cuadernillo_entrenamiento_secundaria_2011.docx

8/17/2019 cuadernillo_entrenamiento_secundaria_2011.docx

1/55

SECRETARÍA DE EDUCACIÓN JALISCOCOORDINACIÓN DE EDUCACIÓN BÁSICA

DIRECCIÓN GENERAL DE EDUCACIÓN SECUNDARIA

DIRECCIÓN GENERAL DE PROGRAMAS ESTRATÉGICOSDIRECCIÓN DE PROGRAMAS DE ACOMPAÑAMIENTO

PEDAGÓGICO

SEGUNDA OLIMPIADA ESTATAL DE MATEMÁTICASEN EDUCACIÓN PRIMARIA Y SECUNDARIA

2ª OEMEPS 2011

CUADERNILLO DE ENTRENAMIENTONIVEL SECUNDARIA

Guaa!a"a#a$ Ja!%&'() *+,#+#( + 2011


2/55

Cuadernillo de Entrenamiento Nivel Secundaria 2ª OEMEPS

2011

ÍNDICE

Pág.PRESENTACIÓN 3

JUSTIFICACIÓN 5

INSTRUCTIVO DE PROCEDIMIENTOS PARA LA APLICACIÓN YEVALUACIÓN DE LOS EXÁMENES

6

PROBLEMARIO 7

SOLUCIONES 11

FUENTES DE CONSULTA !

2


3/55


2011

PRESENTACIÓN

L" S#$%#&"%'" (# E()$"$*+, J"-*$/0 " &%"2 (# -" C//%(*,"$*+, (#E()$"$*+, Bá*$"0 $/, #- %/+*&/ (# 4/%&"-#$#% #- (#"%%/--/ (#

$/#,$*" "á&*$" #, -/ "-),/ (# #()$"$*+, %*"%*" #$),("%*"0 " &%"2 (# ), $/,$)%/ )# *-*)# #- %"8/,"*#,&/ -"$%#"&**("( #, -" %#/-)$*+, (# %/9-#"0 $/,/$" " -" S#g),("O-**"(" E&"&"- (# M"á&*$" #, E()$"$*+, P%*"%*" S#$),("%*"!11 :; OEMEPS- (# Eg%#/ (#E()$"$*+, Bá*$"@

C/#,$*" "%" #- ",#?/ (# -" *,4/%"$*+,. S# %#-"$*/,", $/,@ -"9)#("0 *(#,&*>$"$*+,0 #"-)"$*+,0 #-#$$*+, *"&*8"$*+, (#*,4/%"$*+, #- #,"%0 %##=*/,"%0 "%g)#,&"% #=%#"% ?)*$*/ $%'&*$/","-*8"%0 *,&*8"%0 )&*-*8"% $/"%&*% *,4/%"$*+, #- $/,/$**#,&/ ",#?/(# (*&*,&" -+g*$" (# $/,&%)$$*+, (#- $/,/$**#,&/ #, (*#%" (*$*-*," #, -/ (*&*,&/ á9*&/ $)-&)%"-#. :SEP0 !!0 ág. !1<

A' $// #, -" (#>,*$*+, )# -" SEP :!11$"% -/ )# # ,#$#*&" "9#% "%#,(#% " 9)$"% *(#,&*>$"%0 #"-)"%0#-#$$*/,"%0 /%g",*8"% *"&*8"% *,4/%"$*+, "%/*"%# (# -" *,4/%"$*+,(# ",#%" $%'&*$"0 )&*-*8"% $/"%&*% *,4/%"$*+, $/, #,&*(/ 2&*$/. :SEP0!110 ág. 3G3<

L/ "-),/ "%&*$*", #$%*9*%á, ) %/$#(**#,&/ (# /-)$*+, -/ ?)#$# "*g,"%á, ),&/ #g, #- "",$# -/g%"(/ #, ) %#)#&".

E&" ?/%,"(" (# &%"9"?/ *,,/ ,#$#"%*"#, (#?"%á "%#,(*8"?# (#g%", "-/% #, -/ "-),/ (#"%%/--"%á $/#,$*" %/4#*/,"-# #,-/ (/$#,.

O%g",*8"% ",*"% *&)"$*/,# (# "%#,(*8"?#. S# %#-"$*/,", $/,@ #- $/,/$#% "&%"2 (# )," (*$*-*," (#%*,"("0 -/ $/,,*(/ )# H" )# #,#"% )&%"()$$*+, #, /9?#&*/ (# "%#,(*8"?# &%"9"?"% " "%&*% (# -"%#%##,&"$*/,# (# -/ "-),/ &%"9"?"% " "%&*% (# -/ #%%/%# -//9&á$)-/ #, #- "%#,(*8"?# $/,&%)*% -",*>$"% (*/*&*/ #$)#,$*"

3


4/55


2011

(*(á$&*$" # *-*$"% " -/ "-),/ #, "$&**("(# (# *,#&*g"$*+,0 #,%/#$&/ (# $/,/$**#,&/. :P#%%#,/)(0 !!7<

P"%" #&" ; O-**"(" E&"&"- (# M"á&*$" #, E()$"$*+, P%*"%*" S#$),("%*"0 # H" (#$*(*(/ "%%",$"% (#(# #- *,*$*/ (#- "/ !110 $/, -"

$/,/$"&/%*" -" "$&**("(# %#-"$*/,"(" $/, -" %#/-)$*+, (#%/9-#" )# # %//,#, #, # C)"(#%,*--/ (# E,&%#,"*#,&/. L/#&)(*", /(%á, "%&*$*"% #, -" $"g/%'" #, -" #&"" )# -#$/%%#/,(", (# "$)#%(/ $/, -" 9"# #&"9-#$*(" #, (*$H"$/,/$"&/%*".

P#,",(/ #, "/"% " -/ %/4#/%# #, -" %#"%"$*+, (# )#&)(*", )# "%&*$*"%á, #, -/ (*&*,&/ /#,&/ (# -" O-**"("0# H" #-"9/%"(/ # C)"(#%,*--/ (# E,&%#,"*#,&/0 #, #- )# #%//,#, %/9-#" **-"%# " -/ )# -/ "-),/ #,4%#,&"%á, #,$"(" )," (# -" #&"" (#- $/,$)%/. E */%&", )# #- "#&%/(#(*)# ), &*#/ #=$-)*/ "%" #- &%"9"?/ $/, -/ "-),/ )",(/ #-%/9-#"%*/. S# %#$/*#,(" (#&*,"% "- #,/ )," H/%" " -" #",". L"#&/(/-/g'" (# &%"9"?/ )g#%*(" # -" *" )# # %//,# #, -/%/g%"" />$*"-# (# -" SEP (#- !110 $/%%#/,(*#, " -" "*g,"&)%"(# M"á&*$" #, E()$"$*+, Bá*$".

E, ), "9*#, (# $/,>",8" $%#"(/ /% #- "#&%/0 -/ "-),/(#9#%á, "9/%("% -/ %/9-#" $/, -" H#%%"*#,&" #%/,"-# (# )#(*/,#, # *,,&"% #,$/,&%"% #, $"(" %/9-#"0 "- #,/ )," /-)$*+,*, #- )/ (# -" $"-$)-"(/%"0 "%" $/,4%/,&"% /%*/%#, $/, #- %#&/(# ) $/"#%/ -/ %#)-&"(/ " -/ )# --#g)#,0 ?)&*>$",(/ "%g)#,&",(/ "/ " "/ $"(" )," (# -" %#)#&" ("(" " -/$)#&*/,"*#,&/ )# # -# -",",. C/, -" >,"-*("( (# 4"/%#$#% -"$/,*,$*" $-"%*("( #, -" "%g)#,&"$*+, )# H"g", -/ "-),/0 #*/%&", )# #- %/4#/% -# /-*$* #$%*9*% &/(" -" *(#" )# # -#/$)%%", ()%", #- %/$#/ (# %#/-)$*+,0 *,(##,(*#,#, (# * -/--#"%/, / ,/ " -" /-)$*+, >,"-.

E- %/4#/% %#*"#, (#9#%á %#/-#% -/ %/9-#" )# %//,(%á #,-" #*+, (# &%"9"?/ / %#*"% -" /-)$*/,# )# # %//,#, #, #%/9-#"%*/ %##,&"% "- #,/ )," /-)$*+, #, #- $"/ (# )# -/"-),/ ,/ -/g%#, #,$/,&%"% "-g),". A(#á0 # ,#$#"%*/ )# ()%",-" $/,4%/,&"$*+, (# /-)$*/,#0 /%g",*$# -/ (*4#%#, %#)-&"(/ " -/)# "%%*9#, ) #&)(*",0 "%/#$H# #- /#,&/ "%" H"$#% -"%#$**/,# $/,#,*#, #, $)",&/ " $/,$#&/0 (#>,*$*/,# / %#"/(# "-g/%*&/ )# H"", *(/ ,#$#"%*/ #, -" %#/-)$*+, / %#%##,&"(/"-g)," (*>$)-&"( "%" -/ #&)(*",.

A-g),/ (# -/ %/9-#" *,$-)*(/ #, # $)"(#%,*--/ 4/%"%/, "%(# -/ #=á#,# "-*$"(/ #, -" #(*$*/,# ",%*/%# (# -" OEMEPS0

4


5/55


2011

*/ )# 4)#%/, &/"(/ %*,$*"-#, (# -/ C"-#,("%*/M"á&*$/ !!7!!G !!!1!0 (# -/ 9/-#&*,# U, %#&/ á” 0 (# "-g),/ #=á#,# %/9-#"%*/ (# -" A/$*"$*+, N"$*/,"- (#P%/4#/%# (# M"á&*$" :ANPM


6/55


2011

JUSTI-ICACIÓN

L" ; O-**"(" E&"&"- (# M"á&*$" #, E()$"$*+, P%*"%*" S#$),("%*" :OEMEPS< # )," *,*$*"&*" (# -" S#$%#&"%'" (# E()$"$*+,

J"-*$/ )# 9)$" %//#% #- (#"%%/--/ (# $/#,$*" "á&*$" 4"/%#$#% #- g)&/ # *,%2 /% -" "á&*$" #, -/ "-),/ (##()$"$*+, 9á*$" (# -" #,&*("(0 "%" #-#"% #- %#,(**#,&/ #$/-"%0$/,*(#%",(/ -/ %#)-&"(/ (# -" E"-)"$*+, N"$*/,"- (#- L/g%/A$"(2*$/ #, C#,&%/ E$/-"%# :ENLACE,"-*("( (#- %/9-#"%*/ ,/ # #-#$$*/,"% "- / -/ "-),/ á$/#,0 #" 4),$*+, -# $/%%#/,(# "- #="#, (# -" E&"" (#E$)#-" #%á g%"()"- $/, %##$&/ " -/ %/9-#" )# # "-*)#,0

%#*" #-#$$*+, (# -/ */. E- /9?#&*/ # $/"%&*% $/, -/(/$#,0 #- &*/ (# %/9-#" )&*-*8"(/ $// "% (# -" %#"%"$*+, K#,&%#,"*#,&/0 #, #- $"/ (# -" /-**"("K (# -/ "-),/0%#$/*-",(/ %/9-#" (# -/ #=á#,# (# /&%" /-**"("0 )#"),"(/ " -/ "/% (# -" I,%,#&0 #%*&*%á, $%#"% ), 9",$/ (#%/9-#".

E- %/9-#"%*/ #&á #,4/$"(/ 1!! "- #,&%#,"*#,&/ (# -/ "-),/)# "%&*$*"%á, #, -" ; O-**"(" E&"&"- (# M"á&*$" #,E()$"$*+, P%*"%*" S#$),("%*".

6


7/55


2011

INSTRUCTIVO DE PROCEDIMIENTOS PARA LAAPLICACIÓN Y EVALUACIÓN

DE LOS E.ÁMENES

"< E- #="#, )# # "-*$"%á #, $"(" )," (# -" #&"" $/,&" (# $*,$/%/9-#" #- $)"- # /(%á %#/-#% H"&" #, $)"&%/ H/%".

9< C"(" %/9-#" ,(%á ), "-/% (# 7 ),&/0 (*&%*9)*(/ (# -"*g)*#, ",#%"@ (/ ),&/ /% #- %#)-&"(/ $/%%#$&/ (#- %/9-#" H"&" $*,$/ ),&/ á /% -/ %/$#(**#,&/ (# /-)$*+,)&*-*8"(/. E&/ $*,$/ ),&/ # "*g,"%á, (# "$)#%(/ $/, #- g%"(/(# (#"%%/--/ (# -" $/#,$*" "á&*$" :%#/-)$*+, (#%/9-#"0 "%g)#,&"$*+,0 $/),*$"$*+, ",#?/ (# &2$,*$"</&%"(/ #, ) %/$#(**#,&/ (# /-)$*+, &/",(/ $// 9"#-/ $%*%*/ (# #"-)"$*+, (# $"(" %/9-#" (#- #="#,0 */

)# #%á, (#>,*(/ ", (# -" "-*$"$*+,.

$< C// *(#,&*>$"(/% (# $"(" #="#, +-/ # )&*-*8"%á -" $-"# (#"%&*$*"$*+, (#- "-),/ )# -/ %#)#-#0 "*g,"(" #, ) >$H" (#*,$%*$*+,. L/ #"-)"(/%# ,/ (#9#, $/,/$#% -" *(#,&*("( (#-"-),/ )# # #"-".

(< L/ %/9-#" (#- #="#, (#9#%á, #% #"-)"(/ /% ), ?)%"(/*,g%"(/ "- #,/ /% $*,$/ %/4#/%# (#&"$"(/ (# -" "*g,"&)%".

#< C"(" ),/ (# -/ *#9%/ (#- ?)%"(/ #"-)"%á ), á=*/ (# (/

%/9-#" $"(" %/9-#" (#9#%á #% #"-)"(/ "- #,/ /% (/ ?)#$#. P/% #?#-/ * # (*/,# (#- ',*/ (# ?)#$# :5< -/--""/ A0 B0 C0 D E0 -/ $*,$/ %/9-#" (#- #="#, )#(#, #%#"-)"(/ "'@ ?)#8 A@ %/9-#" 1 ?)#8 B@ %/9-#" 3 ?)#8C@ %/9-#" 3 ?)#8 D@ %/9-#" 5 ?)#8 E@ %/9-#" 5 1.

7


8/55

12 cm2

20 cm2 30 cm2

A B

CD


2011

PROBLEMAS

1. U",(/ ), %#-/? (# "%#," (# *# *,)&/ /&%/ (# /,$# *,)&/.C)á- # -" ",#%" á #,$*--" (# #(*% )*,$# *,)&/ ,#$#"%*/

"%" H#%*% ), H)#/

. C*,$/ ,*/ ?)#g", " -" #$/,(*(" #, #- "&*/ (# ) #$)#-"0 $)"&%/# #$/,(#, /&%/ -/ 9)$". E, ## "&*/ H" +-/ 3 #$/,(*0 -/)# (*"%*/ )",@ "&%á (#- á%9/-0 "&%á (#- 9/ (# 9")%" 9"?/ -"9",$" :#, (/,(# $"9#, (/ ,*/g)%". L" á%#" (# &%# (# #--/ /, -" )# #&á,#$%*&" (#,&%/ :,/ # $/,/$# #- á%#" (#- $)"%&/ %#$&á,g)-/


9/55

C

FE

BD

A

H G

A

75º

50ºD

CB


2011

5. E, -" >g)%" # )#&%" ), $)"(%"(/ (# -"(/ 60 (/,(# A B /, -/),&/ #(*/ (# (/ (# ) -"(/. S"9*#,(/ )# #- á%#" (# CFD #-" %$#%" "% (#- á%#" (#- $)"(%"(/0 $)á,&/ *(# CD

6. U, $/&"- #&á --#,/ (# $",*$" (# ! $/-/%# (*&*,&/. A- "8"% # ",

"$",(/ $",*$" (#- $/&"-. C)á- # #- ',*/ ,#%/ (# $",*$" )#(#9#, "$"%# "%" /(#% g"%",&*8"% )# #, -" $/-#$$*+, &/"("H"9%á "- #,/ 1!! $",*$" (#- */ $/-/% :#, #- $/&"- H" á (#1!! $",*$" (# $"(" ),/ (# -/ ! $/-/%#


10/55

H B

I

EF G

C


2011

&%# (# 6 $. C)á,&/ &%*á,g)-/ (*&*,&/ )#(# 4/%"% C"%-/)&*-*8",(/ &%# "-*--/ (# ) $/-#$$*+,

1!. N)#%2 !1! &"%?#&" (#- 1 "- !1! )*&2 ")2--" )# %*,"9",$/, 7. D#)2 /-' " ,)#%"% -" )# # )#("9", /% -&*/)*&2 -" )# %*,"9", #, 3. A- >,"-0 $)á,&" &"%?#&" #)#("%/,

11. T#,#/ )," *$*," $)"(%"(" %/(#"(" (# $2#(0 $// )#&%" #-

(*9)?/. S* P0 0 R S /, -/ ),&/ #(*/ (# -/ -"(/ (#- $)"(%"(/g%",(# $"(" ),/ (# #&/ -"(/ *(# 1! #&%/0 $"-$)-" #- á%#" (#-" *$*,".

1. C)",(/ # #$%*9#, -/ ,#%/ 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,...

$)á- # #- ('g*&/ )# /$)" -" /*$*+, !! N/&"@ #, -" -*&"",%*/% #- ('g*&/ *# :(# 17< /$)" -" /*$*+, 5.

13. L" >g)%" BF # ), $)"(%"(/ AD G $. S* á%#" (#- %#$&á,g)-/EFID # (# 36 $0 C)á- # #- á%#" (#- %#$&á,g)-/ ABCD

10

A

D


11/55


2011

1. U," -',#" (# $"*/,# H" (#$*(*(/ %#*"% $/, ""?# g%"&* " &/(/-" #%/," )# -" )" (# -" $*4%" (#- ,#%/ )# ""%#$# #, )9/-#&/ (# $"*+, #" 1. L" %//$*+, ()%"%á +-/ /% #- # (#M"%8/0 "' )# ",("%/, *%**% 9/-#&/ )# ", (#- 1 "- !!!.

C)á,&/ 9/-#&/ (# 2&/ ("%á, ""?# g%"&* " -/ ))"%*/

15. C/, -/ ('g*&/ 10 0 30 0 5 6. E,$)#,&%" ), ,#%/ (# #* $*4%"abcdef 0 (# &"- ",#%" )# #- ,#%/ (# &%# $*4%" abc #" -&*-/(# 0 #- ,#%/ (# &%# $*4%" bcd #" -&*-/ (# 50 #- ,#%/ (#&%# $*4%" cde #" -&*-/ (# 3 #- ,#%/ (# &%# $*4%" def #"-&*-/ (# 11.

16. C"-$)-" #- á%#" (# -" 8/," *,&"("0 * #- -"(/ (#- $)"(%"(/ "/%

*(# ! $. :$/,*(#%" / 3.1<

17. C)"&%/ "%#?" (# ,/*/ # %#,#, "%" *% " ), $/,$)%/ (# 9"*-#.S"9*#,(/ )#@

• B#"&%*8 9"*-+ $/, E()"%(/.

• A-*$*" 9"*-+ $/, #- ,/*/ (# C-"%".

• F#(#%*$/ 9"*-+ $/, -" ,/*" (# )&"/.

• D",*#-" 9"*-+ $/, #- ,/*/ (# A-*$*".

• )&"/ 9"*-+ $/, -" ,/*" (# E()"%(/.

:"< C/, )*2, 9"*-+ )9#%&/:9< E,$)#,&%" )*2,# /, "%#?" (# ,/*/.

11


12/55


2011

1G. D# ),/ (# -/ 2%&*$# (# ), %#$&á,g)-/ (# G X 13 "% )," -',#"%#$&" " 5Q. A- &/$"% #- -"(/ (#- %#$&á,g)-/0 -" -',#" %#9/&" 4/%",(/,)#"#, ), á,g)-/ (# 5Q0 $// # )#&%" #, -" >g)%"0$"-$)-" #- á%#" (# -" %#g*+, g%*.

1. )2 4%"$$*/,# # (#9#, )*&"% (# -" )" "%" )# -" )" (# -"4%"$$*/,# %#&", #" *g)"- " 1 E,$)#,&%" &/(" -" /*9*-*("(#.

!. E, )," $*%$),4#%#,$*" H#/ *,$%*&/ ), %#$&á,g)-/ #, 2- ),%/9/0 &/",(/ -/ ),&/ #(*/ (# -/ -"(/ (#- %#$&á,g)-/. S* #-(*á#&%/ (#- $'%$)-/ # (# 1! $0 $)á,&/ *(# #- #%'#&%/ (#-%/9/

1. C)á,&/ $)"(%"(/ #, &/&"- H" #, -" >g)%" :(# 1X10 X0 3X30 X05X50 6X60 7X70 GXG


13/55

Fig. 1 Fig. 2 Fig. 3 Fig. 4


2011

. C)á,&/ ),&/ ,(%á -" >g)%" 35

3. E, -" >g)%" -/ ),&/ P0 0 R0 S (**(#, $"(" -"(/ (#- %#$&á,g)-/#, %"8+, 1@.

S* AB 1 $ BC 1 $0 $)á- # #- $/$*#, #,&%# #- á%#" (#-"%"-#-/g%"/ PRS #- á%#" (# ABCD

. A," M"/ # #&á, %#"%&*#,(/ )," 9/-" (# ()-$#. S# -" ", "

%#"%&*% (# -" *g)*#, ",#%"@ P%*#%/ M"/ &/" ), ()-$# A,"&/" (/ M"/ &/" &%# A," &/" $)"&%/. A' )$#*"#,$"(" ),/ &/" ), ()-$# á (#- )# &/+ #- ",%*/%. A," # -"-&*" )# &/" ()-$# -" 9/-" )#(" #,&/,$# "$'". A," &*#,#! ()-$# á )# M"/. C)á,&/ ()-$# $/,,'" -" 9/-"

5. E- &%*á,g)-/ #)*-á%/ g%",(# &*#,# G $ (# #%'#&%/. E- #%'#&%/(#- #g),(/ &%*á,g)-/ # -" *&"( (# %*#%/ #- #%'#&%/ (#-

%$#%/ # -" *&"( (#- #g),(/. C)á- # #- #%'#&%/ (# -" >g)%"/9%#"("

13


14/55


2011

6. U, "%)#+-/g/ H" (#$)9*#%&/ )# )," ",&*g)" $**-*8"$*+, )"9" 5'9/-/ "%" %#%##,&"% -/ ,#%/@ . E&/ '9/-/

$/%%#/,(#, #, "-g, /%(#, " -/ ('g*&/ !0 10 0 3 . P"%" H"$#%

,#%/ á g%",(# )# # ##8"9", " $/9*,"% '9/-/0 /%#?#-/0 #- ,#%/ )# *g)# (#- # #- 1! :#- '9/-/ (#- ),/ " -"*8)*#%(" (#- '9/-/ (#- $#%/


15/55


2011

C)á- # #- á%#" (# -" >g)%" /9%#"(" (# 2%&*$# AFBCDE

. M"%&"0 A-*$*" # E%*" -##%/, ), */ -*9%/ (# #,/ (# 3!!ág*,". M"%&" -#+ *# ág*," #- %*#% ('" #- %#&/ " (*#8ág*," /% ('". A-*$*" -#+ (/ ág*," #- %*#% ('" #- %#&/ "/,$# ág*," /% ('". E%*" -#+ $*,$/ ág*," #- %*#% ('" #- %#&/" ,)## ág*," /% ('". C)á,&" ág*," &*#,# #- -*9%/

3!. J)", &*#,# 1!! *#8" $)"(%"(" (# 1 $ (# -"(/. U&*-*8",(/ &/("-" *#8" "%" ), %#$&á,g)-/. C)á,&/ %#$&á,g)-/ (*&*,&/ )#(#

"%"% I,(*$" -" -/,g*&)(# (# ) -"(/.

C)á-# (# #&/ %#$&á,g)-/ # )#(#, "%&*% #, $)"(%"(/ (# $(# -"(/

31. M* %#-/? (*g*&"- "%$" $*$-/ (# -" 1@!! " -" 1@5. D)%", $)á,&/*,)&/ "- ('" )#(/ #% #, #- %#-/? ), $)"(%"(/ #%4#$&/0 *g,/%",(/

-/ (/ ),&*&/ )# #"%", -/ *,)&/ (# -" H/%":P/% #?#-/0 ), ,#%/ $)"(%"(/ #%4#$&/ # 10 /%)# 1

1


16/55


2011

16


17/55

11 7

117

11

11


2011

SOLUCIONES

1. U," 4/%" )#(# #% -" *g)*#,@

O&%" /-)$*+,@

17

S# P/,#, " 4),$*/,"% -/ %#-/?# (# "%#," "- */&*#/.C)",(/ # "$"9" -" "%#," (#- %#-/?(# 7 *,)&/0 #, #- (# 11 ",)#(", *,)&/. A' )# ")' *,*$*" #- $/,/ "%" -/ 15 *,)&/

C)",(/ # %*," (# "$*"% -" "%#,"(#- %#-/? (# 11 *,)&/ H"9%á,&%",$)%%*(/ -/ %*#%/ *,)&/(# $/$$*+, (#- H)#/ "%"

$/-#&"% -/ 15 )# /, ,#$#"%*/0+-/ /-"/ # */ %#-/? "%")# # "$'# *(" -/ 11 *,)&/


18/55

Ana

Beto

Carlos

David

Ana

Ana

Beto

Beto

Carlos

Carlos

David

David

David

Ana

Beto

Carlos

Carlos

Carlos

Carlos

Carlos

Carlos

Beto

Beto

Ana

Beto

Ana

David

Beto

Beto

David

David

David

David

Beto

David

Carlos

Ana

Ana

Ana

Ana


2011

. U," /*9-# 4/%" # -" )# *g)#@S# &*#,#, 3 #$/,(* /*9-# :#- á%9/-0 #- 9/ -" 9",$"< (/,(#)#(#, #$/,(#%# ,*/0 #, #- #,,(*(/ (# )# #, ),/ (# #--/

:-" 9",$"< $"9#, (/ ,*/. D#&%á (#- á%9/- )#(# #$/,(#%#$)"-)*#%" (# -/ ,*/0 /% -/ )# # )#(# /$)"%# (# ",#%" (*4#%#,.

U," #8 /$)"(/ #- á%9/- (# #" ",#%" /*9-#0 "&%á (#- 9/+-/ )#(#, #$/,(#%# $)"-)*#%" (# -/ 3 ,*/ %#&",0 /% -/)# $/,*(#%",(/ #&" 3 4/%" (# /$)"% #- 9/ "%" $"(" )," (#-" ",#%" (# /$)"% #- á%9/-0 # &*#,#, 1 /*9-# 4/%"(*4#%#, (# /$)"% #- á%9/- #- 9/.

F*,"-#, &/",(/ #, $)#,&" )# (/ ,*/ #&á, #$/,(*(/),/ (#&%á (#- á%9/- /&%/ "&%á (#- 9/ #, -" 9",$" +-/ )#(#,#$/,(#%# -/ (/ ,*/ %#&",. A' )# -" 9",$" )#(# /$)"%#(# 1 /-" ",#%". P/% -/ )# -" 4/%" (*&*,&" (# #$/,(#%#(#&%á (#- á%9/-0 "&%á (#- 9/ (#9"?/ (# -" 9",$" /,@ 3XX1 1.S* A,"0 B#&/0 C"%-/ D"*( #- *g)*#, (*"g%"" (# á%9/- ,/#%* (#%*,"% 2&" ",#%"@

18


19/55

12 cm2

20 cm2 30 cm2

A B

CDa b

c

d


2011

3. U," 4/%" (# %#/-#% )#(# #% -" *g)*#,@

N/. D#&%á (#-á%9/- A&%á (#-9/ D#9"?/ (# -" 9",$"1 A," B#&/ C"%-/ D"*( A," C"%-/ B#&/ D"*(3 A," D"*( B#&/ C"%-/ B#&/ A," C"%-/ D"*(5 B#&/ C"%-/ A," D"*(6 B#&/ D"*( A," C"%-/7 C"%-/ A," B#&/ D"*(G C"%-/ B#&/ A," D"*( C"%-/ D"*( A," B#&/

1! D"*( A," B#&/ C"%-/11 D"*( B#&/ A," C"%-/1 D"*( C"%-/ A," B#&/

.D#*g,#/ $/, a0 b0 c d -" (*#,*/,# (# -/ %#$&á,g)-/ #,)# #&á (**(*(/ #- %#$&á,g)-/ ABCD (#- $)"- # )*#%# $"-$)-"% #-á%#".

P"%" #--/ H" )# *,#&*g"% -" #(*(" a, b, c d "9*#,(/ )# a x d ! $0 b x d 3! $0 a x c = 12 cm2 b x c

C// a x d ! $0 a )#(# #% !0 10 1!0 0 5 + d )#(# #% 10!0 0 1!0 0 + 50 /%)# ! X 1 1 X ! 1! X X 1! 5 X X 5 !.

P#%/ $// a x c 1 $0 (# -/ "-/%# ",%*/%# a +-/ /(%'" #% + 0 $/%%#/,(*#,(/ " d "-#% 1! + 5 /%)# X 1! ! X5! " c 6 + 3 /%)# 6 X 1 X 3 1.

19


20/55


2011

P#%/ $// "(#á b x d 3! $0 b +-/ )#(# "-#% 3 + 6$/%%#/,(*#, "-/%# (# d 1! 5 (# &"- ",#%" )# # $)-")# 3 X 1! 3! + 6 X 5 3!. A' )# H" (/ /*9*-*("(# "%" -"

#(*(" (# a0 b0 c d@1< a $ b 3 $ c 6 $ d 1! $.

E, # $"/ -" 9"# (#- %#$&á,g)-/ ABCD #%'" " 9 $ 3 $ 5 $ ) "-&)%" $ ( 6 $ 1! $ 16 $0 /% -/ )# #- á%#"(#- %#$&á,g)-/ #%'" *g)"- " 5 $ X 16 $ G! $.

< a $ b 6 $ c 3 $ d 5 $.

E, #&" #g),(" *&)"$*+, -" 9"# (#- %#$&á,g)-/ ABCD #%'" a + b =

4 cm + 6 cm = 1 cm ) "-&)%" c + d = ! cm + " cm = # cm0 /% -/)# #- á%#" (#- %#$&á,g)-/ #%'" *g)"- " 1! $ X G $ G! $0 )## #- */ %#)-&"(/ /9,*(/ ",%*/%#,.

P/% -/ &",&/ #- á%#" (#- %#$&á,g)-/ ABCD # G! $.

O&%" /*9-# 4/%" (# "9/%("% -" /-)$*+, (#- %/9-#" # -"*g)*#,@

S* (#*g,"/ $/, x #- á%#" (#- %#$&á,g)-/ (# (*#,*/,# 9 = (0 #,#/ )#

! $

" = ( 3! $

9 = ( 1 $

" = $ x 9 =$

O9#%#/ )#

D# ")' )# #,&/,$# .

A' )# .

D# (/,(# #- á%#" (#- %#$&á,g)-/ ABCD # *g)"- "

! $ 3! $ 1 $ 1G $ G! $.

5. U," ",#%" (# %#/-#% #- %/9-#" #@

20


21/55


2011

S# *(# $"-$)-"% -" )" (# -" $*4%" (#- ,#%/ N1! .

0 # (#$*% ), 1

#g)*(/ (# $#%/0 ,#%/ "- )# H" )# %#&"%-# (#)2)"% -" $*4%" (#- %#)-&"(/ /9,*(/.

P/(#/ *-*>$"% #- %/9-#" "%" #% * /9,#/ "-g),"%#g)-"%*("( )# ,/ #%*&" $"-$)-"% %á*("#, -" )" /-*$*&"(".

O9#%#/ )# #, &/(/ -/ $"/ #, #- %#)-&"(/ %*," #, !G*,"%*"9-##, " -" *8)*#%(" ""%#$# #- &",&" #$# $// #-,#%/ (#- #=/,#, :)# # *g)"- " -" $",&*("( (# $#%/ )# ",(#)2 (#- 1< )# &*#,# -" /,$*" (# 1! (**,)*(/ #, (/.

P/% #?#-/ $)",(/ # &%"&" (#

0 #- %#)-&"(/ %*," #, !G " -" *8)*#%("

&",&/ ,)## $// #- #=/,#, (# -" /,$*" (# 1! :#, # $"/0< #,/ 0 # (#$*%0 K ! #$#0 /% -/ )# #- ,/ #&á #, #-%#)-&"(/.

E, #- $"/ (# 0

#- %#)-&"(/ %*," &"9*2, #, !G " -" *8)*#%(" ""%#$#, &",&/,)## $// #- #=/,#, (# -" /,$*" (# 1! :#, # $"/0 3<#,/ 0 # (#$*%0 3 K 1 #$#0 /% -/ )# #- #&á 1 #8 #, #-%#)-&"(/.

C)",(/ # &%"&" (#

0 #- %#)-&"(/ ,)#"#, %*," #, !G

" -" *8)*#%(" ""%#$#, &",&/ ,)## $// #- #=/,#, (# -"/,$*" (# 1! :#, # $"/0 < #,/ 0 # (#$*%0 K #$#0/% -/ )# #- # #,$)#,&%" #$# #, #- %#)-&"(/.

21


22/55


2011

L/ */ )$#(# $/,

@ #- %#)-&"(/ %*," /&%" #8 #,

!G " -" *8)*#%(" ""%#$#, &",&/ ,)## $// #- #=/,#, (# -"

/,$*" (# 1! :#, # $"/0 5< #,/ 0 # (#$*%0 5 K 3 #$#0/% -/ )# #- # %#* 3 #$# #, #- %#)-&"(/.

D# -/ ",%*/% /(#/ $/,$-)*% )# #, #- $"/ (# N1! #,(%á )#

0 #, (/,(# #- %#)-&"(/ %*," *g)"- #, !G "

-" *8)*#%(" ""%#$#, &",&/ ,)## $// #- #=/,#, (# -"/,$*" (# 1! :#, # $"/0 < #,/ 0 # (#$*%0 K !#$#0 /% -/ )# #- # %#* ! #$# #, #- %#)-&"(/.

A' )# -" )" (# -" $*4%" (#- ,#%/ N1! # )#(#$"-$)-"% (# #&" 4/%"@ ! X ! G G1! ! G G1G.

22


23/55

C

FE

BD

A

H G


2011

6. U," /*9*-*("( (# %#/-#% #- %/9-#" # 2&"@

7. U," 4/%" (# #,"% -" /-)$*+, (#- %/9-#"# $// *g)#@S* "$"/ "- "8"% 1!! $",*$"0 )," /*9*-*("( # )# -" $*#,4)#%", (#- */ $/-/%0 #%/ ,/ # #g)%/0 /%)# &"9*2, /(%'"

%##,&"%# -" /*9*-*("( #=&%#" (# #=&%"#% #="$&"#, 5 $",*$"(# $"(" ),/ (# -/ ! $/-/%#. S* $/,*(#%"/ #&" -&*" *&)"$*+,0/(#/ *"g*,"% #, #=&%"#% "- "8"% X ! 1G! $",*$"#,",(/ #, )# -" /*9*-*("( #=&%#" "%" -/ )# %#,(#/:/9,#% "- #,/ 1!! $",*$" (#- */ $/-/%< 4)#%" )#&)*2%"/ $",*$" (# $"(" ),/ (# -/ ! $/-/%# ">%"% $/,&/(" #g)%*("( )# #, -" *g)*#, #=&%"$$*+, H"9%á "- #,/ 1!!$",*$" (# ), */ $/-/%. E,&/,$# #- ',*/ ,#%/ (# $",*$")# (#9#, "$"%# "%" /(#% g"%",&*8"% )# #, -" $/-#$$*+, &/"("H"9%á "- #,/ 1!! $",*$" (#- */ $/-/% # X ! 1 1981.

G. U," ",#%" (# %#/-#% #- %/9-#" # "'@S# &*#,#, !!6 ('g*&/ #$%*&/ #, -',#" #- -&*/ (# #--/ " -"(#%#$H" # #- G. A(#á $"(" "% (# ('g*&/ "("$#, " -"*8)*#%(" # (***9-# #,&%# 17 + #,&%# 3. S# (##" $/,/$#% -/%*#%/ 5 ('g*&/ (# -" -*&".

23

E- á%#" (#- $)"(%"(/ EF # 6 X 6 36 ),*("(# $)"(%"(". C// #-á%#" (#- %/9/*(# CFD0 # -" %$#%" "% (# -" (#- $)"(%"(/#,&/,$# # (# 36W31 ),*("(# $)"(%"(". AH/%" 9*#,0 #- %/9/*(#

CFD #&á $/)#&/ /% -/ &%*á,g)-/ CFD DC )# /,$/,g%)#, /% ,#% -" *" 9"# :CD< -" *" "-&)%" :3),*("(# " )# A B /, ),&/ #(*/ (# -/ -"(/ (#- $)"(%"(/ 2&/ *(#, 6 ),*("(#


24/55


2011

C// $"(" "% (# ('g*&/ "("$#, " -" *8)*#%(" # (***9-# /9*#, #,&%# 17 + #,&%# 3 /(#/ #$%*9*% -/ -&*-/ (# 17 3 (# $*4%" #, )," &"9-".

MLTIPLOS DE17

MLTIPLOS DE3

17 3

3 6

51 6

6G

G5

< P/(#/ %"8/,"% (# -" *" ",#%" "%" *,#&*g"% #- ('g*&/",%*/% " 6 #,&/,$# ,/ *,%#" ), -&*-/ (# (/ $*4%" (#17 + 3 %*,"(/ #, 6. E, -" &"9-" #,$/,&%"/ )# +-/ #- 60)# # -&*-/ :/ (***9-# #,&%#< (# 30 %*," #, 6. D# ")' )##- ",#,-&*/ ('g*&/ (# -/ !!6 ('g*&/ # .

3< AH/%" ,/ $/,$#,&%"/ #, 9)$"% ), -&*-/ (# (/ $*4%" (#

17 + 3 %*,"(/ #, . O9#%"/ )# #, -" &"9-" ,*$"#,#- 30 )# # -&*-/ :/ (***9-# #,&%#< (# 170 %*," #, .E,&/,$# #- ",%*/% ('g*&/ " -" *8)*#%(" (# # #- .

< D# 4/%" ",á-/g" *,("g"/ #, -" &"9-" "%" #,$/,&%"% ),-&*-/ (# (/ $*4%" (# 17 + 3 %*,"(/ #, 3 /9#%"/+-/ #- 30 )# # -&*-/ :/ (***9-# #,&%#< (# 30 #- &*#,# #&"$"%"$%'&*$". A' )# #- ",%*/% ('g*&/ " -" *8)*#%(" (# 3 # #- 2.

5< R"8/,",(/ *g)"-0 9)$"/ #&" #8 #, -" &"9-" ), -&*-/ (#(/ $*4%" (# 17 + 3 %*,"(/ #, 0 (á,(/,/ $)#,&" (# )#+-/ #- 0 )# # -&*-/ :/ (***9-# #,&%#< (# 30 &*#,# #&"$)"-*("(0 /% -/ )# #- ",%*/% ('g*&/ " -" *8)*#%(" (# # #- .

6< I,#&*g"/ ,)#"#, #, -" &"9-"0 #%/ "H/%" ), -&*-/(# (/ $*4%" (# 17 + 3 %*,"(/ #, /9#%",(/ )#

24

1< P"%" "#%*g)"% $)á- # #-('g*&/ ",%*/% " G0 /9#%#/)# #- "% 4/%"(/ /% #('g*&/ #- G (#9# #% (***9-##,&%# 17 + 30 # (#$*%09)$"/ ), -&*-/ (# (/$*4%" (# $)"-)*#%" (# #--/%*,"(/ #, G. E, -" &"9-" ,/("/ $)#,&" )# #&/ +-//$)%%# $/, 6G )# # -&*-/ :/

(***9-# #,&%#< (# 17. P/% -/


25/55

A

75º

50ºD

CB


2011

,*$"#, #- 60 )# # -&*-/ :/ (***9-# #,&%#< (# 30 $)-#$/, #&" $/,(*$*+,0 /% -/ )# #- ",%*/% ('g*&/ " -" *8)*#%(" (# # #- 3.

7< E, # ),&/ ,/ ("/ $)#,&" (# )# %/$#(*#,(/ $//H#/ #,*(/ H"$*#,(/0 ", (# 6 ,(%'" )# #&"% ,)#"#,#- ", (#- 0 #- 3 ", (#- 30 #- ", (#- 0 #- (# ")' #, "(#-", -" #$)#,$*" 36 # %#* H"&" $/-#&"% -/!!6 ('g*&/. E (#$*%0 ",$#(*#,(/ "- G0 H" !!5 ('g*&/ #,#$)#,$*" %##&*(" (# 360 #&/ #0 #="$&"#, !1#$)#,$*" 4/%"(" /% -/ ('g*&/ 0 0 30 0 6. E,&/,$# -/%*#%/ $*,$/ ('g*&/ (# -" -*&" /, ?)&"#, 0 0 30 0 6.

. U," /*9-# /-)$*+, # -" *g)*#,@E, -" >g)%" /9#%"/ )# #- á,g)-/ $)" #(*(" *,#&*g"/0 :

# *g)"- " -" )" (# -/ á,g)-/ CAB CAD0 # (#$*%0

:1


26/55

1 c 2 c 3 c 4 c 5 c 6 c


2011

C/, ), %"8/,"*#,&/ /(#/ $/,$-)*% )# #- &%*á,g)-/ ADC

&"9*2, # *+$#-# :ADDC< )# -/ á,g)-/ CAD DAC &*#,#, -"*" #(*(" ("(/ )# &"9*2, #, #- &%*á,g)-/ ADC -" )" (#-" #(*(" (# ) á,g)-/ *,%*/%# # 1G!Q0 ,#/ )#@

:


27/55


2011

:60 60 1


28/55


2011

P/% -/ &",&/ (#- 1 "- 1!!! /, $*#,0 /% -/ )# (#- 1 "- !!! /,(/$*#,&" &"%?#&"0 á -" &"%?#&" ,#%/ !!70 (", ), &/&"- (# !1&"%?#&".

P/% -" )# )#(", !1! K !1 1G! &"%?#&".

L" &"%?#&" %*,"(" #, 3 (#- 1 "- 1!! /, (*#8@ 30 130 30 330 30530 630 730 G3 3.P/% -/ &",&/ (#- 1 "- 1!!! /, $*#,0 /% -/ )# (#- 1 "- 1G!! /, 1G!&"%?#&"0 á -" &"%?#&" ,#%/ 1G!30 (", ), &/&"- (# 1G1 ), &"%?#&".

1. U," /*9-# 4/%" (# %#/-#% $/,* #, *"g*,"% -" >g)%"/%*g*,"- (#- $)"(%"(/ "/%0 (#$/)#&" #, ! #)#/&%*á,g)-/ %#$&á,g)-/ *g)"-# :$/,g%)#,< #,&%# '0 (# &"- ",#%")# #- $)"(%"(/ (# -" *$*," #&"%'" $/)#&/ " ) #8 /% $)"&%/(# #&/ &%*á,g)-/0 # (#$*% -" )*,&" "% (# -/ )# &*#,# #-$)"(%"(/ "/%. C// #- á%#" (#- $)"(%"(/ "/% # 1! X 1!

1!!

0 #- á%#" (#- $)"(%"(/ (# -" *$*," %#)-&" #% 1!!

5 ! .

P/(#/ %#/-#% $// *g)#@ D#- 1 "- :,#%/ (# )," $*4%"< /, ('g*&/0 (#- 1! "- :,#%/ (# $*4%"</, ! = 1G! ('g*&/. L-#"/ 1G "%" #- !1! 4"-&", 1G1('g*&/. C// *g)#, -/ ,#%/ (# 3 $*4%"0 1G1 #,&%# 3 # *g)"- "6!7 ,#%/ #="$&"#,. D#- 1!! "- 6 ", 6!! ,#%/ :1G!!('g*&/< $/, -/ )# # --#"9",0 ", 1G!! 1G 1G ('g*&/ #, #-6. C/, #- 7!5 ", !!7 :1G 6 = 3< >,"-#, $/, #- 403 ",-/ !1! ('g*&/ #="$&/0 /% -/ )# #- ('g*&/ )# /$)" -" /*$*+, !1!# #- 3.

28


29/55

H BA

I

EF G

DC


2011

O&%" "%*", #, -" /-)$*+, #@

V#"/ )# H"&" #- 170 H" ,#%/ (# 1 ('g*&/ G (# #- ('g*&/

7 /$)" -" /*$*+, 50 #%/ # ,/ %#g),&" /% #- ('g*&/ )# /$)" -"

/*$*+, !1!.

E&/ )*#%# (#$*% )# $/,&*,)",(/ -" ,)#%"$*+, (#9#/ $)9%*% !1!

K 5 1G5 -)g"%# á. D#- 1G "- H" G ,#%/ (# (/ ('g*&/

$/, -/ $)"-# # $)9%*%á, /&%/ G X 16 -)g"%#0 )#(",(/ /%

$)9%*% 1G5 K 16 1G1. D#- 1!! H"&" #- H" !! ,#%/ (#

&%# ('g*&/0 -/ $)"-# /$)"%'", !! X 3 7!! -)g"%#0 #%/ +-/

4"-&", /% $)9%*% 1G10 "' )# /(#/ (**(*% 1G1 #,&%# 3 "%" #%

$)á,&/ (# #&/ !! ,#%/ /, ,#$#"%*/@ $// 6!7 X 3 1G10

-/ *g)*#, 6!7 ,#%/ (#)2 (#- :H"&" #- 7!6< $)9%*%á, -/

*g)*#, 1G1 -)g"%# 4"-&",0 *#,(/ #- ('g*&/ 3 (#- 7!3 #- )#

/$)" -" /*$*+, 5 16 1G1 !1!

13. P/(#/ %#/-#%-/ "'@

D# -" >g)%" /(#/ /9#%"% )#@

Á%#" (#- R#$&á,g)-/ EFID Á%#" (#- &%*á,g)-/ AFE Á%#" (#- &%*á,g)-/FCI Á%#" (#- C)"(%"(/ BF Á%#" (#- &%*á,g)-/ AF Á%#" (#-&%*á,g)-/ FC :1


30/55


2011

C BC K B %#)-&" )# C G $ K 6 $ $

A(#á@Á%#" (#- R#$&á,g)-/ EFID DI X C 36 $ # (#$*%0 DI X $ 36 $ DI 36 $ $ DI 1G $

30


31/55


2011

T"9*2,@ IC F B 6 $ DC DI IC0 # (#$*%0 DC 1G $ 6 $ $Á%#" (#- R#$&á,g)-/ ABCD DC X BC $ X G $ 1$

1. U," ",#" (# %#/-#%-/ #%'"@P"%" (#%*,"% $)á,&/ (# -/ !!! 9/-#&/ ("%á, ""?# g%"&* "-/ ))"%*/0 ,#$#*&"/ #,$/,&%"% &/(/ -/ ,#%/ (#- 1 "- !!!)# (", 1 "- )"% ) ('g*&/.

P/(#/ %/$#(#% "'@S* /%(#,"/ -/ !!! 9/-#&/ (#- ,#%/ #,/% "- "/%0 (#- 1 "- H" ,#%/ (# 1 ('g*&/ (# 2&/ #- "/% # #- /% -/ )#,*,g),/ (# #--/ #%á %#*"(/. L/ *g)*#, ! ,#%/ &*#,#, ('g*&/ *#,(/ #- á g%",(# . C// -" )" (# ) ('g*&/ # 1G0 ,*,g),/ (# 2&/ &"/$/ #%á %#*"(/.

D#- 1!! "- # &*#,#, !! ,#%/ (# 3 ('g*&/. P"%" "9#%$)á,&/ (# #--/ ("%á, ""?# g%"&* " ) /%&"(/%#0 ,#/ )#(#%*,"% $)á,&" %," (# ('g*&/ )", 1.

L" %," (# ('g*&/ )# )", 1 /,@

30 G0 570 5GG0 660 67G0777.30 0 $/, -" )# # 4/%", 3 ,#%/ (*&*,&/@ 30 3 3.0 G0 $/, -" )# # 4/%", 6 ,#%/ (*&*,&/@ G0 G0 G0G0 G G.

50 70 $/, -" )# # 4/%", 6 ,#%/ (*&*,&/@ 570 570 750 75057 75.50G0G $/, -" )# # 4/%", 3 ,#%/@ 5GG0 G5G GG5.6060 $/, -" )# # 4/%", 3 ,#%/@ 660 66 66.60 70 G $/, -" )# # 4/%", 6 ,#%/ (*&*,&/@ 67G0 6G70 76G0 7G60G67 G76.70 70 7 $/, -" )# # 4/%" ,*$"#, #- ,#%/ 777.E (#$*%0 (# -/ ,/#$*#,&/ 9/-#&/ )# H" (#- 1!! "- 0 3 6 6 3 3 6 1 G0 ,/ "g"%á, 9/-#&/.

F*,"-#, (#- 1!!! "- !!!0 ,#/ 1!!1 ,#%/ (# ('g*&/.

D#%*,"/ #,&/,$# -" $)"%&" )# )", 1@10 0 0 $/, -" )# # 4/%", 3 ,#%/ (*&*,&/ #,/%# /*g)"-# )# !!!@ 10 1 1.10 30 G0 $/, -" )# # 4/%", 6 ,#%/ (*&*,&/ #,/%# /*g)"-# )# !!!@ 13G0 13G0 1G30 1G30 13G 1G3.10 0 G0 G $/, -" )# # 4/%", 3 ,#%/ (*&*,&/ #,/%# /*g)"-# )# !!!@ 1GG0 1GG 1GG.10 0 70 $/, -" )# # 4/%", 6 ,#%/ (*&*,&/ #,/%# /*g)"-# )# !!!@ 170 170 170 170 17 17.

31


32/55


2011

10 50 60 $/, -" )# # 4/%", 6 ,#%/ (*&*,&/ #,/%# /*g)"-# )# !!!@1560 1560 1650 1650 156 165.10 50 70 G $/, -" )# # 4/%", 6 ,#%/ (*&*,&/ #,/%# /

*g)"-# )# !!!@ 157G0 15G70 175G0 17G50 1G57 1G75.10 50 G0 G $/, -" )# # 4/%", 3 ,#%/ (*&*,&/ #,/%# /*g)"-# )# !!!@ 15GG0 1G5G 1GG5.

10 60 70 7 $/, -" )# # 4/%", 3 ,#%/ (*&*,&/ #,/%# /*g)"-# )# !!!@ 16770 1767 1776.

E (#$*%0 (# -/ 1!!1 9/-#&/ )# H" (#- "- !!!0 3 6 3 6 6 6 3 3 360 ,/ "g"%á, ""?#.

D# -/ ",%*/% /(#/ $/,$-)*% )# (# -/ !!! 9/-#&/ +-/ G

36 6 /, -/ %#*"(/.O&%" 4/%" (# #,"% -" /-)$*+, # -" *g)*#,@D#- 1 "- 3!!0 ,/ H" ,#%/ $)/ ('g*&/ )", 1.D#- 3!1 "- !!0 H" 1 ,#%/@ 3.D#- !1 "- 5!!0 H" ,#%/@ G G.D#- 5!1 "- 6!!0 H" 3 ,#%/@ 570 57 5GG.

C/,&*,)",(/ (# -" *" ",#%"0 (#- !1 "- 1!!! H" 7 (# #&/,#%/@ 30 G0 570 660 750 G 3.

E (#$*%0 (#- 1 "- 1!!!0 #, &/&"- # &*#,#, 1 3 5 6 7 G ,#%/ $/, #&" $"%"$%'&*$".

D#- 1!!1 "- 1!! ,/ ,#/ ,#%/ $)/ ('g*&/ )", 1.D#- 1!1 "- 13!!0 H" 1 ,#%/@ 1.D#- 13!1 "- 1!!0 H" ,#%/@ 13G 13G.D#- 1!1 "- 15!!0 H" 3 ,#%/@ 170 1GG 17.

C/,&*,)",(/ (# *g)"- 4/%"0 (#- 1!1 "- !!! H" G (# #&/,#%/@ 10 13G0 170 1560 1650 170 1G3 1.

E&/ #0 (#- 1!!1 "- !!!0 #, &/&"- # &*#,#, 1 3 5 6 7 G 36 ,#%/ $/, #&" $"%"$%'&*$".

A' )# (# -/ !!! 9/-#&/ G 36 6 ,/ "g"%á, ""?#.

15. P/(%'"/ %#/-#%-/ "'@E- ,#%/ (# &%# $*4%" bcd (#9# #% -&*-/ (# 50 /% -/ )# ( 50" )# ), -&*-/ %*," #, ! + #, 50 #%/ $// abcdef # 4/%",/-"#, $/, 10 0 30 0 5 60 -" ,*$" /*9*-*("( "%" ( # 5.

32


33/55

Q

P

R


2011

AH/%" 9*#,0 5#4 (#9# #% -&*-/ (# 110 # (#$*%0 # &%"&" (# ),-&*-/ (# 11 #,&%# 5!! 6!!@ 5!60 5170 5G0 530 55!0 5610 5705G3 5. D# 2&/ +-/ 561 # -")*9-# " )# &/(/ -/ (#á

$/,&*#,#, "-g, ('g*&/ (*4#%#, (#- $/,?),&/ %/)#&/@ 10 0 30 0 5 6. D# #&" ",#%"0 # 6 4 1.

P/% /&%" "% $56 (#9# #% -&*-/ (# 30 #&/ #0 -" )" $ 5 60 &*#,# )# #% -&*-/ (# 3. C// $ ,*$"#, )#(# #% 0 3 0 $ 0 /%)# 5 6 15 )# # -&*-/ (# 30 #, &",&/ )# 5 6 3 5 6 ,/ -/ /,.

F*,"-#, $// "9 (#9# #% -&*-/ (# 0 9 &*#,# )# #%&"9*2, -&*-/ (# 0 "' )# 9 /%)# ' # -&*-/ (# *#,&%" )# 30 ,/. Y $// +-/ )#(" (*/,*9-# #- ('g*&/ 30 2(#9# $/%%#/,(#% ,#$#"%*"#, "- "-/% (# ".

E,&/,$# #- ,#%/ abcdef 9)$"(/ #@ 3561.

16. U," /-)$*+, /(%'" #%@

17.U," 4/%"

(#%#/-#%

#-%/9-#")#(# #%

2&"@"$#/ (/ &"9-" (# (/9-# #,&%"("@ -" %*#%" "%" *,#&*g"% -"/*9-# "%#?" (# 9"*-# -" #g),(" "%" (#%*,"% -" /*9-#"%#?" (# ,/*/0 -" "/ " *% --#,",(/ $/, ), SZ / $/, ), NO ""%&*% (# -" *,4/%"$*/,# )# (", -/ 5 #,),$*"(/ #=%#"(/ #,#- %/9-#". E&á $-"%/ )# )," #8 )# #&2, $/-#&"0 -" %*#%"&"9-" ,/ #%*&*%á $/,&"% -" %#g),&" (#- *,$*/ "< -" #g),("0

-" (#- *,$*/ 9%" )# B#"&%*8 E()"%(/ 4)#%/, "%#?" (#9"*-# /% -/ )#0 /% -/ &",&/0 B#"&%*8 ,/ 9"*-+ $/, F#(#%*$/0 $/,)&"/ ,* $/, )9#%&/ (# *g)"- ",#%" E()"%(/ ,/ )(/ H"9#%9"*-"(/ $/, A-*$*"0 $/, C-"%" ,* $/, D",*#-" -" &"9-" (# "%#?" (#9"*-# )#(" "'@

33

O9#%#/ )# -" /$&"" "% (# -" 8/," *,&"("%#)-&" (# %#&"% " -" $)"%&" "% (#- $'%$)-/ #- á%#"(#- &%*á,g)-/ PR.E- á%#" (# -" $)"%&" "% (#- $'%$)-/ #

á%#" (#- &%*á,g)-/ PR %#)-&" #% #, -" >g)%" -" /$&"" "% (#-#" (#- $)"(%"(/ (# -"(/ ! $0 # (#$*%0 :! X !< G !! G $.

' )# -" /$&"" "% (# -" 8/," *,&"(" # *g)"- 7G.5 K 5! G.5.


34/55


2011

E- )*,&/ #,),$*"(/ ">%" )# )&"/ 9"*-+ $/, -" ,/*" (# E()"%(/0"' )# B#"&%*8 ,/ )#(# #% ,/*" (# E()"%(/ /% H"9#% 9"*-"(/ $/, 2- ,/ $/, )&"/. E- #g),(/ #,),$*"(/ (*$# )# A-*$*" 9"*-+ $/, #-,/*/ (# C-"%"0 #%/ A-*$*" ,/ 9"*-+ $/, E()"%(/ "' )# C-"%" ,/ )#(##% ,/*" (# E()"%(/. E- $)"%&/ #,),$*"(/ ">%" )# D",*#-" 9"*-+ $/,

#- ,/*/ (# A-*$*"0 #%/ D",*#-" ,/ 9"*-+ $/, E()"%(/0 -)#g/ A-*$*"&"/$/ )#(# #% ,/*" (# E()"%(/. D# -/ ",%*/% # &*#,# )# -",/*" (# E()"%(/ ,/ )#(# #% á )# D",*#-" #,&/,$# ,* B#"&%*80 ,*A-*$*"0 ,* C-"%" )#(#, #% ,/*" (# E()"%(/0 (# -" *" 4/%" )# ,*F#(#%*$/0 ,* )&"/0 ,* )9#%&/ )#(#, #% ,/*/ (# D",*#-". A' )#-" &"9-" (# "%#?" (# ,/*/ )#(" "'@

D# ,)#/ #, #- )*,&/ #,),$*"(/ # "#g)%" )# )&"/ 9"*-+ $/, -",/*" (# E()"%(/ #,&/,$# )&"/ 9"*-+ $/, D",*#-" (# #&/ #(#%#,(# )# )&"/ ,/ 9"*-+ ,* $/, A-*$*"0 ,* $/, C-"%" )# D",*#-"&"/$/ -/ H*8/ $/, F#(#%*$/0 ,* $/, )9#%&/ #, #- $)"%&/ #,),$*"(/(*$# )# D",*#-" 9"*-+ $/, #- ,/*/ (# A-*$*" #&/ *g,*>$" )# )&"/# ,/*/ (# A-*$*" #,&/,$# )&"/ ,/ # ,/*/ ,* (# B#"&%*80 ,* (#C-"%" $// &"/$/ A-*$*" # ,/*" (# F#(#%*$/0 ,* (# )9#%&/. L)#g/-" (/ &"9-" )#(",@

E, #- %$#% #,),$*"(/ # "##%" )# F#(#%*$/ 9"*-+ $/, -" ,/*" (#)&"/0 #%/ A-*$*" # -" ,/*" (# )&"/0 #,&/,$# %#)-&" )#

34

PAREJAS DE BAILE B#"&%*8 A-*$*" C-"%" D",*#-"E()"%(/ SZ N/ N/ N/F#(#%*$/ N/)&"/ N/

)9#%&/ N/

P"%#?" (# ,/*/ B#"&%*8 A-*$*" C-"%" D",*#-"E()"%(/ N/ N/ N/ S' F#(#%*$/ N/

)&"/ N/)9#%&/ N/

P"%#?" (# 9"*-# B#"&%*8 A-*$*" C-"%" D",*#-"E()"%(/ S* N/ N/ N/F#(#%*$/ N/ N/)&"/ N/ N/ N/ S' )9#%&/ N/ N/

P"%#?" (# ,/*/ B#"&%*8 A-*$*" C-"%" D",*#-"E()"%(/ N/ N/ N/ S' F#(#%*$/ N/ N/)&"/ N/ S' N/ N/)9#%&/ N/ N/


35/55


2011

F#(#%*$/ 9"*-+ %#$*"#, $/, A-*$*" ,/ -/ H*8/ $/, C-"%"0 "' $//A-*$*" &"/$/ -/ )(/ H"9#% H#$H/ $/, )9#%&/. A(#á #, #-#g),(/ #,),$*"(/ # ">%" )# A-*$*" 9"*-+ $/, #- ,/*/ (# C-"%" $// A-*$*" 9"*-+ $/, F#(#%*$/0 #,&/,$# F#(#%*$/ # #- ,/*/ (# C-"%"

$/, -/ )# F#(#%*$/ ,/ # ,/*/ (# B#"&%*8 ,* C-"%" # ,/*" (#)9#%&/. S* %#g*&%"/ -" *,4/%"$*/,# ",%*/%# #, -" (/ &"9-")#(", "'@

35


36/55


2011

P/% -&*/ (# -" &"9-" # )#(# #% )# )9#%&/ +-/ )(/ H"9#%9"*-"(/ $/, C-"%" #% ,/*/ (# B#"&%*8. A' )# -" %#)#&" /,@)9#%&/ 9"*-+ $/, C-"%" -" "%#?" (# ,/*/ /,@ E()"%(/ D",*#-"0)&"/ A-*$*"0 F#(#%*$/ C-"%" )9#%&/ B#"&%*8.

L" &"9-" $/-#&" /,@

1G.

U," ",#%" (# %#/-#% #- %/9-#" $/,* #, #,"% #- á%#" (# -"%#g*+, g%* $// -" (*4#%#,$*" (#- á%#" (#- %#$&á,g)-/ ABCD #,/-" á%#" (# -/ &%*á,g)-/ AED0 EBF FC. E- á%#" (#- %#$&á,g)-/ #(# 1! ),*("(# $)"(%"(". P"%" *,#&*g"% #- á%#" (# -/ &%*á,g)-/

(#9#/ /9#%"% %*#%/ )# 2&/ /, %#$&á,g)-/ #, A0 B C :/,2%&*$# (#- %#$&á,g)-/< "(#á *+$#-# :D"(/ )# -/ á,g)-/ADE0 BEF CF /, -/ á,g)-/ (# %#9/ " 5/@ #, #- &%*á,g)-/ AED0

# (#$*%0

#,&/,$#

. D# 4/%" ",á-/g" "%" -/

&%*á,g)-/ BEF CF


37/55

A E B

CG

F

D

H


2011

A' )#@$l rea del trin&lo AD$ es i&al a # &nidades - # &nidades . 2 = !2&nidades c&adradas/

$l rea del trin&lo B$0 es i&al a " &nidades - " &nidades . 2 =12/" &nidades c&adradas/

$l rea del trin&lo C0 es i&al a ! &nidades - ! &nidades . 2 =4/" &nidades c&adradas/

3nalmente el rea de la rein ris es i&al a 14 5 '!2 + 12/ " +4/"( = 14 5 4 = "" &nidades c&adradas/

T"9*2, /(#/ $"-$)-"% #- á%#" /-*$*&"(" # "'@ * &%"8"/ #-#g#,&/ F "%"-#-/ "- #g#,&/ DC #- á%#" (# -" %#g*+, g%* # *g)"-

" -" )" (# -" á%#" (#- $)"(%*-á%/ FD #- &%*á,g)-/ EF0 $//# "%#$*" #, -" >g)%" (# "9"?/@

7in embaro, res&lta %&e el c&adriltero 80D es &n paraleloramo p&esto %&e9

1/ $l lado 80, por constr&ccin, es paralelo a s& lado op&esto D ) 2/ $l lado D8 tambi:n lo es a s& lado op&esto 0 por%&e los n&los

08 ) 08$ son correspondientes e i&ales entre s; 'ambos miden4"o (9 el n&lo 08 es alterno interno con el n&lo 0C entre lossementos paralelos 80 ) D cortados por el sementotransversal 8 por lo %&e son i&ales entre s;, con medida 4"o 'esfcil ver el n&lo 0C mide 4"o9 en el trin&lo C0,

pero

el n&lo C0 es &no de los n&los del rectn&lo ) C0 es eln&lo de rebote, es decir,

) entonces

( ) el n&lo 08$ es correspondiente al

37


38/55


2011

n&lo D8 entre los mismos sementos paralelos cortados por elsemento transversal 0 ) entonces los dos tambi:n son i&alesentre s;, con medida 4"o 'tambi:n no es dif;cil observar %&e D8mide 4"o9 , pero el

n&lo 8DA es el n&lo de rebote ) el n&lo DA es &no de losn&los del rectn&lo, es decir, )

entonces (/

A*ora bien como los trin&los A$D, $B0 ) 0C son rectn&los eissceles 'A, B ) C son v:rtices de los trin&lo ) tambi:n delrectn&lo ) los n&los AD$, B$0 ) C0 son los n&los de rebote a4"o9 en el trin&lo A$D, por e


39/55


2011

L/ ,)#/ ,)#%"(/%# /,@

-" $/9*,"$*/,# (# 2&/ )# )", 1! /,@

• T/",(/ (/ ,)#%"(/%#0 ,*,g)," (# -" 1 $/9*,"$*/,#/*9-# )" 1! :6!0 !


40/55


2011

• T/",(/ ,)#%"(/%# +-/ 1 (# -" 35 $/9*,"$*/,# /*9-#)" 1!:6!0 !0 3!0 !


41/55


2011

!. U," 4/%" (# %#/-#%-/ )#(# #% -" *g)*#,@N// )# #- #g#,&/ ),"(/ # #- %"(*/(#- $'%$)-/ /% -/ )# ) -/,g*&)( # *g)"- " -"*&"( (# -" -/,g*&)( (#- (*á#&%/0 # (#$*%0 5

$. N// &"9*2, )# # #g#,&/),"(/ &*#,# *g)"- -/,g*&)( (#- -"(/ (#-%/9/0 )#&/ )# "9/ /, (*"g/,"-# (# ),*/ %#$&á,g)-/. D# ")' )# #- #%'#&%/ (#-%/9/ # *g)"- X 5 $ ! $.

41


42/55


2011

1. P/(%'" %#/-#%# (# #&" ",#%"@S* $/,&"/ -/ $)"(%"(/ (# 1X10 2&/ /, 6 * $/,&"/ -/ (#X /, -/ (# 3X3 /, 36 -/ (# X /, 5 -/ (# 5X50 /,16 -/ (# 6X6 /, -/ (# 7X7 /, - (# GXG # +-/ 1. E, &/&"-

6 36 5 16 1G

7

6

5

3

1 !.

E, -" >g)%" (# "9"?/0 #- $)"(%"(/ (# 1X1 /(#/ (*9)?"% #, G#$# H"$*" -" (#%#$H" G H"$*" "9"?/0 #, &/&"- GXG #$#.A,á-/g"#, #- $)"(%"(/ (# X # )#(# (*9)?"% 7 #$# H"$*" -"(#%#$H" /&%" 7 H"$*" "9"?/0 #, &/&"- 7X7 #$#. D# -" *"",#%" /(#/ H"$#% $/, -/ $)"(%"(/ (# 3X30 X0 5X50 6X607X7 GXG.

. U," %*#%" 4/%" ) #-##,&"- (# %#/-#% )# /(%'" #g)*%#$/,* #, /9#%"% )# -" $",&*("( (# ),&/ ,#$#"%*/ "%" -">g)%" ")#,&" #, #, %#-"$*+, $/, -" >g)%"1 #, -" >g)%" 3")#,&" 7 %##$&/ " -" >g)%" (# -" >g)%" 3 " -" >g)%" ")#,&"1! # (#$*%0 -/ *,$%##,&/ # (", #g, -" )$#*+, 0 70 1!0 13. [-" $)"- ")#,&" (# 3 #, 3. E&/ *g,*>$" )# * -" >g)%" &*#,# ),&/ "%" -" >g)%" 5 # ,#$#*&"%á, 13 35 ),&/0 "%" -">g)%" 60 # %#)#%*%á, 35 15 5! ),&/ * $/,&*,)"/ (# #&"

42


43/55


2011

4/%" H"&" -" >g)%" 35 $/,$-)*%#/ )# # ,#$#*&"%á, 1G!),&/.

O&%" ",#%" (# %#/-#% # #% -" $",&*("(# (# ),&/ ,#$#"%*/"%" $"(" >g)%" "'@ C"(" >g)%" # 4/%" "g%#g",(/ " -" ",%*/% #-

*g)*#, -&*-/ (# 3 (**,)*(/ #, ),*("(# $// # )#&%" #, -"&"9-"

F*g)%"P),&/

S#$)#,$*" (#

4/%"$*+,

S#$)#,$*" (# 4/%"$*+,*,%%#&"(" $//

-&*-/ (# 3(**,)*(/ #,

),*("(#

S*-*>$"$*+, (#-" #$)#,$*" (#

4/%"$*+,

1 1 1 3:1< K 3:1< K :1< 5 1 3:1< K 3:< K 3:1< K :<3 1 1

7

3:1< K 3:< K 3:3<

K

3:13< K :3<

1 7 1!

3:1< K 3:< K 3:3<K 3:<

3:13< K :<

35 1 7 [ 1!

3:1< K 3:< K 3:3<K 3:< K [ 3:35< K

3:13[335< K :35<1G!

D#- -&*/ %#,g-+, (# -" &"9-" /(#/ /9#%"% )# -/ ),&/,#$#"%*/ "%" 4/%"% -" >g)%" 35 /, 1G!.

T"9*2, /(#/ %#/-#% #$%*9*#,(/ #- ,#%/ (# ),&/ (# $"(">g)%" $// -/ ),&/ (# -" >g)%" ",%*/% á -/ ),&/ )# #*,$%##,&", /9#%"% -" #$)#,$*" (# 4/%"$*+, )# ""%#$#@

:1


44/55


2011

E- á%#" (#- "%"-#-/g%"/ PRS # *g)"- "- á%#" (#- %#$&á,g)-/ ABCD#,/ -" á%#" (# -/ &%*á,g)-/ APS0 PB0 CR SRD.

S# &*#,# )#@Á%#" (#- &%*á,g)-/ APS :AP$" )# -" #$# )# &/"%/, ()-$# ),/ /&%/ 4)#%/,&"9*2, ! /(#/ $/,$-)*% )# " M"/ -# &/$"%/, 1 3 5 [ 37 3 !! ()-$# *#,&%" )# " A," -# $/%%#/,(*#%/, 6 [ 3G ! ! ()-$#. A' )# #, -" 9/-" H"9'" !! ! G! ()-$#.

S* # /9#%"0 -" $",&*("( (# ()-$# )# H"9'" #, -" 9/-" #(#%*," " "%&*% (# -/ )# -# &/$+ " M"/ " A,"0 #&/ #@ :1 3 5 [ 37 3< : 6 [ 3G !< 1 3 [ 3 ! G! ()-$#. C// # # # &%"&" #, %#"-*("( (# -" )"

44

T/"("

1; ; 3;1;

!;

A," 6 [ 3G !M"/ 1 3 5 [ 37 3D*4#%#,$*" $"("#8

1 1 1 [ 1 1

D*4#%#,$*""$))-"("

1 3 [ 1 !


45/55

!

" #

$

%


2011

(# &/(/ -/ ,#%/ ,"&)%"-# $/,#$)&*/ (#(# #- 1 H"&" #- !.E&" )" # )#(# $"-$)-"% &"9*2,0 #%/ (# 4/%" á %á*("0)-&*-*$",(/ #- -&*/ ,#%/ (# -" )" :!< /% #- *g)*#, :1< (**(*#,(/ #,&%# @ :! X 1< G!. 1<

5. U," ",#%" (# #,"% -" /-)$*+, (#- %/9-#" )#(# #% -"*g)*#,@ #- #%'#&%/ (# -" >g)%" /9%#"("# *g)"- " -" )" (#- #%'#&%/ &%*á,g)-/g%",(# :MN MR< -" -/,g*&)(# ),/ (# -/-"(/ (#- &%*á,g)-/ #(*",/ :NP N< ),/(#- &%*á,g)-/ #)#/. C// #- #%'#&%/ (#-&%*á,g)-/ #(*",/ # -" *&"( (#- g%",(#0 )-"(/ *(#@ 3 G $. E- #%'#&%/ (#-&%*á,g)-/ #)#/ # -" *&"( (#- #(*",/0 #,&/,$# ) -"(/ *(#1 3 $. A' )# #- #%'#&%/ (# -" >g)%" /9%#"(" # *g)"- "@

G G 6! $.

45


46/55


2011

6. U," 4/%" (# %#/-#%-/ /(%'" #%@" )# /9#%"% )# $// -/ &%# ,#%/ /, $/,#$)&*/ -"$/%%#/,(#,$*" (# -/ '9/-/ /, $/, -/ ,#%/ !0 10 0 3 0-/ '9/-/ 0 \0 ] :'9/-/ $/, -/ )# %*,", #&/ ,#%/<

/(%'", %#%##,&"% -" %,"9A< !0 1 B< 10 3C< 0 3 D< 30 0 !E< 0 !0 1

A,"-*$#/ )# ,(%'" )# /$)%%*% $/, -" #$%*&)%" (# -/ &%#,#%/ $/,#$)&*/ #, $"(" ),/ (# #&/ $*,$/ $"/@

A< S* #- %*#%/ (# -/ &%# ,#%/ $/,#$)&*/ %*,"%" #, $#%/0

#- #g),(/ %*,"%'" #, 1 #- %$#%/ %*,"%'" #, . C// -"),*("(# #, $"(" ),/ (# #--/ # /(%'", #$%*9*% $/, '9/-/*-# (#- *" (# ,)#%"$*+,0 #- #g),(/ '9/-/ (# -/,#%/ $/,#$)&*/ ,/ # "4#$&"%'". E,&/,$# #, -/ &%#,#%/ $/,#$)&*/ #- #g),(/ (# ) '9/-/ ,(%'" )##% #- */.

B< S* #- %*#% ,#%/ $/,#$)&*/ %*,"%" #, 10 #- #g),(/%*,"%'" #, #- %$#%/ #, 3. D# *g)"- 4/%" )# #- $"/",%*/%0 #- #g),(/ '9/-/ (# -/ ,#%/ $/,#$)&*/ ,/#%'" "4#$&"(/ /%)# -" ),*("(# /(%'", #% #$%*&" $/, -/

'9/-/ *-#. E,&/,$# #, -/ &%# ,#%/ $/,#$)&*/ #-#g),(/ (# ) '9/-/ *g)"-0 ,(%'" )# #% #- */.

C< S* #- %*#% ,#%/ $/,#$)&*/ %*,"%" #, 0 #- #g),(/%*,"%'" #, 3 #- %$#%/ #, . N)#"#, -" ),*("(#/(%'", #% #$%*&" $/, -/ '9/-/ *-# #- #g),(/'9/-/ (# -/ ,#%/ $/,#$)&*/. E,&/,$# #, -/ &%#,#%/ $/,#$)&*/ #- #g),(/ (# ) '9/-/ ,(%'" )##% #- */ &"9*2,.

D< S* #- %*#% ,#%/ $/,#$)&*/ %*,"%" #, 30 #- #g),(/

%*,"%'" #, #- %$#%/ #, !. C// (#- #g),(/ "- %$#%,#%/ $/,#$)&*/ H" )# ""% (# "- *g)*#, ,#%/0 #,#- *" (# ,)#%"$*+, 2 # ,(%'" )# #$%*9*%@ 1!. L/",%*/% *-*$" )# #- #g),(/ '9/-/ (#- %*#%/ #g),(/,#%/ $/,#$)&*/ ,/ # "4#$&"%'"0 #%/ #- (#- %$#%/ '0 /%-" ,#$#*("( (# #$%*9*% #- *g)*#, (# $// 1!0 )#(",(/ !$// ) -&*" $*4%" #- 1 )"(/ "- "-/% (#- #g),(/ (# )'9/-/. E,&/,$# #, -/ %*#%/ ,#%/ $/,#$)&*/ #-

46


47/55


2011

#g),(/ (# ) '9/-/ ,(%'" )# #% #- */0 #%/ #%'"(*4#%#, #, #- %$#%/ (# #--/.

E< S* #- %*#% ,#%/ $/,#$)&*/ %*,"%" #, 0 #- #g),(/

%*,"%'" #, ! #- %$#%/ #, 1. C// (#- %*#%/ "- #g),(/,#%/ $/,#$)&*/ H" )# ""% (# "- *g)*#, ,#%/0 #,#- *" (# ,)#%"$*+, 2 # ,(%'" )# #$%*9*% &"9*2,@1!. E&/ -&*/ *-*$"0 (# ",#%" **-"% "- $"/ ",%*/%0 )#(#- %*#%/ "- #g),(/ ,#%/ $/,#$)&*/ H"9%'" ), $"9*/ #,#- #g),(/ '9/-/ )# 2 -&*/0 " /(*>$"(/0#%",#$#%'" *, $"9*/ "- ""% (#- #g),(/ "- %$#% ,#%/$/,#$)&*/. E,&/,$# #, -/ -&*/ ,#%/ $/,#$)&*/ #-#g),(/ (# ) '9/-/ #%'" #- */0 #%/ #%'" (*4#%#, #,#- %*#%/ (# #--/.

S* #="*,"/ -" #$%*&)%" (# -/ &%# ,#%/ $/,#$)&*/#,$/,&%"(" /% #- "%)#+-/g/0 -" ,*$" /*9*-*("( "%" -"%*,"$*+, (# -/ ,#%/ # -" (#- *,$*/ D0 #&/ #0 -/,#%/ ,(%'", )# %*,"% #, 30 !. A' )# #- '9/-/ $/%%#/,(# " 30 \ $/%%#/,(# " ] $/%%#/,(#,#$#"%*"#, " !. P"%" (#>,*% )2 '9/-/ $/%%#/,(#, " 1 0 /(#/ /9#%"% )# #- '9/-/ (#- %#$&á,g)-/ g%",(# ,/)#(# "-#% /%)# "- ""% (#- #g),(/ "- %$#% ,#%/$/,#$)&*/0 #- 1 (# 1!0 )# # #- *g)*#, (# #, #- *" (#,)#%"$*+,0 # ,(%'" )# )"% " ) "-/% # $/,#%&*%'" #,30 #%/ 3 # $/%%#/,(# $/, :%#$&á,g)-/ #)#/< 2

,(%'" )# #% #- #g),(/ '9/-/ #, #- %$#% ,#%/$/,#$)&*/ #, -)g"% (# ^ 0 )# # >,"-#, #- )# ""%#$#.A' )# #- '9/-/ ^ $/%%#/,(# " 1 #- '9/-/ :%#$&á,g)-/ g%",(#< $/%%#/,(# " .

0inalmente los tres n>meros consec&tivos en el sistema den&meracin anti&o son9

!130 !1 !!

7. P"%" %#/-#% (# "-g)," 4/%"0 $/#,$#/"$//(",(/ %*#%/ -/ "9/%# )# &*#,#,%#&%*$$*+,@ &",&/ #- &2 ,#g%/ $// #- #%(# +-/)#(#, *% #, -" $"*--" (# -" #)*,". S*##8"/ /,*#,(/ #- &2 ,#g%/0 /%#?#-/0 #=*, 4/%" "%" H"$#%-/0 ?)&"#,-" $)"&%/ #)*," (# -" $"?"@

47

?

?

?

?


48/55


2011

A$//("(/ #- ,#g%/0 * #,#g)*(" H"$#/ -/ */ $/, #- #%(# ("(/ )# ,/ )#(# *% (# #$*,/ $/, #- ,#g%/0 "%" $"(" )," (# -"$)"&%/ /*9*-*("(# (#- ,#g%/0 +-/ /(#/ /,#%-/ #, (/ (# -" &%##)*," %#&",@

P/% -/ ,(%'"/ X G ",#%"(# "$//("% #- &2 ,#g%/ #- &2#%(#.

C// -/ (#á "9/%# ,/ &*#,#,%#&%*$$*/,# (# "$//(/0

/$)"(/ (/ (# -/ 6 #"$*/(*/,*9-# #, -" $"?"0 #- *g)*#, "9/%0/% #?#-/0 ",8",*--"0 "%" $"(" ),/ (#-/ /$H/ "$//(/ (#- ,#g%/ (#- #%(# ,(%'" 4/%" (*&*,&" (# /,#%#0 # (#$*%0 #%'", #,&/&"- G X 3 ",#%" (*4#%#, (# $/-/$"% ?),&/ -/ "9/%# ,#g%/0 #%(# ",8",*--".

O$)"(/ &%# (# -/ 6 -)g"%# (# -" $"?"0 "%" #- *g)*#, "9/% /%

#?#-/0 H*#%9"9)#,"0 "%" $"(" ),/ (# -/ 3 "$//(/ (#- ,#g%/0#%(# ",8",*--"0 # ,(%'", 3 4/%" (*4#%#, (# "$//("%-/0 /#"0 3 X 3 6 ",#%" (*&*,&" (# /,#% ?),&/ #&/ "9/%#.

P)#&/ " $)"&%/ "9/%# #, $)"&%/ (# -/ 6 #"$*/ (*/,*9-# #, -"$"?"0 "%" $"(" ),/ (# -/ 6 "$//(/ (*&*,&/0 #- *g)*#, "9/%0/% #?#-/0 $",#-"0 ,(%'" ,"(" á ",#%" (# $/-/$"%#0 #, &/&"-6 X 1 4/%" (*4#%#, (# /,#% 5 "9/%# ?),&/.

F*,"-#, /$)"(/ 5 (# -/ 6 -)g"%# (# -" $"?"0 "%" $"(" ),/ (# -/1 "$//(/ ,*$"#, )#(" 1 4/%" (# "$//("% #- -&*/"9/%0 /% #?#-/0 -*+,. P/% -/ )# -/ #* "9/%# ?),&/ # )#(#,/,#% (# 1 X 1 1 ",#%" (*&*,&".

48

@

?

@

?? @

@

?

?

@

@ ?

?

@

?

@


49/55


2011

G. P/(#/ %#/-#%-/ (# #&" ",#%"@

E- á%#" (# -" >g)%" /9%#"(" # *g)"- "- á%#" (# -/$)"&%/ &%*á,g)-/ DOE0 OC0 OB0 FOA0 $)" 9"#

"-&)%" /,@ :OD OE$*#,$"-$)-"% #- á%#" (# ),/ (# #--/0 #- OB0 /% #?#-/0 )-&*-*$"%-" /% $)"&%/ "%" (#%*,"% #- á%#" (#-" >g)%" /9%#"(". A' )#@ AAFBCDE :AOB< :OB$" )# :OC


50/55


2011

P"%" #,$/,&%"% -" "-&)%" (#- &%*á,g)-/ OB0 %#$/%(#/ )# OB OC 1 $ # 3 (# OF. E&/ #0

3 OF 1 $ #,&/,$# OF :


51/55


2011

L/ -&*-/ (# #,/%# )# 3!! %*,"(/ #, /,@ X G 70 X 1G 16 X G 5. L" /*9-# ág*," (#- -*9%/ /,@770 167 57.

L" ,*$" $/*,$*(#,$*" ("(" #, #- ,#%/ (# ág*," # #, 167. S* "# ,#%/ -# %#&"/ 7 :-" ág*," -#'(" /% M"%&" #- %*#%('"< %#)-&" 16!0 )# # ), -&*-/ (# 1! :-" )# ág*," )# 4)#-##,(/ (*"%*"#, " "%&*% (#- #g),(/ ('"


52/55


2011

S* *g,/%"/ -/ (/ ),&*&/ #, #- %#-/? (*g*&"- # /*9-# #% -/,#%/ #,&%# 1!! 150 #=$#&/ -/ $/%#,(*(/ (#- 16! " 10(#- 6! " 0 ...0 (#- 1!6! "- 1! (#- 116! "- 110 /%)# -"(/ -&*" $*4%" /, -/ *,)&/ 2&/ ""%#$#, #, #- %#-/? (*g*&"-

,*$"#, H"&" #- 5.L/ ,#%/ $)"(%"(/ )# # &*#,#, #,&%# 1!! 15 /,@1!!0 110 10 160 160 50 560 G0 30 3610 !!0 10 G050 5760 650 6760 70 7G0 G10 !!0 610 1!0 1!G0 1156015.

P#%/ /% -/ )# # (*?/ ", ("(/ )# #- ('" &*#,# H/%"0 +-/#&"%'", **9-# ()%", (/ *,)&/ #, #- ('" $"(" ),/@ 1!!0 11010 50 560 30 !!0 10 50 650 70 G10 !!0 1!0 1156015. E, &/&"- /(#/ #% ), ,#%/ $)"(%"(/ #, #- %#-/? (*g*&"-

()%", X 16 3 *,)&/.3. P/(#/ %/$#(#% "'@

E- #%'#&%/ (#- &%"#$*/ AOEF # *g)"- " -" )" (# -" -/,g*&)(#(# -/ #g#,&/ FA0 FE0 EO AO.

C/#,$#/ $"-$)-",(/ $"(" )," (# #&" -/,g*&)(#. P)#&/ )#ACDF # ), %#$&á,g)-/ FA CD 15 $ /% #% -" -/,g*&)(# (#-"(/ /)#&/. E,#g)*(" AC :1! K :15


53/55


2011

-UENTES CONSULTADAS

P#%%#,/)(0 P. :!!7


54/55


2011

DIRECTORIO

J(&5 A67(6%( G!(#%a M(#a!+&S#$%#&"%*/ (# E()$"$*+, J"-*$/

P+#( D%a8 A#%a&C//%(*,"(/% (# E()$"$*+, Bá*$"

Sa!9a(# R(#:;u+8 L%8(!aD*%#$&/% #,#%"- (# E()$"$*+, S#$),("%*"

G%!,+#7( T%6a"+#( D:a8D*%#$&/% #,#%"- (# P%/g%"" E&%"&2g*$/

M%;u+! Á6;+! Ca&%!!a& C+#6a

D*%#$&/% (# P%/g%"" (# A$/""*#,&/ P#("g+g*$/

COMITÉ ORGANI%'a C(>%&%=6 O?+#a7%9aL)* A-#?",(%/ R/(%'g)#8 A$##L)* M*g)#- R"'%#8 P)-*(/ T#%#" F/,#$" Cá%(#,"*/",,* R*g/9#%&/ R*$/ L+#8

V'$&/% M",)#- R/(%'g)#8 T%#?/L*-*"," L*8#& L+#8R"8$+,S",&/ A%%#g)', R",g#-O-g" /(',#8 )8á,

C(>%&%=6 + L(;:&7%'a C(>%&%=6 + D%*u&%=6S*-*" E&H#-" R*#%" A-$"-áL)* J"*#% E&%"(" /,8á-#8%"$*#-" B%"/ R*$/E-*8"9#&H Á-"%#8 R. J)", J/2 Á-"%#8 L+#8L)* A-4"%/ R/(%'g)#8

A," M"%'" D'"8 C"&*--/"9%*#-" F%",$/ .

C(!a,(#a(#+&A'a5>%'(&@C2"% O$&"*/ P2%#8C"%%*8"-# J/2 J"*#% )&*2%%#8 P*,#("CH%*&" A-#?",(%" A#8$)"E$$*)P#(%/ J"*#% B/9"(*--" T/%%#

54


55/55


2011

P"9-/ A-9#%&/ M"$'"M"%&',#8 J)-*/ R/(%'g)#8 #%,á,(#8

Documents

cuadernillo_entrenamiento_secundaria_2011.docx