HH OHoro o6m;ero 3JIeMeHTa. B TaKOM CJIae
MHomeCTBa A H X Ha3bIBaIOTCH Jl,enepeceKaJOUfuMucB
MHomeCTBaMH. IIepeceqeHHeM MHomeCTB A H X HBJIHeT-
CH nYCToe MHomeCTBO: A n X = 0 (pHC. 3).
. Oo""e8uJI,eJl,ue
ECJIH H3 3JIeMeHTOB MHomeCTB X H Y COCTaBHTb HOBoe
MHomeCTBO, COCTOHm;ee H3 Bcex 3JIeMeHTOB 3THX MHO-
meCTB H He coepmam;ee pyrHx 3JIeMeHTOB, TO nOJIY-
qHTCH oo""e8uJI,eJl,ue MJI,O:HCeCm8 X H Y, KOTopoe 0603Ha-
qaIOT C nOMOm;bIO 3HaKa u:
X u Y = {a, 6, 6, z, iJ, e, :He) (pHC. 4).
06'beHHeHHe MHomeCTB A H X He HBJIHeTCH nYCTbIM:
A u X = {I, 3, 5, a, 6, 6, z, iJ) (pHC. 5).
X Y
A
XuY
PHe. 4
x
AuX
Pile. 5-

MHO>KeCTSa
. Pa31£OCmb
Pa31£OCmb MHOmeCTB X H Y - 3TO MHOmeCTBO BCeX
3JIeMeHTOB H3 X, He HBJIHIOID;HXCH 3JIeMeHTaMH H3 Y.
Pa3HOCTL 0603HaqaIOT X\ Y = {a, 6, 6} (pHC. 6).
Pa3HOCTL Memy MHomeCTBOM H ero nOMHomeCTBOM
Ha3LIBaIOT iJonol£l£el£ueM.
IIycTL MHomeCTBO B = {3, 5} HBJIHeTCH nOMHomeCT-
BOM A = {I, 3, 5}. MHomecTBo 3JIeMeHTOB A, KOTopLIe He
BOmJIH B B, Ha3LIBaIOT OnOJIHeHHeM B o A H 0603Haqa-
IOT B = {I} (pHC. 7).
X Y
B = A \B
PIIC. 7
38A8""
27. 3anHmHTe C nOMOIO CPHrypHLIX cKo6oK MHomeCTBa:
a) OH03HaqHLIX HeqeTHLIX qHCeJI;
6) BY3HaqHLIX qeTHLIX qHCeJI, MeHLmHX 20;
B) OH03HaqHLIX qeTHLIX qHCeJI;
r) BY3HaqHLIX HeqeTHLIX qHCeJI, MeHLmHX 20.
KaKHM o6ID;HM CBOHCTBOM 06JIa.n;aIOT Bce qHCJIa, npH-
HaJIemaID;He BceM 3THM MHomecTBaM?
28. CKOJILKO 3JIeMeHTOB B MHomeCTBe:
a) Hcpp eCHTHqHOH CHCTeMLI CqHCJIeHHH;
6) CTOpOH TpeyrOJILHHKa;
B) 6YKB B PYCCKOM aJIcpaBHTe?
29. MHomecTBo Hcpp qHCJIa A - {I, 0, 7, 3}. MHomecTBo
Hcpp qHCJIa B - {I, 0, 7, 3, 4}. MomHo JIH YTBep-
maTL, qTO qHCJIO A MeHLme qHCJIa B?
17

rAasa II
30. KambIH eHHK KJIaCCa H3yqaeT HJIH aHrJIHHCKHH,
HJIH HClIaHcKHH, HJIH 06a H3bIKa. AHrJIHHCKHH H3Y-
qaIOT 25 qeJIOBeK; HClIaHcKHH - 27 qeJIOBeK; H aHr-
JIHHCKHH H HClIaHcKHH - 18 qeJIOBeK. CKOJIbKO yqe-
HHKOB B KJIaCce?
31. B OHOM H3 ropOOB rpY3HH mHTeJIH rOBopHT HJIH
IIO-rpY3HHCKH, HJIH IIO-PYCCKH, HJIH Ha BYX H3bI-
Kax. IIo-rpY3HHcKH rOBopHT 850/0 Bcex mHTeJIeH, a
IIO-PYCCKH - 750/0 . CKOJIbKO IIpOII.eHTOB Bcex mHTe-
JIeH rOBopHT Ha 060HX H3bIKax?
32. B KJIaCCe 500/0 yqeHHKOB HrpaIOT B 6aCKeT60JI,
40% - B TeHHHC, 100/0 - B 06e HrpbI. KaKaH qaCTb
KJIaCCa He HrpaeT HH B 6aCKeT60JI, HH B TeHHHc?
33. 1I3 40 yqaHXCH KJIaCCa 32 BbIIIHCbIBaIOT ra3eTY,
21 - mypHaJI, 15 yqaHXCH - H ra3eTY H mypHaJI.
CKOJIbKO yqaHXCH He BbIIIHCbIBaIOT HH mypHaJIa,
HH ra3eTbI?
34. B KJIaCCe 35 yqeHHKOB. 20 qeJIOBeK IIOCeaIOT MaTe-
MaTHqeCKHH KpymoK, 11 - 6HOJIOrHqeCKHH. 10 qe-
JIOBeK He IIOCeaIOT KpymKOB. CKOJIbKO 6HOJIOrOB
YBJIeKaIOTCH MaTeMaTHKoH?
35. CpeH qHCeJI IIepBOH TbICHqH HaHHTe KOJIHqeCTBO
qHCeJI, He eJIHHXCH HH Ha 3, HH Ha 5.
36. 1I3 38 yqaHXCH KJIaCCa 28 IIOCeaIOT XOpOBOH
KpymoK H 17 - JIbImHYIO CeKII.HIO. CKOJIbKO JIbImHH-
KOB IIOCeaeT XOpOBOH KpymoK?
37. 1I3 100 qeJIOBeK 85 3HaIOT aHrJIHHCKHH H3bIK, 80 -
HClIaHcKHH, 75 - HeMeII.KHH. Bce BJIaeIOT 110 KpaH-
HeH Mepe OHHM HHOCTpaHHbIM H3bIKOM. CpeH HHX
HeT TaKHX, KOTopbIe 3HaIOT TOJIbKO Ba HHOCTpaH-
HbIX H3bIKa, HO eCTb BJIaeIOHe TpeMH H3bIKaMH.
CKOJIbKO qeJIOBeK 3HaIOT TPH HHOCTpaHHbIX H3bIKa?
18

MHO>l<eCTSa
38. Bce yqaCTHHKH noe3KH BJIaeIOT no KpaHHeH Mepe
OHHM HHOCTpaHHbIM H3bIKOM.
6 H3 HHX BJIaeIOT aHrJIHHCKHM;
6 - HeMeIJ;KHM;
7 - cppaHIJ;Y3cKHM;
4 - aHrJIHHCKHM H HeMeIJ;KHM;
3 - HeMeIJ;KHM H cppaHIJ;Y3cKHM;
2 - cppaHIJ;Y3cKHM H aHrJIHHCKHM;
1 - aHrJIHHCKHM, cppaHIJ;Y3cKHM H HeMeIJ;KHM.
CKOJIbKO yqacTHHKOB noe3KH?
39. B OTqeTe rOBOpHJIOCb, qTO Bce 100 yqaCTHHKOB KOH-
cpepeHIJ;HH BJIMeIOT no KpaHHeH Mepe OHHM HHO-
CTpaHHbIM H3bIKOM:
5 3HaIOT aHrJIHHCKHH, HeMeIJ;KHH H cppaHIJ;Y3cKHH;
10 - aHrJIHHCKHH H HeMeIJ;KHH;
8 - cppaHIJ;Y3cKHH H aHrJIHHCKHH;
20 - HeMeIJ;KHH H cppaHIJ;Y3cKHH;
30 - aHrJIHHCKHH;
23 - HeMeIJ;KHH;
50 - cppaHIJ;Y3cKHH.
CocTaBHTeJIHM OTqeTa 6bIJIO YKa3aHO Ha OIIIH6KH.
IIoqeMY?
40. B JIarepe OTbIxa 70 eTeH. KambIH nocem;aeT no
KpaHHeH Mepe O,l{HH KpymOK.
27 qeJIOBeK nocem;aIOT cpoToKpymoK;
32 - nOIOT B xope;
22 - 3aHHMaIOTCH cnopTOM.
B cpoToKpymKe 10 pe6HT H3 xopa H 8 cnopTCMeHOB;
B xope 6 cnopTCMeHOB;
3 cnopTCMeHa nocem;aIOT H cpoToKpymoK H Xop.
CKOJIbKO pe6HT 3aHHTbI TOJIbKO cnopToM?
41. OTpH H3 92 IIIKOJIbHHKOB c06paJICH B noxo.
47 qeJIOBeK B3HJIH C C060H 6YTep6pobI C KOJI6acOH;
38 - C CbIpOM;
42 - C BeTqHHOH;

rJ\asa 1/
28 - C KOJI6acOH H C CbIpOM;
31 - C KOJI6acOH H C BeTqHHOH;
26 - C CbIpOM H C BeTqHHOH;
25 - 6YTep6pobI Bcex Tpex COpTOB.
HeKoTopbIe B3HJIH TOJIbKO IIO 6YTbIJIKe MOJIOKa.
CKOJIbKO 6bIJIO TaKHX, KOTopbIe B3HJIH TOJIbKO MO-
JIOKO?
42. B KJIaCCe 38 qeJIOBeK.
1I3 HHX 16 YMelOT HrpaTb B 6aCKeT6oJI;
1 7 - B xOKKeH;
18 - B BOJIeH6oJI.
YBJIeKalOTCH TOJIbKO BYMH BHaMH CIIopTa:
6aCKeT6oJIOM H XOKKeeM - 4;
6aCKeT6oJIOM H BOJIeH6oJIOM - 3;
BOJIeH6oJIOM H XOKKeeM - 5.
3 He HrpalOT HH B 6aCKeT6oJI, HH B XOKKeH, HH B BO-
JIeH6oJI.
CKOJIbKO qeJIOBeK YBJIeKaIOTCH TpeMH BHaMH cIIopTa?
43. YqHTeJIb 3aaJI Ha ypOKe CJIOmHYIO 3aaqy. B pe-
3YJIbTaTe KOJIHqeCTBO MaJIbqHKOB, peIIIHBIIIHX 3TY
3aaqy, OKa3aJIOCb paBHO KOJIHqeCTBY eBOqeK, ee
He peIIIHBIIIHX. Koro B KJIaCCe 60JIbIIIe - peIIIHBIIIHX
3aaqy HJIH eBOqeK?
44. B IIOXO OTIIpaBHJIHCb yqaHeCH 5-x H 6-x KJIaCCOB.
MarrbqHKOB 6bIJIO 16; IIIeCTHKJIaCCHHKOB H IIIeCTH-
KJIaCCHHIJ; Bcero 24; IIHTHKJIaCCHHIJ; CTOJIbKO, CKOJIb-
KO IIIeCTHKJIaCCHHKOB (MaJIbqHKOB H3 6 KJIaCca).
CKOJIbKO Bcero eTH OTIIpaBHJIOCb B IIOXO?

rpatP - aTO Ha60p TOqeK, HeKOTopbIe H3 KOTOpbIX CO-
e,n;HHeHbI JIHHHHMH.
.
3TH TOqKH Ha3bIBaIOTCH 8ep-
zuunaMU.
Coe,n;HHHIOW;He HX JIHHHH Ha-
3bIBalOTCH pe6paMU zpatPa.
IIpHMepbI rpacpOB rrpHBe,n;e-
HbI Ha pHC. 8.
Ipacp Ha3bIBaeTCH lWnet£nb£M,
ecJIH t.IHCJlO ero pe6ep KOHeqHO, H
6ecKonel£nb£M - B rrpOTHBHOM
CJIyqae. MbI 6y,n;eM paCCMaTpH-
BaTb TOJIbKO KOHeqHbIe rpacpbI.
qHCJIO pe6ep, BbIXO,n;HW;HX
H3 BepIIIHHbI rpacpa, Ha3bIBa- PHe. 8
eTCH cmenenblO aTOM BepIIIHHbI (pHC. 9).
BepIIIHHa rpacpa, HMelOaH HeqeTHYIO CTerreHb, Ha3bI-
BaeTCH HeqeTHOM, a HMelOa.H qeTHyro CTerreHb -l£emnou.
quCltO nel£emnb£X 8epzuun zpatjJa - l£emno.
l{ltl£ mozo, l£mo6b£ naumu KOltUl£eCm80 pe6ep zpa-
tPa, nY3ICno npocYMMupo8amb cmenenu 8epzuun.u no-
ltYl£ennb£u peayltbmam paaOeltUmb na o8a (pHC. 10).
A
rna8a III
rPAcDbl
,
A
CTeneHH
BepmJIH:
A-I
B-3
C-2
D - 2
B
C
D
PJIe. 9
. 2 BeplllIiHbI
H 1 pe6po
3 BeplllJIHhI
H 3 pe6pa
4 BepIIlHHbI
Ii 5 pe6ep
6 BepIIllfHbI
Ii 6 pe6ep
B
D
(1 +3+2+3+2+1) :2=6
PJIC. 10

PRC. 11
rpacp Ha3bIBaeTCH C8.R,3nb£M, eCJIH ero HeJIb3H pa3-
6HTb Ha ,l];Ba, He pa3pbIBaH B KaKoH-HH6y,l];b BepIIIHHe. B
CBH3HOM rpacpe MomHO, ,l];BHraHCb no pe6paM, nepeHTH
H3 JII060H BepIIIHHbI B JII06ylO ,l];pyrylO (pHC. 11).
U"JI,OM Ha3bIBaeTCH 3a.MKH}'TbIH nyTb (pHC. 12).
J(epe8oM Ha3bIBaeTCH CBH3-
HbIH rpacp, He HMelOHH K-
JIOB (pHC. 13).
TeopeMa:
!pacp MomHO 060HTH, npoM-
,l];H no KroK,l];OMY pe6py TOJIbKO
O,l];HH pa3 B TOM CJIyqae, eCJIH
rpacp CBH3HbIH H HeqeTHbIX
BepIIIHH y Hero 0 HJIH 2. IIpH
aTOM eCJIH HeqeTHbIX BepIIIHH
,l];Be, TO MapIIIpyr HaqHHaeTCH
u
B O,l];HOH H3 HHX, a 3aKaHqHBa-
eTCH B ,l];pyrOH. ECJIH HeqeTHbIX
u
BepIIIHH HeT HH O,l];HOH, TO Map-
IIIPYT MomeT HaqaTbCH B JIIO-
60H BepIIIHHe H B HeM me
OKOHqHTbCH.
IIOJIb3YHCb aTOM TeopeMOM,
MomHO YCTaHoBHTb, HBJIHeT-
CH JIH ,l];aHHaH cpHrypa ynu-
...
"ypCaJl,bnOU, TO eCTb MomHO
JIH aTY cpHrypy HapHCOBaTb,
'"asa III
CB.H3HbIH rpa<lJ
rpa(p
c I1HK.."10M
rpa<t>
6ea HKJIa
PRC. 12
,LJ:epeBbH
\1\11\
PRC. 13
HeCBH3HhIH rpacp


rpaQ>bl
HeYHIIKypCaJIbHaH
KplIBaH
y IIHKypCaJIbliaH IpI1BaSi
PRC. 14
He OTpbIBaH KapaH,n;ama OT 6YMarH H He npOBo,n;H O,n;HY H
TY me JIHHHIO ,n;Bam,n;bI.
OCHOBbI TeopHH rpacpOB KaK MaTeMaTHqeCKOH HaYKH
3aJIOmHJI B 1 736 r. JI. 8HJIep, pemaH 3a,n;aqy 0 MOCTax
KeHHrc6epra (CM. 3aqy N2 75).
A CJIOBO «rpacp» nepBbIM CTaJI HCnOJIb30BaTb aHrJIHH-
CKHH MaTeMaTHK ,n:meHMc ,n:m. CHJIbBeCTp (1814-1897).
IIpHMepaMH rpacpoB MorYT CJIymHTb JII06aH KapTa ,n;o-
u
por, cxeMa JIHHHH MeTpOnOJIHTeHa, 3JIeKTpOCxeMa, qep-
Tem MHoroyrOJIbHHKa H T. ,n;.
OqeHb qaCTO nOCTpoeHHe rpacpa nOMoraeT npH peme-
HHH 3a,n;aq.
38A8I1H
45. BCTpeTHJIHCb TPH ,n;pyra - BeJIOB, CepoB H qepHoB.
qepHoB CKa3aJI ,n;pyry, o,n;eTOMY B cepbIH KOCTIOM:
«MHTepecHo, qTO Ha O,n;HOM H3 Hac 6eJIbIH KOCTIOM, Ha
u u
,n;pyrOM - cepbIH H Ha TpeTbeM - qepHbIH, HO Ha KroK-
,n;OM KOCTIOM BeTa, He COOTBeTCTByroero cpaMH-
JIHH». KaKOH BeT KOCTIOMa y Kam,n;oro H3 ,n;pY3eH?
46. Arrema, BOpH H BHTH yqaTCH B O,n;HOM KJIaCce. O,n;HH
e3,n;HT ,n;OMOH H3 IIIKOJIbI Ha aBTo6yce, ,n;pyroH - Ha
TpaMBae, TpeTHH - Ha TpOJIJIeH6yce. O,n;Ham,n;bI rroc-
JIe ypOKOB Arrema nomeJI rrpOBo,n;HTb ,n;pyra ,n;o OCTa-
HOBKH aBTo6yca. Kor,n;a MHMO HHX rrpoxo,n;HJI TpOJI-
JIeH6yc, TpeTHH ,n;pyr KPHKHYJI H3 OKHa: «BOPH, TbI
3a6bIJI B mKOJIe TeTpa,n;b!» KTO Ha qeM e3,n;HT ,n;OMOH?
23

rAasa III
47. A, B, B H r - ,n;PY3bH. O,n;HH H3 HHX - Bpaq, ,n;pyroH -
mypHaJIHCT, TpeTHH - TpeHep crropTHBHOH IlIKOJIbI H
qeTBepTbIH CTpOHTeJIb. JKYPHaJIHCT HarrHCaJI CTaTbH
06 A Hr. TpeHep H mypHaJIHCT BMeCTe C B XO,n;HJIH B
rroxo,n;. A H B 6bIJIH Ha rrpHeMe y Bpaqa. Y Koro Ka-
KaH rrpocpeccHH?
48. B 6yrbIJIKe, CTaKaHe, KYBIIIHHe H 6aHKe HaJIHTbI MOJIO-
KO, JIHMOHa.,D;, KBac H Bo,n;a. M3BecTHo, qTO Bo,n;a H MO-
JIOKO HaxO,n;HTCH He B 6yrbIJIKe, B 6aHKe - He JIHMOHa.,D;
H He Bo,n;a, a cocy,n; C JIHMOHa.,D;OM CTOHT Mem,n;y KYBIIIH-
HOM H COCy,n;OM C KBaCOM. CTaKaH CTOHT OKOJIO 6aHKH H
cocy,n;a C MOJIOKOM. Orrpe,n;eJIHTe, r,n;e KaKaH mH,n;KOCTb.
49. Ha CTOJIe rrOCTaBJIeHbI B pH,n; 6yrbIJIKa, KpymKa, qaIII-
Ka, CTaKaH H KYBIIIHH, rrpHqeM B TaKOM rropH,n;Ke, B
KaKOM OHH rrepeqHCJIeHbI. B HHX Haxo,n;HTCH pa3JIHq-
HbIe HanHTKH: Kocpe, qaii, MOJIOKO, KBac H Bo,n;a, HO He
H3BeCTHO, KaKOH HarrHTOK B KaKOH rrocy,n;e. ECJIH CTa-
KaH rrOCTaBHTb Mem,n;y qaeM H MOJIOKOM, TO rro coce,n;-
CTBY C MOJIOKOM 6y,n;eT KBac, a Kocpe 6y,n;eT TOqHO B ce-
pe,n;HHe, KBac He B 6YTbIJIKe. Orrpe,n;eJIHTe, B KaKyro rro-
cy,n;y qTO HaJIHTO.
50. B O,n;HOM ,n;Bope mHBYT qeTbIpe ,n;pyra. Ba,n;HM H IIIO-
cpep CTapIIIe CepreH; HHKOJIaH H CJIeCapb 3aHHMaroT-
CH 60KCOM; 3JIeKTpHK - MJIa,n;IIIHH H3 ,n;pY3eH; rro Be-
qepaM AHTOH H TOKapb HrparoT B ,n;OMHHO rrpOTHB
CepreH H 3JIeKTpHKa. Orrpe,n;eJIHTe rrpocpeccHro Kam-
,n;oro H3 ,n;pY3eH.
51. B p03bIrpbIIlIe rrepBeHCTBa rro BOJIeH60JIY KOMaH,n;a A
OTCTaJIa OT KOMaH,n;bI B Ha TPH MeCTa, KOMaH,n;a E
orrepe,n;HJIa B, HO OTCTaJIa OT J(, KOMaH,n;a B orrepe,n;H-
JIa KOMaH,n;y r. KaKoe MeCTO 3aHHJIa Kam,n;aH H3 3THX
IIIeCTH KOMaH,n;?
52. BoceMb IIIKOJIbHHKOB BbICTpOHJIHCb TaK, qTO A 6bIJI
Brrepe,n;H B H B; B - rrOCJIe K qepe3 O,n;HOFO; JI - Brre-
24

rpatPbl
pe,l);H A, HO IIOCJIe p,; B - IIOCJIe M qepe3 O,l);HOro; p, -
Mem,l);Y B H r; K - cpa3Y 3a M, HO BIIepe,l);H B. B Ka-
KOM IIOpH,l);Ke BbICTpOHJIHCb IIIKOJIbHHKH?
53. IIeTp, reHHa,l);HH, ArreKceH H BJIa,l);HMHp 3aHHMaroT-
CH B O,l);HOH CIIOpTHBHOH IIIKOJIe B pa3HbIX CeKII.HHX:
rHMHaCTHKH, JIerKOH aTJIeTHKH, BOJIeH6oJIa H 6ac-
KeT6oJIa. IIeTp, ArreKceH H BOJIeH6oJIHCT yqaTcH B
O,l);HOM KJIaCce. IIeTp H reHHa,l);HH Ha TpeHHpOBKH xo-
,l);HT IIeIIIKOM BMeCTe, a rHMHaCT e3,l);HT Ha aBTo6yce.
JIerKOaTJIeT He 3HaKOM HH C BOJIeH6oJIHCTOM, HH C
6aCKeT6oJIHCTOM. KTO B KaKOH CeKII.HH 3aHHMaeTcH?
54. IIHTb qeJIOBeK mHBYT B O,l);HOM ropO,l);e. Mx HMeHa -
JIeoHH,l);, BJIa.,D;HMHp, HHKOJIaH, OJIer H IIeTp. Mx cpa-
MHJIHH - CTerraHOB, BOPHCOB, K03HH, ,n:p03,l);OB H Mc-
TOMHH. BOpHCOB 3HaKOM TOJIbKO C ,l);BYMH, a C K03H-
HbIM 3HaKOM TOJIbKO O,l);HH qeJIOBeK. IIeTp 3HaKOM co
BceMH, KpOMe O,l);HOro. HHKOJIaii: H MCTOMHH 3HaroT
,l);pyr ,l);pyra C ,l);eTCTBa. BJIa.,D;HMHp, HHKOJIaH H OJIer
3HaKOMbI Mem,l);Y C060H. ,n:p03,l);OB H BJIa.,D;HMHp He3Ha-
KOMbI. OJIer, HHKOJIaH H BOpHCOB qaCTO BMeCTe XO,l);HT
B KHHO. Ha30BHTe HMeHa H cpaMHJIHH Kam,l);Oro.
55. KOMaH,l);bIA, B, B, r HP, yqaCTBOBaJIH B 3cTacpeTe.,n:o
copeBHoBaHHH IIHTb 60JIeJIbHKOB BbICKa3aJIH Ta-
KHe rrporH03bI:
1) KOMaH,l);a p, 3aHMeT l-e MeCTO, KOMaH,l);a B-2 -e;
2) KOMaH,l);a A 3aHMeT 2-e MeCTO, r - 4-e;
3) B - 3-e MeCTO, p, - 5-e;
4) B - l-e MeCTO, r - 4-e;
5) A - 2-e, .B - 3-e.
B KamOM IIporH03e O,l);Ha qaCTb IIO,l);TBep,l);HJIaCb, ,l);PY-
raH - HeT. KaKoe MeCTO 3aHHJIa Kam,l);aH H3 KOMaH,l);?
56. TpH IIO,l);pyrH 6bIJIH B 6eJIOM, KpaCHOM H rOJIy6oM
IIJIaTbHX. Mx TYCPJIH 6bIJIH Tex me II.BeTOB. TOJIbKO y
TaMapbI II.BeTa rrJIaTbH H TycpeJIb COBrra,l);aJIH. BaJIH
25

rAasa III
6bIJIa B 6eJIbIX TYCPJIHX. HH IIJIaTbe, HH TYCPJIH JIHbI
He 6bIJIH KpaCHbIMH. OIIpeeJIHTe n;BeT IIJIaTbH H TY-
cpeJIb KamOH H3 IIoPyr.
57. TpH TOBapHm;a - BJIaHMHp, Mropb H CepreH -
OKOHt.lHJIH OHH H TOT me IIearOrHt.leCKHH HHCTHTYT
H IIpeIIoaIOT MaTeMaTHKY, CPH3HKY H JIHTepaTYPY B
llIKOJIaX TyJIbI, PH3aHH H HpOCJIaBJIH. BJIa,n;HMHp
pa60TaeT He B PH3aHH, Mropb - He B TYJIe. PH3aHen;
IIpeIIoaeT He CPH3HKY, Mropb - He MaTeMaTHKY, TY-
JIHK IIpeIIoaeT JIHTepaTYPY. KaKOH IIpeMeT H B Ka-
.
KOM ropoe IIpeIIoaeT KambIH H3 HHX?
58. qeTbIpe yqeHHn;bI - Mama, JIHa, jKeHH H KaTH -
YMeJIH HrpaTb Ha pa3HbIX HHCTpYMeHTax (apcpe, pOH-
JIe, rHTape H cKpHIIKe), HO KamaH TOJIbKO Ha OHOM.
8TH me yqeHHn;bI BJIa.,n;eJIH pa3HbIMH HHOCTpaHHbIMH
H3bIKaMH (aHrJIHHCKHM, cppaHn;Y3CKHM, HeMen;KHM H
HCIIaHCKHM), HO KamaH TOJIbKO OHHM H3 3THX H3bI-
KOB. M3BeCTHO, t.lTO Ta, KOTopaH HrpaeT Ha rHTape, ro-
BOpHT IIO-HCIIaHCKH. JIHa He HrpaeT HH Ha CKpHIIKe,
HH Ha apcpe H He 3HaeT aHr JIHHCKOro H3bIKa. Mallia He
HrpaeT HH Ha CKpHIIKe, HH Ha rHTape H He 3HaeT aHr-
JIHHCKOro H3bIKa. Ta, KOTopaH rOBopHT IIO-HeMen;KH,
He HrpaeT Ha apcpe. jKeHH 3HaeT cppaHn;Y3CKHH H3bIK,
HO He HrpaeT Ha CKpHIIKe. KTO Ha KaKOM HHCTPYMeHTe
HrpaeT H KaKOH H3bIK 3HaeT?
59. B KYIIe OHoro H3 BarOHOB IIoe3a MocKBa-Oecca
eXaJIH MOCKBHt.I, rreTep6ypmen;, TYJIHK, KHeBJIHHHH,
xapbKOBt.laHHH H oeCCHT. Mx cpaMHJIHH Hat.lHHa-
JIHCb 6YKBaMH A, B, B, r, /l, H E. B opore BbIHCHH-
JIOCb, t.lTO A H MOCKBHt.I - Bpat.lH; /l, H IIeTep6ypmen; -
Yt.lHTeJIH, a TYJIHK H B - HHmeHepbI. B H E - Yt.laCT-
HHKH BOHHbI, a TYJIHK B apMHH COBceM He CJIymHJI.
XapbKOBt.laHHH CTapllIe A, oeCCHT CTapllIe B. B H
MOCKBHt.I COllIJIH B KHeBe, a B H xapbKOBt.laHHH B
26

rpacpbl
BHHHHe. OlIpeeJIHTe IIpOcpeCcHro KamOrO H3 3THX
IIIeCTH IIaCCamHpOB H MeCTO mHTeJIhCTBa KamOrO
H3 HHX.
60. CKOJIhKO HaroHarreH Y TpHaTH'tIeThIpeXyroJIhHHKa?
61. ,l];ecHTh 'tIeJIOBeK IIpHBeTCTBOBaJIH Pyr Pyra PYKO-
IIOmaTHHMH. IIHTh 'tIeJIOBeK CeJIarrH IIO ceMh PYKO-
IIOmaTHH, Tpoe - IIO IIHTh H Boe - IIO 'tIeThIpe.
CKOJIhKO Bcero 6hIJIO CeJIaHO PYKolIomaTHH?
62. MomHo JIH opraHH30BaTh CPYT60JIhHhIH TYPHHP e-
BHTH KOMaH TaK, 'tITo6hI KamaH KOMaHa IIpOBeJIa
IIO 'tIeThIpe BCTpe'tlH?
63. MomHo JIH 15 TeJIec}x>HoB coeHHHTh Mem.zw C060H TaR,
'tIT06hI KamhIH H3 HHX 6hIJI CBH3aH pOBHO C 11 PY-
rHMH?
64. B rocyapcTBe 100 ropOOB, a H3 Kamoro H3 HHX BhI-
XOHT 4 oporH. CKOJIhKO Bcero opor B rocyapcTBe?
65. MomeT JIH B rocyapcTBe, B KOTOpOM H3 Kamoro ro-
poa BhIXOHT 3 oporH, 6hITh pOBHO 100 ,n;opor?
66. B rocyapcTBe 24 ropoa. MomeT JIH 6hITh TaK, 'tITO
8 H3 HHX coeHHeHhI C TpeMH ropoaMH, 11 - C IIH-
Thro rbpoaMH, a 5 - C 'tIeThIphMH ropoaMH?
67. B KJIaCCe 30 'tIeJIOBeK. MomeT JIH 6hITh TaK, 'tITO 9 H3
HHX HMeroT IIO 3 pyra B 3TOM KJIaCCe, 11 - IIO 4 py-
ra, a 10 - IIO 5 pY3eH?
68. ,l];eBHTh IIIKOJIhHHKOB, pa3'he3maHCh Ha KaHHKYJIhI,
u
,n;orOBOpHJIHCh, 'tITO KamhIH H3 HHX IIOIIIJIeT OT-
KphITKH IIHTephIM H3 OCTaJIhHhIX. MomeT JIH OKa-
3aThCH, 'tITO KamhIH H3 HHX IIOJIy'tlHT OTKphITKH
HMeHHO OT Tex pY3eH, KOTOphIM HalIHIIIeT caM?
69. B KJIaCCe 60JIhIIIe 30, HO MeHhIIIe 40 'tIeJIOBeK. JIro-
60H MaJIh'tlHK pymHT C TpeMH eBO'tlKaMH, a JIro6aH
,n;eBO'tlKa - C IIHThro MaJIh'tlHKaMH. CKOJIhKO B KJIaCCe
eBO'tleK H CKOJIhKO MaJIh'tlHKoB?
27

rAasa III
70. ,l];oKamHTe, qTO qHCJIO JIIOeii, KOra-JIH60 mHBIlIHX
Ha 3eMJIe H CeJIaBIlIHX HeqeTHOe qHCJIO PYKonoma-
THH, qeTHO.
71. MomHo JIH nOCTpOHTb rpacp, B KOTOpOM 6YYT TOJIb-
KO BepIlIHHbI CTeneHH 3 H 5, a qHCJIO pe6ep 6yeT
paBHO 7?
72. MomHo JIH nOCTpOHTb rpacp, Y KOToporo TPH BepIlIH-
HbI CTeneHH 5, nHTb BepIlIHH CTeneHH 7 H ceMb Bep-
IlIHH CTeneHH 3?
73. MorYT JIH CTeneHH BepIlIHH B rpacpe 6bITb paBHbI:
a) 8, 6, 5, 4, 4, 3, 2, 2;
6) 7, 7, 6, 5, 4, 2, 2, 1;
B) 6, 6, 6, 5, 5, 3, 2, 2?
74. B rpacpe H3 JII060ii BepIlIHHbI BbIXOHT no 3 pe6pa.
MomeT JIH B HeM 6bITb 2000 pe6ep?
75. PaccMoTpHM TY caMYIO 3aaqy, JIH peIlIeHHH KOTO-
poii JI. 8iiJIep BnepBbIe npHMeHHJI rpacpbI, - 3TO 3a-
aqa 0 MOCTax KeHHrc6epra (ceiiqac ropo Ha3bIBa-
eTCH KaJIHHHHrpa).
B XVIII BeKe ropo KeHHrc6epr 6bIJI pacnOJIO-
meH Ha 6eperax peKH H BYX OCTpOBax. Pa3JIHqHbIe
qaCTH ropoa 6bIJIH coeHHeHbI ceMblO MOCTaMH, KaK
nOKa3aHO Ha pHC. 15. MomHo JIH COBepIlIHTb npo-
rYJIKY, npoHH no KamOMY MOCTY pOBHO OHH pa3?
A
c
PIlC. 15
A
D
c

rpa Q> bl
76. ,l];aH KYCOK npOBOJlOKH ,l];JlHHOH 120 CM. MomHo JIH,
He JlOMaH npOBOJlOKH, H3rOTOBHTb KapKac Ky6a C
pe6poM 10 CM?
77. BaHKa HMeeT CPOPMY Ky6a. CMomeT JlH Myxa nocJle-
,l];OBaTeJlbHO 060HTH Bce 12 pe6ep Ky6a, He npOXO,l];H
,l];Bam,l];bI no 1 pe6py?
78. MomHo JlH HapHCOBaTb cpHrypy O,l];HHM pOCqepKOM,
He OTpbIBaH KapaH,l];arna OT 6YMarH H He npOBO,l];H no
O,l];HOH JlHHHH ,l];Bam,l];bI (pHC. 16)?
a)
6)
79. MomHo JlH o6BecTH KapaH,l];arnOM, He OTpbIBaH ero OT
6YMarH H He npOXO,l];H no O,l];HOH JlHHHH ,l];Bam,l];bI,
npaBHJlbHbIH nHTHyroJlbHHK C ,l];HarOHaJlHMH?
)
PRe. 16
29

rna8a IV
KOM&HHATOPHKA
KOM6HHaTopHKa - 3TO pa3eJI MaTeMaTHKH, nOCBH-
m;eHHbIH peIIIeHHK> 3aq BbI60pa H pacnOJIOmeHHH 3JIe-
MeHTOB HeKOToporo MHomecTBa B COOTBeTCTBHH C 38HHbI-
MH npaBHJIaMH. KOM6HHaTOpHKa H3yqaeT KOM6HHaIJ;HH H
nepeCT8.HOBKH npeMeToB, pacnOJIOmeHHe 3JIeMeHTOB, 06-
JI81Om;ee 3aHHbIMH CBOHCTBaMH. 06bNHhIH Bonpoc B
KOM6HHaTopHbIX 3aqax: CKOJIbKHMH cnOC06aMH...
K KOM6HHaTopHbIM 3aaqaM OTHOCHTCH TaKme 3a-
qH nOCTpoeHHH MarHqeCKHX KBapaTOB, 3aaqH pac-
IIIHcppOBKH H KOHpOBaHHH.
PomeHHe KOM6HHaTopHKH KaK pa3eJIa MaTeMaTH-
KH CBH3aHO C TpyaMH BeJIHKHX cppaHIJ;Y3cKHX MaTeMa-
THKOB XVII BeKa BJIe3a IIacKaJIH (1623-1662) H IIbepa
qJepMa (1601-1665) no TeopHH a3apTHbIX Hrp. 9TH TPy-
bI coepmaJIH npHHHnbI onpeeJIeHHH qHCJIa KOM6H-
.
HaIJ;HH 3JIeMeHTOB KOHeqHOrO MHomeCTBa. C 50-x rOOB
XX BeKa HHTepec K KOM6HHaTopHKe B03pomaeTCH B
CBH3H C 6YPHhIM pa3BHTHeM KH6epHeTHKH.
OCHOBHbIe npaBHJIa KOM6HHaTopHKH - 3TO npa8UltO
CYMMh£ H npa8UltO npou38eiJenUIL.
. npa8UltO CYMMh£
ECJIH HeKOTopbIH 3JIeMeHT A MomHO BbI6paTb m cno-
c06aMH, a 3JIeMeHT B - n cnoc06aMH, TO BbI60p «JIH60 A,
JIH60 B» MomHO CeJIaTb m + n cnoc06aMH.
HanpHMep, eCJIH Ha TapeJIKe JIemaT 5 H6JIOK H 6 rpYIII,
TO OHH nJIO MomHO BbI6paTh 5 + 6 = 11 cnoc06aMH.
. npa8UltO npou38eiJenuIL
ECJIH 3JIeMeHT A MomHO BbI6paTb m cnoc06aMH, a 3JIe-
MeHT B MomHO BbI6paTb n cnOC06aMH, TO napy A H B Mom-
HO BbI6paTb m . n cnOC06aMH.
30

HanpHMep, eCJIH eCTb 2 pa3HbIX KOHBepTa H 3 pa3HbIe
MapKH, TO BbI6paTb KOHBepT H MapKY MomHO 6 cnoco6a-
MH (2 . 3 = 6).
IIpaBHJIo npOH3BeeHHH BepHO H B TOM CJIyqae, KOra
paCCMaTpHBaIOTCH aJIeMeHTbI HeCKOJIbKHX MHOmeCTB.
HanpHMep, eCJIH eCTb 2 pa3HbIX KOHBepTa, 3 pa3HbIe
MapKH H 4 pa3HbIe OTKpbITKH, TO BbI6paTb KOHBepT, Map-
KY H OTKPbITKY MomHO 24 cnOC06aMH (2 . 3 . 4 = 24).
IIpoH3Be,IJ;eHHe Beex HaTypaJIbHbIX 'IHCeJI OT 1 ,IJ;O n
BKJIIO'IHTeJIbHO Ha3bIBaeTCJI n-4JaKTOpHaJI H 0603Ha-
'IaeTCJI CHMBOJIOM n!
n ' - 1 . 2 · 3 . . n
. - ....
HanpHMep, 51 = 1 . 2 . 3 · 4 . 5 = 120.
IIpHHJlTO C'IHTaTb O! paBHbIM 1.
qHCJIO rrepeCTaHOBOK H3 n aJIeMeHTOB paBHO n!.
HanpHMep, eCJIH eCTb 3 mapHKa - KpacHbIH, CHHHH H
3eJIeHbIH, TO BbIJIOmHTb HX
B pH MomHO 6 cnoco6aMH
(3 . 2 · 1 = 31 = 6).
MHora KOM6HHaTopHaH
3aaqa pemaeTCH C nOMO-
m;bIO nOCTpoeHHH iJepe6a
603M03ICUb£X 6apuaumo6.
HanpHMep, pemHM npe-
bIym;yIO 3aaqy 0 3-x ma-
pHKax rrOCTpoeHHeM epe- 3
Ba (pHC. 1 7).
38A8I1H
KOM6a.tHaTopMKa
AAA
C 3K 3IC C
C 3 ICC
PRe. 17
K
80. B Ba3e 6 H6JIOK, 5 rpym H 4 CJIHBbI. CKOJIbKO BapHaH-
TOB BbI60pa OHoro nJIoa?  -1.- :--..J:'/ __ /<.-:
o I .-' . I "
81.CKOJIbKO cym;ecTByeT BapHaHTOB nOKynKH OHOH
p03bI, eCJIH npoaIOT 3 aJIbIe, 2 6eJIbIe H 4 meJITbIe
p03bI?
7 I (' ' /
')7 ..; 
....' L-) !
o
-J
31

rAaaa IV
I
--I
82. M3 ropoa A B ropo B BeYT nHTb opor, a H3 ropo-
a B B ropo C - TPH oporH. CKOJIbKO nYTeH, npo-
XOHHX tIepe3 B, BeYT H3 ABC? :;.? -- 1,,-'
83. CKOJIbKHMH cnoco6aMH MomHO BbI6paTb rJIaCHYro H
COrJIaCHYro 6YKBbI B CJIOBe «nJIaToK»? J,  :- 2
84. CKOJIbKO TaHeBaJIbHbIX nap MomHO cocTaHTb H3
8 roHomeH H 6 eBymeK? 8. (, - '1 2
85. B CTOJIOBOH eCTb 4 nepBbIX 6JIroa H 7 BTOpbIX.
CKOJIbKO pa3JIHtIHbIX BapHaHTOB o6ea H3 BYX
6JIro MomHO 3aKa3a Tb ? l/ ? ../1 _ - l [)
/ I .. - _ l '. ("
86. CKOJIbKO pa3JIHtIHbIX BY3HatIHbIX tIHCeJI MomHO co-
CTaBHTb, HCnOJIb3YH H<PPbI 1, 4 H 7, eCJIH H<PPbI
MorYT nOBTopHTbCH? 3 '  '-- 0:: /
f) ----
87. CKOJIbKO pa3JIHtIHbIX TpeX3Ha tIHbIX tIHCeJI MomHO
COCTaBHTb, HCnOJIb3YH H<PPbI 3 H 5, eCJIH H<PPbI
MorYT nOBTopHTbCH? (/? J) ,.) 2
88. CKOJIbKO pa3JIHtIHbIX BY3HatIHbIX tIHCeJI MomHO co-
CTaBHTb H3 H<PP 0, 1, 2, 3, eCJIH H<PPbI MorYT no-
BTOpHTbCH? 7 ' 'j  / f
- .....--
(" I
\ 89. CKOJIbKO cyecTByeT TpeX3HatIHbIX tIHCeJI, Y KOTO-
pbIX Bce H<PPbI HetIeTHble? t; -'-, r" _...... /: \._ - '_
90. CKOJIbKO CY CT BY qeTHbIX TP3aqHX qHCeJI? .
!J 1 ,I 0 , - f.t > ,) , ,
91. CKOJIbKO pa3JIHtIHbIX TpeX3HatIHbIX tIHCeJI MomHO
COCTaBHTb H3 Tpex pa3HbIX H<PP 4, 5, 6? 3,   .J
92. CKOJIbKHMH cnoco6aMH MomHO 0603HatIHTb BepmH-
HbI peyrOJIbHHKa, HCnOJIb3YH 6YKBbI A, B, C H D?
-. 01 . l t
. "
93. CKOJIbKO pa3JIHtIHbIX TpeX3HatIHbIX tIHCeJI MomHO
COCTaBHTb H3 H<PP 1, 2, 3, 4, 5 npH YCJIOBHH, tITO HH
O,ll;Ha IJ;HCppa He nOBTopHeTCH? j.   ::: r "
94. CKOJIbKO pa3JIHtIHbIX TpeX3HatIHbIX tIHCeJI, MeHb-
mHX 400, MomHO COCTaBHTb H3 H<PP 1, 3, 5, 7,9,
32
1"rb
 ,t I
0ti

KOM6a.tHaTopMKa
eCJIH JIro6aH H3 3THX Hcpp MOmeT 6bITb HCIIOJIb30-
BaHa TOJIbKO OHH pa3? !,?, U ;),:-. _/ t/
oL -' l..
95. CKOJIbKHMH cIIoc06aMH MomHO COCTaBHTb cpJIar,
COCTOHIIJ;HH H3 Tpex ropH30HTaJIbHbIX IIOJIOC pa3-
JIHtIHbIX BeTOB, eCJIH HMeeTCH MaTepHH meCTH
BeToB? C I ,\--, .. ---. I J.J
96. 113 KJIaCCa BbI6HparoT 8 tIeJIOBeK, HMeroIIJ;HX JIYtI-
mHe pe3YJIbTaTbI 110 6ery. CKOJIbKHMH cIIoc06aMH
MomHO COCTaBHTb H3 HHX KOMaHy H3 3-x tIeJIOBeK
,lI;JUI yqaCTHSI BaCT acpeTe? J I ::; , 6. ;:- 3J (;)
97. B tIeTBepr B IIepBOM KJIaCCe OJImHO 6bITb tIeTbIpe
ypOKa: IIHCbMO, tITeHHe, MaTeMaTHKa H cpH3KYJIb-
Typa. CKOJIbKO pa3JIHtIHbIX BapHaHTOB paCIIHca-
HHH MomHO COCTaBHTb Ha 3TOT TTeHb? L./,;; " I' . ./
""" / 1,../ '
98. B IIHTOM KJIaCCe H3ytIaroTcH 8 IIpeMeTOB. CKOJIbKO
pa3JIHtIHbIX BapHaHTOB paCIIHcaHHH MomHO COCTa-
.
BHTb Ha IIOHeeJIbHHK, eCJIH B 3TOT eHb OJImHO
6bITb 5 YPOKOB H Bce ypOKH pa3HbIe? f} _I f( /. '"
99. illHcpp JIH ceHcpa COCTaBJIHeTCH H3 IIHTH pa3JIHtI-
HbIX Hcpp. CKOJIbKO pa3JIHtIHbIX BapHaHTOB co-
? IA / J '-' h ) ,J' I
CTaBJIeHHH mHcppa. J g r -(-{, ,,&:.f " - ' {'
100. CKOJIbKHMH cIIoc06aMH MomHO pa3MeCTHTb 6 tIeJIO-
BeK 3a CTOJIOM, Ha KOTOpOM IIOCTaBJIeHO 6 IIpH60-
?  I t
pOB (C)' -:,1' 1.:'
101. CKOJIbKO BapHaHTOB ceMH3HatIHbIX TeJIecpoHHbIX
HOMepOB MomHO COCTaBHTb, eCJIH HCKJIrotIHTb H3
HHX HOMepa, HatIHHaroIIJ;HeCH C HYJIH H 9?
102. TeJIecpoHHaH CTaHHH 06CJIymHBaeT a60HeHToB, y
KOTOpbIX HOMepa TeJIecpoHoB COCTOHT H3 7 Hcpp H
HatIHHaroTcH C 394. Ha CKOJIbKO a60HeHToB paCCtIH-
TaHa 3Ta cTaHHH?
103.I1MeeTcH 6 IIap IIeptIaTOK pa3JIHtIHbIX pa3MepOB.
CKOJIbKHMH cIIoc06aMH MomHO BbI6paTb H3 HHX o-
33
. //
../
'j r
../' ') (I
'.; /';
/'.1-
/' ;_J

rABsa IV
HY nepqaTKY Ha JIeBYIO PYKY H OHY nepqaTKY Ha
npaBYIO PyKY TaK, qT06bI 3TH nepqaTKH 6bIJIH pa3-
JIHqHbIX pa3MepoB?
104. H3 H<PP 1, 2, 3, 4, 5 COCTaBJISlIOTCSI nHTH3HaqHbIe
qHCJIa, B KOTOpbIX Bce H<PPbI pa3HbIe. CKOJIbKO Ta-
KHX qeTHbIX qHCeJI?
105. CKOJIbKO cyecTByeT TpeX3HaqHbIX qHCeJI, Y KOTO-
pbIX Bce H<PPbI qeTHbIe?
106. CKOJIbKO cyecTByeT qeTbIpeX3HaqHbIX qHCeJI, co-
CTaBJIeHHbIX H3 pa3HbIX HeqeTHbIX H<PP H eJISI-
HXCSI Ha 5?
107. CKOJIbKO cyecTByeT nSlTH3HatlHbIX qHCeJI, Y KOTO-
pbIX TpeTbSl Hcppa - 7, a nOCJIeHSISI - qeTHaSl?
108. CKOJIbKO cyecTByeT meCTH3HatlHbIX qHCeJI, Y KO-
TOpbIX BTopaSl H<ppa - 2, qeTBepTaSl - 4, meCTaSl-
6, a Bce OCTaJIbHbIe - HeqeTHbIe?
109. CKOJIbKO pa3JIHqHbIX qHCeJI, MeHbmHX MHJIJIHOHa,
MomHO 3anHcaTb C nOMOblO H<PP 8 H 9?
110. B <PYT6oJIbHOH KOMaHe 11 qeJIOBeK. HymHo BbI-
6paTb KanHTaHa Hero 3aMeCTHTeJISI. CKOJIbKHMH
cnoco6aMH 3TO MomHO CeJIaTb?
111. B KJIaCCe yqaTcSI 30 qeJIOBeK. CKOJIbKHMH cnoco6a-
MH H3 HHX MomHO BbI6paTb CTapOCTY H OTBeTCTBeH-
Horo 3a npoe3HbIe 6HJIeTbI?
112. B noxoe yqacTBYIOT 12 MaJIbqHKOB, 10 eBOqeK H
2 yqHTeJISI. CKOJIbKO BapHaHTOB rpynn emypHbIX
H3 Tpex qeJIOBeK (1 MaJIbqHK, 1 eBOqKa H 1 yqH-
TeJIb) MomHO cocTaBHTb?
113. CKOJIbKO KOM6HHaHH H3 qeTbIpex 6YKB pYCCKoro
aJI<paBHTa (B aJIcpaBHTe Bcero 33 6YKBbI) MomHO co-
CTaBHTb npH YCJIOBHH, qTO 2 coceHHe 6.YKBbI 6YYT
pa3HbIMH?
34

rJl,464 v
nPHHQHn AHPHXAE
IIpH pemeHHH MHOrHX 38)1;aq HCnOJIh3yeTcH TeOpeMa -
npHHHn HpHXJIe.
IIpHHHn Ha3BaH no HMeHH ero aBTOpa - HeMeKoro
yqeHoro IIeTepa rycTaBa JIemeHa HpHXJIe (1805-1859).
npuJtlfun lfupuxne HJIH npHHHn «HHKOB» - 3TO
nOJIOmeHHe, YTBepmalOee, qTO eCJIH no N JIHKaM
pa3JI03tHTb npeAMeTbl, 'IHCJIO KOTOpblX 60JIbWe N, TO
HaHAeTCH JIHK, B KOTOpOM HaXOAHTCJI 60JIbWe OAHO-
ro npeAMeTa.
CJIOBa «HHKH» H «npeMeThI» CJIeyeT nOHHMaTh B
o606eHHOM CMhICJIe.
HanpHMep, eCJIH 4 KpOJIHKa pa3MeCTHTh B 3 KJIeTKaX,
TO HaHeTCH XOTH 6hI OHa KJIeTKa, B KOTOpOH 6yeT He
MeHee 2 KpOJIHKOB.
oKa3aTeJIhCTBO npOCToe. eHCTBHTeJIhHO, paccym-
aH OT npOTHBHoro, onYCTHM, qTO He cyecTByeT KJIe-
TOK, re 60JIee OHoro KpOJIHKa. Tora B 3 KJIeTKaX OKa-
meTCH He 6oJIee,3 KpOJIHKOB, a HX 4 - npOTHBOpeqHe
(pHC. 18).
CnpaBeJIHB TaKme 0606eHHb1H npHHHn HpHX-
JIe: eCJIH no N HHKaM pa3JI03tHTb npeAMeTbl, 'IHCJIO
KOTOpblX 60JIbWe N · k (rAe k - HaTYpaJIbHOe 'IHCJIO),
TO HaHAeTCJI JIHK, B KOTOpOM HaXOAHTCJI 60JIbWe k
npeAMeTOB.
PRe. 18

rAasa v
B CJIyqae k = 1 o606eHHb1H rrpHHl1;Hrr ,n:HpHXJIe rrpe-
BpaaeTCSI B 06b1qHb1H rrpHHl1;Hrr ,n:HpHXJIe.
HarrpHMep, eCJIH 10 KpOJIHKOB pa3MeCTHTb B 3 KJIeT-
Kax, TO HaHeTCSI XOTSI 6b1 OHa KJIeTKa, B KOTOpOH 6y-
eT He MeHee 4 KpOJIHKOB.
,n:oKa3aTeJIbCTBO OT rrpOTHBHoro. ,n:orrycTHM, qTO He cy-
ecTByeT KJIeTOK, re 60JIee 3 KpOJIHKOB. Tora B 3 KJIeT-
Kax OKameTCSI He 60JIee 3 . 3 = 9 KpOJIHKOB, a HX 10 - rrpo-
THBOpeqHe.
38A8..H
114. B MemKe JIemaT mapHKH BYX pa3Hb1X l1;BeTOB. Ka-
Koe HaHMeHbmee qHCJIO mapHKOB HymHo Bb1HYTb
H3 MemKa, qTo6b1 cpeH HHX o6S13aTeJIbHO OKa3a-
JIHCb Ba llIapHKa OHoro l1;BeTa?
115. B SIHKe JIemaT HOCKH qeTb1pex l1;BeTOB. KaKoe
HaHMeHbllIee KOJIHqeCTBO HOCKOB Hao Bb1TaHTb,
qTo6b1 H3 HHX MomHO 6b1JIO COCTaBHTb XOTSI 6b1 o-
HY rrapy?
116. B Kopo6Ke JIemaT KapaHallIH: 7 KpaCHb1X H 5 CH-
HHX. B TeMHOTe 6epYT KapaHamH. CKOJIbKO Hao
B3S1Tb KapaHameH, qTo6b1 cpeH HHX 6b1JIO He Me-
Hee 2 KpaCHb1X H He MeHee 3 CHHHX?
117. B TeMHoH KJIaOBOH JIema T 60THHKH OHoro pa3Me-
pa: 10 rrap qepHb1X H 10 rrap KOpHqHeBb1X. HaHH-
Te HaHMeHbmee qHCJIO 60THHOK, KOTopoe HymHo
B3S1Tb H3 KJIaOBOH, qTo6b1 cpeH HHX OKa3aJIaCb
XOTSI 6b1 OHa rrapa (JIeBb1H H rrpaBb1H) OHoro l1;Be-
Ta. B TeMHOTe HeJIb3S1 orrpeeJIHTb He TOJIbKO l1;BeT
60THHKa, HO H JIeBb1H OT rrpaBoro.
118. B KJIaCCe 34 yqeHHKa. ,n:oKamHTe, qTO cpeH HHX
o6S13aTeJIbHO HaHYTcSI rro MeHbmeH Mepe Boe, y
KOTOpb1X cpaMHJIHH HaqHHalOTCSI C OHOH H TOH me
6YKBb1. -
36

npMHUH" A"pwxJ\e
119. IIpH KaKOM HaHMeHbmeM KOJIHtIeCTBe yqeHHKoB
mKOJIbI cpeH HHX 06S13aTeJIbHO HaHYTcSI Boe,
y KOTOpbIX eHb H MecSI pomeHHSI cOBnaaIOT?
120. MMeeTcSI 25 tIeTbIpeX3HatIHbIX HaTYPaJIbHbIX tIHCeJI
C pa3JIHtIHbIMH H<ppaMH, KOTopbIe 3anHCbIBaIOTCSI
C nOMOIIJ;bIO H<PP 1, 2, 3 H 4. BepHo JIH, tITO cpeH
HHX 06S13aTeJIbHO HaHYTCSI Ba paBHbIX tIHCJIa?
121. B JIecy paCTeT MHJIJIHOH eJIOK. M3BecTHo, tITO Ha
KamoH H3 HHX He 60JIee 600 000 HrOJIOK. ,l];oKa-
mHTe, tITO B JIecy HaHYTcSI XOTSI 6bI Be eJIKH
C oHHaKoBbIM tIHCJIOM HrOJIOK.
122. B ropoe CaHKT-IIeTep6ypre mHBeT 60JIee 4 MHJI-
JIHOHOB tIeJIOBeK. ,l];oKamHTe, tITO y KaKHX-TO BYX
H3 HHX OHHaKOBoe tIHCJIO BOJIOC Ha rOJIOBe, eCJIH
H3BeCTHO, tITO y JII060ro tIeJIOBeKa Ha rOJIOBe He 60-
JIee MHJIJIHOHa BOJIOC.
123.15 MaJIbtIHKOB C06paJIH 100 opeXOB. ,l];oKamHTe,
tITO KaKHe-TO Ba H3 HHX C06paJIH OHHaKOBoe tIHC-
JIO opeXOB.
124. B Mara3HH npHBe3JIH 25 SlIIJ;HKOB C Sl6JIOKaMH Tpex
COpTOB, npHtIeM B KamOM SlIIJ;HKe JIemaJIH Sl6JIOKH
KaKorO-TO OHoro copTa. MomHo JIH HaHTH 9 SlIIJ;H-
KOB C Sl6JIOKaMH OHoro copTa?
125. B KJIaCCe 37 tIeJIOBeK. HaHeTcSI JIH TaKoH MecSI B
roy, B KOTOpOM OTMetIalOT CBOH eHb pomeHHSI He
MeHee, tIeM 4 ytIeHHKa 3Toro KJIaCca?
126. B mKOJIe 30 KJIaCCOB H 1000 ytIaIIJ;HXCSI. ,l];oKa3aTb,
tITO eCTb KJIaCC, B KOTOpOM He MeHee 34 ytIeHHKOB.
127. B OMe mHBYT 5 KomeK. Y HHX 16 KOTHT. ,l];oKamH-
Te, tITO XOTSI 6bI Y OHOH KOlliKH He MeHee tIeTbIpex
KOTSIT.
37

fAasa V
128. Ha nJIOID;Ke 20 co6aK BOCbMH pa3HbIX nopo. o-
KamHTe, t.lTO cpeH HHX eCTb He MeHee Tpex co6aK
u
OHOH nopObI.
129. B HID;HKe 25 6eJIbIX mapOB, 25 t.lepHbIX, 20 CHHHX H
10 KpaCHbIX. Ha oID;ynb mapbI HeOTJIHt.lHMbI pyr
OT pyra. illapbI BbIHHMalOT H3 SlID;HKa B TeMHOTe.
KaKoe HaHMeHbmee t.lHCJIO mapOB HymHo BbIHYTb,
t.lTo6bI cpeH HHX o6S13aTeJIbHO OKa3aJIOCb:
1) 10 mapOB OHoro BeTa;
2) 10 6eJIbIX mapoB?
130. B HID;HKe JIemaT BeTHbIe KapaHamH: 10 Kpac-
HbIX, 8 CHHHX, 8 3eJIeHbIX H 4 meJITbIX. B TeMHOTe
6epYT H3 SlID;HKa KapaHamH. KaKoe HaHMeHbmee
t.lHCJIO KapaHameH Hao B3S1Tb, t.lTo6bI cpeH HHX
o6S13aTeJIbHO OKa38.JIOCb:
1) He MeHee t.leTbIpeX KapaHameH OHoro BeTa;
2) XOTSI 6bI OHH KapaHam Kamoro BeTa;
3) He MeHee 6 CHHHX KapaHameH?
131. OKa3aTb, t.lTO cpe mecm JIr06bIX HaTYP8JIbHbIX'tfiI-
ceJI Haii,lQ'TCSI Ba, pa3HOCTb KOTOpbIX eJIHTCSI Ha 5.
132,. oKa3aTb, t.lTO H3 JII06bIX Tpex HaTYP8JIbHbIX t.lHCeJI
MomHO HaHTH Ba, CYMMa KOTOpbIX eJIHTCSI Ha 2.
133. aHbI 12 pa3JIHt.lHbIX BY3Hat.lHbIX HaTypaJIbHbIX
t.lHCeJI. oKamHTe, t.lTO H3 HHX MomHO BbI6paTb Ba
t.lHCJIa, pa3HOCTb KOTOpbIX - BY3Hat.lHoe t.lHCJIO,
3anHCbIBaeMoe BYMSI OHHaKOBbIMH H<ppaMH.

(85-x)+x+(80-x)+
+(75-x)=100 H
PRe. 21
I
I OrBe,..
I 1. 1, 2, 10, 11, 12, 20, 21, 22, 100,
I 101. 2. a) B TpoHHOH; 6) B llIeeTe-
pIftlHOH; B) B BOCLMepwmoH. 3. a) tIHe-
I JIapaBHLI; 6) 60JILllIe BTopoe; B) 60JIL-
I llIeBTOpoe.4. 333. 5. 1000.6.43210.
I 7. 102345. 8. a) 1000000; 6) 102220;
B) 34311; r) 6002. 9. 10000010;
I 11211; 2002; 1010; 334; 244; 202;
I 154. 10. a) 37; 6) 136;B)517;r)324.
11. 31; BOeLMepHHaH eHeTeM8.
I 12. PHC. 19. 13. HanpHMep: 1023 +
+ 2112 = 3201. 14. PHe. 20. 15. a) B
'rpOIftlHOH; 6) B llIecrepwmOH; B) B eT-
BepINHOH; r) B nHTepHHoH. 16. a) B
6 pa3; 6) B 36 pa3. 17. a) B 3 pa3a; 6) B
27 pa3. 18. a) B nHTepHHoH; 6) B
BoeLMepHHOH; B) B BoeLMepHHOH;
r) B eeMeplNHoH. 19. ,l];a. 20. HeT.
21. HanpHMep: 199010 = 22012013.
22. B ,n;BOwmOH; 11 CTOpoH. 23. ,l];BO-
HHaH eHeTeMa; 15 JIeT; B 7 KJIaeee.
24. B llIeeTepHHoH eHeTeMe. 25. B
nHTepHHoH eHeTeMe. 26. B ,n;BOH-
HOH eHeTeMe. 27. Bee HeJIa HaTY-
p8.JILHbIe, MeHLIlIHe 20. 28. 10; 3;
33. 29. HeT. 30. 34 eJI. PHe. 21.
31.600/0.32. 1/5.33. 2. 34. 6 eJI.
35. 533. 36. Or 7 ,n;o 17 . 37. 70 eJI.
PHe. 22. 38. 11 . 39. IIoeTpoHTL
x
PRe. 21
+ 0 1 2 3 4 5
0 0 1 2 3 4 5
1 1 2 3 4 5 10
2 2 3 4 5 10 11
3 3 4 5 10 11 12
4 4 5 10 11 12 13
5 5 10 11 12 13 14
PRe. 19
x 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 11 13 15
3 0 3 6 12 15 21 24
4 0 4 11 5 22 26 33
5 0 5 13 21 26 34 42
6 0 6 15 24 33 42 51
PRe. 20
39

OTBeTbl
16 + (8 -x)+5+(10-x)+3 = 38
x=2
PRe. 23
M
A
(16 -x) +x+(24 -x) +x=40
PRe. 24
KpyrH 3HJIepa H HaHTH OllIH6KY. 40. 11 eJI. 41. 25 eJI. 42. 2 eJI. PIle. 23.
43. lIopoBHY. 44. 40 eJI. PHe. 24.45. BeJIOB - B eepOM, CepoB - B ep-
HOM, qepHoB - B 6eJIOM. 46. Anellia - Ha TpaMBae, BOpH - HaaBTo6yee,
BHTH - Ha TpOJIJIeH6yee. 47. A - TpeHep, B - eTpOHTeJIL, B - mypHa-
JIHCT, r - Bpa. 48. B 6YTLIJIKe - JrnMOHM, B eTaKaHe - Bo,n;a, B K}TBllIHHe -
MOJIOKO, B 6aHKe - KBae. 49. B 6YTLIJIKe - Bo,n;a, B KpymKe - aH, B am-
Ke - MOJIOKO, B eTaKaHe - KoetJe, B KYBllIHHe - KBae. 50. BMHM - TOKapL,
CepreH - eJIeCapL, HHKOJIaH - 3JIeKTpHK, AHTOH - llIoetJep. 51. 1-,
2- E, 3- B, 4- B, 5- r, 6-A. 52. r  JI AM K BB. 53. lIeTp-6aeKeT-
6oJIHeT, reHHMHH - BOJIeH60JIHeT, AneKeeH - mMHaeT, BJIMHMHp-
JIerKOaT JIeT. 54. JIeoHH,n; K03HH, BJIMHMHp BoPHCOB, HHKOJIaH OrenaHoB,
OJIer HCTOMHH, lIeTp ,l];p03,n;OB. 55. A - 5 MeeTO, B-2 MeeTO, B-3 Me-
eTO, r - 4 MeCTO,  -1 MecTO. 56. TaMapa - B KpaeHOM IIJIaTLe H Kpae-
HLIX TYJUIX, BarrH - B ro.ny60M nJIaTLe H 6eJIbIX TYetJJIHX, JlH,n;a - B 6eJIOM
MaThe H ro.ny6LIX TY.HX. 57. BnMHMHI> - JIHTepaTYPY B TyJIe, HroPL-
etJH3HKY B HpOCJIaBJIe, CepreH - MaTeMaTHKY B PH3aHH. 58. Mallia - He-
MeIJ;KHH H3LIK, pOHJIL; meHH - etJpaHIJ;Y3cKHH H3LIK, apetJa; JIH,n;a - He-
naHCKHH H3LIK, rHrapa; KaTH - aHrJIHHCKHH H3LIK, CKpHnKa. 59. A -
Bpa, O,n;ecea; B - }"IHTeJIL, lIeTep6ypr; B - HHmeHep, KHeB; r - HHme-
Hep, TyJIa; l1. - }"IHTeJIL, XapLKoB; E - Bpa, MocKBa. 60. 34 . 31 : 2 =
= 527. 61. (5 . 7 + 3 . 5 + 2 . 4) : 2 = 29. 62. ,l];a, 9 . 4 : 2 = 18. 63. HeT, 15 . 11 -
HeeTHOe HCJIO. 64. 100 .4 : 2 = 200. 65. Herr, 200 He ,n;eJIHTCH Ha 3. 66. HeT,
(8.3+ 11.5+ 5. 4) -HeeTHoe'lllCJIo. 67. Her, (9.3+ 11 .4 + 10 . 5)- HeeT-
HOe'lHCJIO. 68. He'!', 9 . 5 - He'le'!'Hoe '1HCJIO. 69. 12 AeBOqeK H 20 MaJIL'IH- X
40

OmeTbl
KOB, 12.5 = 20.3. 70. n . m: 2 - IJ;eJIoe HCJIO, eCJIH m - HeeTHO, TO n-
eTHO. 71. HeT. n .3 + m . 5 * 14 . 72. HeT, (3.5+5.7+7.3) - HeeTHoe
HCJIO. 73. a) a; 6)a; B) HeT. 74. HeT. n. 3*4000. 75. HeT. qHCJIOHe-
errHLIX BepIIIHH 60JILIIIe BYX. 76. Herr. qHCJIO HeeTHLIX BepIIIHH BO-
ceML. 77. HeT. qHCJIO HeeTHLIX BepIIIHH BoceML. 78. a) HeT; 6) a; B) ,n;a;
r) HeT; ,n;) HeT. 79. ,l];a. BO. 6+5+4=15. 81. 3+2+4=9. 82. 5.3=15.
83.2.4=8.84.8.6=48.85.4.7=28.86.3.3=9.87.2.2.2=8. 88.3.4=12.
89.4.5.5=100.90.9.10.5=450.91.3.2.1=6.92.4.3.2=24. 93.5.4.3=
=60.94. 2.4.3=24.95.6.5.4=120.96.8.7.6=336.97.4.3.2.1=24.
98. 8.7.6.5.4=6720. 99. 5.4.3.2.1=120. 100. 6.5.4.3.2.1=720.
101. 8.10.10.10.10.10.10=8000000. 102. 10.10.10.10=10000.
103.6.5 =30.104. 2.4.3.2.1 =48.105.4.5.5 =100.106.1.4.3.2 =24.
107.9.10.1.10.5=4500.108. 5.1.5.1.5.1=125.109. O,n;H03HaHLIx-
2, ,n;BY3HaHLIX - 4, TpeX3HaHLIX - 8, eTLIpeX3HaHLIX - 16, nHTH-
3HaHLIX - 32, IIIeCTH3HaHLIX - 64. Bcero -126.110.11.10 = 110.
111.30.29 = 870.112.12.10.2 = 240.113.33.32.32.32 = 1081344.
114. 3 IIIapHKa. 115. 5 HOCKOB. 116. 10 KapaH,n;aIIIeH. 117. 21 60TH-
HOK. 118. B PYCCKOM 8.JIq>aBHTe 33 6YKBLI, a }"IeHHKOB 60JILIIIe - 34.
3,n;ecL 6YKBLI HrpalOT pOJIL HHKOB, a }"IeHHKH - pOJIL npe,n;MeTOB, pac-
KJIa)J;LIBaeMLIX no HHKaM. lIoCKOJILKY npe,n;MeTOB 60JILIIIe, eM HH-
KOB, TO no npHHIJ;Hny ,l];HpHXJIe HaH,n;eTCH HHK, B KOTOpOM HaxO,n;HTCH
60JILIIIe o,n;Horo npe,n;MeTa, T. e. HaiiA}TTCH no MeHLIIIeH Mepe ,n;Ba yqeHHKa,
Y KOTOpLIX q>aMHJIHH HaHHaIOTCH C O,n;HOH H TOH me 6YKBLI. 119. B He-
BHCOKOCHLIH ro,n; - 366, B BHCOKOCHLIH - 367. 120. BepHo. lIo npaBHJIY
npOH3Be,n;eHHH KOJIHeCTBO pa3JIHHLIX HCeJI paBHO 4.3.2.1 = 24 .
121. ECJIH npe,n;nOJIOmHTL, TO Y Bcex eJIOK pa3Hoe HCJIO HroJIOK OT 0
,n;o 600000, TO Bcero TaKHX eJIOK He 60JIee 600001, a no YCJIOBHIO HX
MHJIJIHOH. lIOJI}"IHJIH npoTHBopeHe, 3HaHT, npe,n;nOJIOmeHHe HeBep-
HO, H eCTL XOTH 6LI 2 eJIKH C paBHLIM HCJIOM HrOJIOK. 122. ECJIH npe,n;-
nOJIOmHTL, TO Y Bcex JIIO,n;eH pa3Hoe KOJIHeCTBO BOJIOC OT 0 ,n;o
1000000, TO HCJIO TaKHX JIIO,n;eH 1000001, a no YCJIOBHIO HX 60JIee
4000000. lIpoTHBopeHe, 3HaHT, npe,n;nOJIOmeHHe 6LIJIO HeBepHO.
123. ECJIH 3TO He TaK, TO MaJILHKH C06paJIH He MeHee, eM 0 + 1 + 2 +
+ 3 +...+14=1050pexoB. lIpoTHBopeHe.124. TaKKaKCOpTOB3, aHH-
KOB 25, TO XOTH 6LI O,n;HOro copTa HMeeM He MeHee 9 HHKOB. 8.3 + 1 =
= 25. 125. ,l];onycTHM - He 60JIee Tpex, Tor,n;a 3.12 = 36. lIpoTHBope-
He, B YCJIOBHH 37 eJIOBeK. 126. ECJIH 6LI B Kam,n;OM KJIaCCe 6LIJIO Me-
Hee, eM no 34 }"IeHHKa, TO B 30 KJIaccax }"IHJIHCL 6LI He 60JIee, eM
33.30 = 990 }"IaHXCH, a He 1000, KaK B YCJIOBHH. 127. ,l];onYCTHM, He
60JIee 3 KOTHT Y Kam,n;oH KOIIIKH, Tor,n;a He 60JIee, eM 3.5 = 15 KOTHT
X Bcero. IIpoTHBope'lHe, B YCJIOBHH 16 KOTJlT. 128. AonycTHM - He 60JIee
41

OmeTbl
BYX c06aK KaOH nOpObI, Tora 2.8 = 16 - MeHLIIIe 20. IIpoTHBope-
,
He, B YCJIOBHH 20 c06aK. 129. 1) 9.4 + 1 = 37; 2) 25 + 20 + 10 + 10 = 65.
130.1) 3.4 + 1 = 13; 2) 10+8+8+ 1 = 27; 3) 10+8+4+ 6= 28.131. IIpH
eJIeHHH Ha 5 B03MomHO 5 pa3HLIX OCTaTKOB: 0, 1, 2, 3,4. TaK KaK H-
ceJI IIIecTL, TO HaHYTCH Ba HCJIa C OHHaKOBbIMH OCTaTKaMH; HX pa3-
HOCTL 6yeT eJIHTLCSI Ha 5. 132. CpeH Tpex HCeJI o6H3aTeJILHO HaH-
YTCH Ba HCJIa oHHaKoBoH eTHOCTH, HX CYMMa eJIHTCH Ha 2.
133. .n;BY3HaHoe HCJIO, 3anHCbIBaeMoe OHHaKOBbIMH Hq>PaMH, e-
JIHTCH Ha 11.

COA8P)l(8HII8
IIpeHcJIOBHe. .
. . . . . . .
. . . . . 3
r JIaBa I
CHCTEMbI CqHCJIEHHH.
. . . . . . . . . . .
. 4
r JIaBa II
MHOmECTBA . . . . . . . . . . . . . . . . . 14
rJIaBa III
rPAbI . . .
. . . . . . . . . . . . . . 21
rJIaBa IV
KOMBHHATOPHKA . . . . . . . . . . . . . . 30
rJIaBa V
IIPHHQHII HPHXJIE . .
. . . . 35
. 39
OTBeTId. . . . . . . . . . .