15-second question
1. Quadrilateral ABCD is inscribed in a circle. If
and
, what is
? [85 degree degrees] s]
. !he "alue #f a certain ancient artifact a$$reciates 1%& e"er' 'ear. !w# 'ears ag#, its "alues was set at 5%%,%%% $es#s. (#w )uch is the artifact n#w? [* +%5,%%%] . -ind the real r##ts #f the euati#n . In
with
? [/10,%]
, if
and
, what is
? [% degre degress] ss]
5. If 2 is 1%& # f', and ' is %& #f 3, then 3 is h#w )an' $ercent #f 2? [5%%&] +. In a class #f % students, % ha"e br#thers, 15 ha"e sisters, and 5 ha"e b#th br#thers and sisters. (#w )an' students in the class are single child in the fa)il'? [1%] 4. In the gra$h #f
, f#r what "alues #f 2 is the gra$h situated ab#"e the 2/a2is? [ 2%]
8. A de)#graher $redicts that the $#$ulati#n #f a c#untr' t 'ears fr#) n#w can be )#deled b' the functi#n
. 6sing this )#del, after h#w )an' 'ears will the curr current ent $#$ulati#n
d#uble? [%] 7. if
, what is
in ter)s #f a? [a0]
1%. hat is the re)ainder when
is di"ided b'
? [40]
30-second questions
1. In a circle centerred centerred at 91,/1:, #ne end$#int #f a dia)eter lies at 9,:. -ind -ind the c##rdinates #f the #ther end$#int #f this dia)eter? [9/,:] . -act#r c#)$lete c#)$letel' l' the e2$ressi#n
#"er the set #f real nu)bers. [292;1:
92<;1:] . !he =rst three ter)s #f an arith)etic seuence are
,
, and
. hat is the =fth ter)
#f the seuence? [%] . If
, what is
? [14]
5. 5 $#ints are a re #n the c##rdinate #f $lane, n# three #f which are c#llinear. c#llinear. (#w )an' $#l'g#ns can be drawn with these $#ints as "ertices? [1+]
1-minute question
1. >et f be a uadratic functi#n such that f9/1:%, and f9%:f91:/8. hat is f9:? [1+] . -ind the nu)ber #f $er)utati#ns #f letters #f the w#rd C(A>>@@. @"aluate fact#rials in '#ur answer. [7%,4%] . -ind all 2 between % degrees and 7% degrees such that
[% degrees,
5 degrees] . !hree "ertices #f a $arallel#gra) are 9%,%:, 9,%:, and 91,:. -ind the c##rdinates #f the f#urth "erte2 s# that the resulting $arallel#gra) has the largest $#ssible $eri)eter. [9/,:] 5. If
and
, =nd the "alues #f
and . [9,,:]
15/sec#nd uesiti#n 1. hat is the su) #f all )ulti$les #f between 1% and %? [1%%] . >et ABC be an acute triangle inscribed in a circle with center . If
, what is angle
ABC? [54 degrees] . In r#lling a $air #f dice, what is the $r#babilit' #f getting a su) #f 5? [107] . If 5 arith)etic )eans are inserted between /15 and 7, what is the largest #f such )eans? [5] 5. -act#r c#)$letel' the e2$ressi#n
. [292;1:92;:]
+. If 7<21%, what is 4<2E? [1%%%] 4. F#l"e f#r 2 in the ineualit' 92;5:<Gsrt52E/1Gle % [/50, 105] 8. !he =rst ter) #f a ge#)etric seuence is , and its c#))#n rati# is . hat is the 5th ter)? [] 7. If 92/: is a fact#r #f
, what is H? [/]
1%. A snail crawls 1+ )eters #n the =rst h#ur, 1 )eters #n the sec#nd h#ur, and e"er' succeeding h#ur, it crawls #nl' 0 the distance it crawled the $re"i#us h#ur. Can it reach a t#tal distance #f +% )eters fr#) where it started after a =nite nu)ber #f h#urs? [@F]
%/sec#nd uesti#n 1. In an in"est)ent that $r#)ises a rate #f 1%& c#)$#unded annuall', h#w )uch are '#u g#ing t# in"est n#w in #rder t# recei"e *h$ 5,%% in 'ears? [ *h*5%,%%%] . If
and %J2J5, what is the c#rres$#nding #f "alues #f '? [ ' is less than + but
greater than #r eual t# /] . If g92:2/ and f9g92::/+2, what is f9:? [/5] . In a circle, the ch#rds *Q and KF intersects at L. If *L, QL5, and KLFL;, what is KF? [M1] 5. A c#))ittee #f is t# be f#r)ed fr#) a gr#u$ #f 5 )en and w#)en. In h#w )an' wa's can this be d#ne if the c#))ittee )ust ha"e at least 1 w#)an? [ +5] +%/sec#nd uesti#ns 1. hat is the c#nstant ter) in the e2$ansi#n #f 92;2
. [M, cube r##t #f ]
Question 8:
(#w )an' 5/digit e"en nu)bers can be f#r)ed #ut #f the digits ,,5,8,and 7 9with#ut re$etiti#n:? Solution:
C#nsider the 5/digit nu)ber as blanHsO P P P P P
!he last digit )ust be e"en. Fince we ha"e three e"en nu)bers 9,,8:. !he last digit can be =lled in wa's. Fince re$etiti#n is n#t all#wed, we can =ll #ut the rest blanHs in wa's, wa's, wa's, and 1 wa'. 1 2 2 2 2 72 wa's. Question 24:
If a fair c#in is t#ssed ="e ti)es, =nd the $r#babilit' that e2actl' t#sses sh#w heads. Solution:
!his t#$ic is under $r#babilit' #f re$eated trials. athe)aticians #ften call it RBin#)ialS because this is an a$$licati#n #f bin#)ial the#re) in $r#babilit'. >et ( be the (ead #utc#)e #f a c#in >et ! be the !ail #utc#)e #f a c#in ne wa' t# arrange the gi"en c#nditi#n is the seuenceO (1T(T(T!1T! But the f#ll#wing seuence als# )eets the c#nditi#nO (T(1T(T!1T! #r (T(T!1T!T(1. C#unting all $#ssible #utc#)es is the sa)e techniue used in c#unting the nu)ber #f $er)utati#ns in the w#rd BU.
All $#ssible #utc#)es can be calculated in
wa's. @ach c#in has wa's t# land. !hus 5
c#ins ha"e 2222 wa's t# land. (ence, the desired $r#babilit' is Question 32:
A b##Hshelf has 8 hist#r' b##Hs and 1% c##Hing b##Hs. #u will select 1% b##Hs 9 (ist#r' b##Hs and 8 c##Hing b##Hs: t# bring #n a tri$. (#w )an' ch#ices are $#ssible? Solutions:
In the language #f c#)binati#ns, we can select hist#r' b##Hs #ut #f 8 in 8C 8 wa's. >iHewise, we can select 8 c##Hing b##Hs #ut #f 1% c##Hing b##Hs in there are 8V5 1260$#ssible ch#ices. Question 41:
1%
C8 5 wa's. B' c#unting $rinci$le,
!he dial #n a c#)binati#n l#cH c#ntains three wheels, each #f which is labeled with a digit fr#) % t# 7. (#w )an' $#ssible c#)binati#ns d#es the l#cH ha"e if digits )a' n#t be re$eated? Solution:
>et A,B,C stands f#r digit fr#) %/7 as a l#cH c#)binati#n. A can =lled in 1% wa's 9%/7:. B can be =lled in 7 wa's, C can be =lled in 8 wa's. B' c#unting $rinci$le, there are 1%2728 720 t# $#ssible c#)binati#ns #f l#cH. Question 46:
If f#ur/nu)ber c#des are f#r) rand#)l' fr#) the digits % t# 7, what is the $r#babilit' that the tw# )iddle digits are the sa)e? Solution:
C#nsider the seuence A/B/B/C. !here is n# gi"en c#nditi#n that the digits canWt be re$eated s# we assu)e that we can re$eat each digit. @ach letter can re$resent 1% nu)bers 9%/7:. But BWs is c#unted as #ne. !hus the t#tal $#ssible wa's is 1%21%21%1,%%%. !he t#tal nu)ber #f $#ssible c#)binati#ns is 1%21%21%21%1%,%%%. !he $r#babilit' that the tw# )iddle digits are the sa)e is 1,%%%01%,%%% 1/10.
1.
hat is
in degrees?
.
C#n"ert 5X t# radians.
.
@"aluateO sin1Xc#s1+X;c#s1Xsin1+X
.
-ind the length #f an arc subtended b' an angle #f +%X in a circle if the radius is 5c).
5.
@"aluate O sin15%X
+.
If sinL0 and L is in sec#nd uadrant. hat is c#sL?
4.
!he angle #f ele"ati#n #f an #bYect fr#) the t#$ #f the tree is %X. If the height #f the tree is 15). (#w far is the #bYect fr#) the tree?
8.
!wice the su$$le)ent #f the angle is tw# )#re than the c#)$li)ent #f the angle. hat is the angle?
7.
!he length #f the diag#nal #f the rectangle is % c). If the diag#nal )aHes an angle #f %X with the l#nger side. hat is the area #f rectangle?
1%.
F#l"e f#r L in 9%,+%X:O c#sL/1%.
11.
hat is the a)$litude and $eri#d #f the 'sin2
1.
hat is the range #f ';sin2?
1.
Z#hn walHed %) fr#) $#int A t# B. -r#) B he )ade a left )aHing an angle #f +% degrees with his $re"i#us $ath, after reaching 5 ) he decided t# g# bacH t# $#int A. (#w l#ng will he need t# walH again?
1.
!w# sides #f triangle )easures + c) and 1% c). hat is the area #f the s)allest triangle with integral sides that can be )ade #ut #f these sides?
15.
!he angle #f ele"ati#n #f the car fr#) the t#$ t#wer is %X. !he car tra"elled with a unif#r) s$eed #f % H$h, after % )inutes the angle #f ele"ati#n #f the car fr#) the t#$ #f the t#wer is 5X. hat is the height #f the t#wer?
1+.
An euilateral triangle is inscribed in a circle, if the radius #f the circle is 1%c). hat is the area #f the triangle?
14.
Zane has 8 shirts and 7 $ants. (#w )an' cl#thing c#)binati#ns d#es Zane ha"e, if she d#esnWt wear s$eci=c shirts with s$eci=c $ants?
18.
ut #f a classr##) #f + b#'s and girls the teacher $icHs a $resident f#r the student b#ard, a "ice $resident and a secretar'. hat is the $r#babilit' that #nl' girls will be elected?
17.
!w# dice are r#lled. hat is the $r#babilit' the su) will be greater than 1%?
%.
In anila, the chance f#r rain #n an' gi"en da' is %&. hat is the $r#babilit' that it rains #n 5 #ut #f 4 c#nsecuti"e da's in anila?
1.
In h#w )an' wa's $e#$le can be arrange in a circle?
.
(#w )an' diag#nals d#es decag#n ha"e?
.
(#w )an' /digit nu)bers can be f#r)ed with the digits 1,,4,8 and 7 if the digits are n#t re$eated.
.
(#w )an' $er)utati#ns d#es the w#rd C(AC(A ha"e?
5.
i"en letters and addressed en"el#$es, in h#w )an' wa's can the letters be $laced in the en"el#$es s# that n# letter is in the c#rrect en"el#$e?
