Metrobank-MTAP-DepEd Math Challenge 2013, Fourth Year, Category A 1 !ne o" t#o $o%ple%entary $o%ple%entary angle& added to one one-ha -hal" the the oth other y'eld 'eld&& (2) F'nd 'nd the the %ea&ure& o" the t#o angle&
|
3x + 1
1( 5ole the 'ne+ual'ty
1 - 2x
|
≤ 1.
2 F'nd F'nd the the a% a%pl pl't 'tud udee and and per' per'od od o" the the θ
"un$t'on y * -3 &'n 5 . 3 F'nd F'nd the the e+ua e+uat' t'on on o" the the l'ne l'ne pa&& pa&&'n 'ng g through 3, -10 and #ho&e &lope '& hal" the &lope o" the l'ne ./ 2y 1 * 0 . A $lo& $lo&ed ed re$t re$tan angu gula larr bo/ bo/ '& o" un'" un'"or or% % th' th'$kne& ne&& / 'n$ 'n$he& he& The bo/ bo/ ha& ha& oute uter d'%en&'on& ( 'n$he&, . 'n$he&, and 3 'n$he& and and 't ha& ha& an 'n&' 'n&'de de olu olu%e %e o" 30 $ub' $ub'$$ 'n$he& F'nd / F'nd the $enter and rad'u& o" the $'r$le / 2 ./ y2 1.y .4 * 0 ( F'nd the e/a$t alue o" $o& 4) -1
5
2 x y
4 5'%pl'"y 3x -3 y -4 3 6 7o# %any -d'g't een nu%ber& $an be "or%ed out o" the d'g't& 2, ., , 6, and 8 #'thout repet't'on9 repet't'on9 8 F'nd the 'ner&e o" y * log 3 / 2 10 :n parallelogra% ;5T<, angle ; '& 3/y), angle 5 '& y-.), and angle T '& ./-6) F'nd angle < R U 11 5uppo& po&e "t "t * A 2002t rep repre& re&ent& ent& the the T ba$ter'a pre&ent 'n a S $ulture t %'nute& "ro "ro% the &tart o" an e/pe e/per' r'%e %ent nt ='e ='e an e/pr e/pre& e&&' &'on on "or "or the the nu%ber o" %'nute& needed "or the ba$ter'a to double
12 !ne &'de o" a re$tangle '& ( $%, and 't& ad>a ad>a$e $ent nt &'de &'de %e %ea& a&ur ure& e& oneone-th th'r 'rd d o" the the re$tangle?& per'%eter F'nd the d'%en&'on& o" the re$tangle 13 :" the area o" a re$tangle '& (/ 2/ 3 ., and 't& #'dth '& 2 /, "'nd 't& length 1. The d'a%eter o" a ba&ketball '& 8 'n$he& :" 't '& $o%p $o%ple lete tely ly &ub% &ub%er erge ged d 'n #ate #aterr, ho# ho# %u$h %u$h #ate #aterr #'ll #'ll be d'&p d'&pla la$e $ed9 d9 ='e ='e your your an&#er 'n ter%& o" @ 1 F'nd all 10 &'n / * (
x
∈
[ ] π ,π 2
&u$h that 3 $o& 2 /
14 F'nd 'nd the the t#o t#o po'nt& 'nt& that hat d' d''de the the &eg%ent >o'n'ng -2, and ., -4 'nto three e+ual part& 16 :"
f
(
x - 3 5 - x
)
= 3x - 2, find find f ( x ) .
18 orker& A and B, #ork'ng together, $an "'n'&h a >ob 'n 6 hour& :" they #ork together "or ( hour& a"ter #h'$h orker A leae&, then orker B need& 8 %ore hour& to "'n'&h the >ob 7o# long doe& 't take orker orker A to do the >ob alone9 alone9 20 F'nd the $oe""'$'ent o" the ter% 'nol'ng /2 'n the e/pan&'on / 3/ -26 21 :n the "'gure, ar$ ;5 * 11), ar$ ;T * 60), and angle ! * ) F'nd ar$ T< 22 :n the "'gure, ABC '& &'%'lar to ADB :" AD * 3 and AB * 6, "'nd AC 23 :n the "'gure, $on$entr'$ $'r$le& #'th rad'' . and hae $enter P F'nd AC, g'en that 't '& tangent to the 'nner $'r$le and '& a $hord o" the outer $'r$le 2. :" a "a'r $o'n '& to&&ed "'e t'%e&, "'nd the proba probab'l b'l'ty 'ty that that e/a$ e/a$tly tly three three to&&e to&&e&& &ho# &ho# head& 2 F'nd the alue o" a &o that po'nt& -1, -2, (, a, and -10, 2 l'e on a &tra'ght l'ne 2( 2( F'nd F'nd the the prod produ$ u$tt o" the the root root&& o" the the +uadrat'$ e+uat'on 3/ / 2 1 * 0 24 M'randa tra'ned $on&'&tently, &o that &he $an "'n'&h a ra$e 'n 1 hour Dur'ng the ra$e, &he ran at the rate o" 6 kph 7o#eer, upon rea$h'ng the hal"#ay po'nt o" the ra$e, &he real real' 'ed ed &he &he need needed ed to run run "a&t "a&ter er &o &he &he 'n$rea&ed her &peed to 10 kph :" &he rea$hed her goal >u&t 'n t'%e, ho# long #a& the ra$e9 26 5ole "or r 'n ter%& o" &, #here r = 1 + s. 2s-r
28 The e/pre&&'on / 3 a/2 b/ ( ha& the &a%e re%a'nder re%a'nder #hen d''ded by / 1 or by 2
/ :" the re%a'nder #hen the e/pre&&'on '& d''ded by / 3 '& -(0, "'nd a and b
.0 5'%pl'"y to a &'ngle "ra$t'on 2a + 5 2
30 A &+uare '& 'n&$r'bed 'n&'de a $'r$le o" rad'u& 10 $% F'nd the per'%eter o" the &+uare 20 cm
31 r'te log34log.3log4( a& &'ngle logar'th% 32 A book&hel" ha& 6 h'&tory book& and 10 $ook'ng book& You #'ll &ele$t 10 book& 2 h'&tory book& and 6 $ook'ng book& to br'ng on a tr'p 7o# %any $ho'$e& are po&&'ble9 33 Fa$tor $o%pletely 2/ ( 3/ 6/. 12/3 3. :n parallelogra% ABCD, AC %eet& BD at ! 5uppo&e that !A * 3/ 2, !C * 13 (/, and !B * 3/ 2 F'nd !D 3 5ole "or / log/ . log/ 4 * 1 3( :" /2. * y-1 * 1(, "'nd the po&&'ble alue& o" / and y 34 :" "/ * 2/ 3 and g/ * 1G/ 1, "'nd the do%a'n o" " o g/ 36 The angle& o" a +uadr'lateral are 'n the rat'on .2(3 F'nd %ea&ure o" the large&t angle 38 5uppo&e that P/ '& a polyno%'al &u$h that the re%a'nder o" P/ H / 2 '& - and the re%a'nder o" P/ H / 3 '& 4 :& 't po&&'ble "or P/ to hae a e/a$tly one root bet#een 2 and 39 b t#o root& bet#een 2 and 39 $ no root bet#een 2 and 39
6a + 1 3a - 5
+
a - 2 2
4a - 17a + 18
.1 The d'al on a $o%b'nat'on lo$k $onta'n& three #heel&, ea$h o" #h'$h '& labeled #'th a d'g't "ro% 0 to 8 7o# %any po&&'ble $o%b'nat'on& doe& the lo$k hae '" d'g't& %ay not be repeated9 .2 F'nd and 'dent'"y the a&y%ptote o" the graph o" y * . / 1 3 .3 5ole "or /
(
x -
3 x
) ( 2
+
x -
3 x
)
– 6 = 0
.. Iet C be a $'r$le o" rad'u& 6 'n$he&, ha'ng a $hord o" length 3 'n$he& F'nd the $entral angle oppo&'te th'& $hord . :" p * log 2, + * log , and r * log 4, e/pre&& log 0 log 40 log 2 4 'n ter%& o" p, +, and r .( :" "our-nu%ber $ode& are "or%ed rando%ly "ro% the d'g't& 0 to 8, #hat '& the probab'l'ty that the t#o %'ddle d'g't& are the &a%e9 .4 For #hat alueG& o" k doe& the graph o" y * 3/2 k/ k hae a %'n'%u% alue o" 39 .6 F'nd the do%a'n o" the "un$t'on "/ * / 3G / 6 .8 The /- and y-'nter$ept o" a l'ne are -8 and ( re&pe$t'ely F'nd the po'nt on the l'ne #ho&e ord'nate '& 0 5uppo&e that an a'rplane $l'%b& at an angle o" 30) :" 't& &peed '& %a'nta'ned at 0 k'lo%eter& per hour, ho# long #'ll 't take to rea$h a he'ght o" 1 k'lo%eter&9
