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8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 1/123
MATHCOUNTS
1990-1991
State Countdown Round
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 2/123
How many of the even counting
numbers less than 100 areprime?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 3/123
nswer! 1
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 4/123
"he probability of rain on anygiven day in tlanta is #0$%
fter how many days would youe&pect it to have rained on '0
days?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 5/123
nswer! 1(0 )days*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 6/123
"wo fair tetrahedral )four-sided*dice are thrown% +f the faces of
each are numbered ', (, , and 9,what is the probability that the sumof the numbers on the bottom faces
is a prime?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 7/123
nswer! 0
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 8/123
.hat is the shortest distance that canbe traveled if you start at any point, , /,
C, or and visit the other three pointsonce?
( ' /
2 (
C
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 9/123
nswer! 1'
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 10/123
How many positive integer pairs
)&, y* satisfy 3 & 4 y,'when 3 #?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 11/123
nswer! ( )pairs*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 12/123
+f the digit is placed to the rightof a three-digit number, forming
a four-digit number, the value ofthe original number increases by,000% .hat was the originalnumber?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 13/123
nswer!
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 14/123
5ifty cards numbered 1 to (0 areplaced in a bo&% +f a card isselected at random, what is the
probability that the card is aprime number and a multiple of
seven? 6&press your answer asa common fraction%
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 15/123
1(0
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 16/123
.hat is the last digit in the
product of all natural numbersbetween 1 and 100?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 17/123
nswer! 0
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 18/123
+f (7 is e&pressed as the product
of its prime factors in e&ponentialform, what is the sum of itse&ponents?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 19/123
nswer! (
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 20/123
"his table shows the distribution of scores of 8r%Sampsons 10-point :ui;% .hat is the median score?
Test Scores Frequency10 I
9 II
8 III7 IIII
6 IIII
5
4 I
3
2 II
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 21/123
nswer!
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 22/123
certain type of < => by 11 paperhas a thic@ness of 0%0# cm% +f
1(,000 sheets of paper are stac@edon top of each other, what is the
height, in meters, of the stac@?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 23/123
nswer! ' )meters*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 24/123
Aoints , /, C, and are located on /such that / 3 ' 3 /C% +f a point is
selected at random, what is theprobability that it is between C and ?6&press your answer as a common
fraction%
C /
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 25/123
1#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 26/123
"hree identical s:uares are
placed side by side to form arectangle with a perimeter of
102 inches% .hat is the area, ins:uare inches, of each s:uare?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 27/123
nswer! 19 )s:uare inches*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 28/123
/rad is less than '0 years old%
His age is a multiple of (% Be&tyear his age will be a multiple of% How old is /rad now?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 29/123
nswer! #0 )years old*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 30/123
How many different triangles are
in the figure?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 31/123
nswer! 1' )triangles*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 32/123
How many even numbers arebetween #01 and 20(?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 33/123
nswer! 101
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 34/123
.hat is the positive s:uare rootof 1&1?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 35/123
nswer! 2&<
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 36/123
.hen n is divided by (, the
remainder is 1% .hat is theremainder when 'n is divided by(?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 37/123
nswer! '
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 38/123
6&press 10$ of '0$ of (0 as adecimal%
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 39/123
nswer! 1%(
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 40/123
3 1, #, ', 2, (, , , <, 9, DE%
.hat fractional part of the numbersin set are not divisible by '?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 41/123
#'
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 42/123
Si& softball teams are to play
each other once% How manygames are needed?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 43/123
nswer! 1( )games*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 44/123
"wo ropes, 1< meters in lengthand #2 meters in length, need to
be cut into pieces which are allthe same length% .hat is thegreatest possible length of eachpiece?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 45/123
nswer! )meters*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 46/123
.hat value of n ma@es thissentence true?
n 3 n 4 1
9
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 47/123
nswer! #
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 48/123
+f f)&* 3 '& 4 # and g)&* 3 )& F 1*#
,what is f)g)-#**?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 49/123
nswer! #9
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 50/123
+f 8arch 1 is a 8onday, what day ofthe wee@ will it be #0 days later?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 51/123
nswer! 5riday
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 52/123
Given & 3 ' and y 3 #, simplify!
#&' F 'y#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 53/123
nswer!
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 54/123
Simplify! 727'7
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 55/123
nswer! '(
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 56/123
+n how many different orders
can a group of si& people beseated around a round table?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 57/123
nswer! 1#0 )ways*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 58/123
How many different
arrangements are there usingthe letters in the wordAR6?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 59/123
nswer! ''0 )arrangements*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 60/123
.hat is I#00 to the nearest tenth?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 61/123
nswer! 12%1
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 62/123
.hat is the positive geometricmean of < and 1<?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 63/123
nswer! 1#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 64/123
5ind the midpoint of the line
segment with endpoints )(, -(* and) -(, (*%
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 65/123
nswer! )0, 0*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 66/123
Give the decimal e&pression for :$ of:%
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 67/123
nswer! 0%01:#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 68/123
Solve for &! #& 4 # 3 1#<
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 69/123
nswer! (
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 70/123
"he area of a triangle is 00 s:uare
feet% 5ind the altitude, in feet, ofthe triangle if the length of the baseis '0 feet%
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 71/123
nswer! 20 )feet*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 72/123
"he longest side of a right triangle
is ( feet and the shortest side is 'feet% .hat is the area of thetriangle in s:uare meters?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 73/123
nswer! )s:uare meters*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 74/123
+f each side of a s:uare is
increased by (0$, what is thepercent of increase in the area ofthe s:uare?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 75/123
nswer! 1#( )percent*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 76/123
5our lines are drawn in a plane%
.hat is the ma&imum number ofdistinct regions in which thelines could divide the plane?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 77/123
nswer! 11
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 78/123
Simplify! 0%D 4 0%'''D0%999D
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 79/123
nswer! 1
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 80/123
.hat percent of <0$ of a
number will lave the numberunchanged?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 81/123
nswer! 1#( )percent*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 82/123
5ind the number of s:uare units inthe area of the shaded region%
)0, '0* )#0, '0* )'0, '0*
)'0, #0* )0, 10*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 83/123
nswer! (00 )s:% units*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 84/123
+f y 3 - 4 I)( F &*, find & when y 3'%
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 85/123
nswer! -9(
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 86/123
+n :uadrilateral /C, angle /and angle C are trisected asshown% .hat is the degree
measure of angle 5? / C
5
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 87/123
nswer! <0 )degrees*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 88/123
5ind the number of s:uare units inthe area of the triangle%
)-#, *
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 89/123
nswer! '# )s:uare units*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 90/123
.hat is the smallest integer thatsatisfies K#& F 'KL <?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 91/123
nswer! -#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 92/123
How many two-digit wholenumbers are divisible by 12?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 93/123
nswer!
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 94/123
.hat percent of the whole
numbers less than 100 have noremainders when divided by (?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 95/123
nswer! #0 )percent*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 96/123
.hat is the 100
th
even wholenumber?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 97/123
nswer! 19<
/C is a s:uare 2 inches on a side,
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 98/123
and each of the inside s:uares is
formed by Moining the midpoints of theouter s:uares sides% .hat is the areaof the shaded region?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 99/123
nswer! 2 )s:uare inches*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 100/123
+f all multiples of ' and allmultiples of 2 are removed from
the list of numbers 1 through100, how many whole numbers
are left?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 101/123
nswer! (0 )whole numbers*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 102/123
FINALS
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 103/123
ivide 2abc by #a#b 'd#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 104/123
cd#
a
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 105/123
.hat is the simplest radicalform of 'I2 ÷ I#?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 106/123
nswer! I#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 107/123
"wo fair cubical dice are tossed%.hat is the probability,
e&pressed as a commonfraction, that the sum of thenumbers showing on the dice
will be four?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 108/123
11#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 109/123
+f a N b is defined as a 4 b,
#what is the value of N )' N (*?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 110/123
nswer! (
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 111/123
+f & and y are positive integers
such that &y 3 10<, what is theleast possible value of & 4 y?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 112/123
nswer! #1
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 113/123
5ind II#(,000%
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 114/123
nswer! 20
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 115/123
6&press the value of1 of 10$ of 0%001 in scientific notation%
100
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 116/123
nswer! 1 or)1%0* J 10-
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 117/123
5ind the sum of all solutions to
this e:uation!&# 4 # 3 10#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 118/123
nswer! 0
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 119/123
.hat is the sum of the lengths, incentimeters, of the two legs of a
'0-0-90 right triangle, if the lengthof the hypotenuse is #I cm?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 120/123
nswer! I 4 'I#
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 121/123
How many diagonals can bedrawn for a he&agon?
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 122/123
nswer! 9 )diagonals*
8/20/2019 1991 State Countdown Round
http://slidepdf.com/reader/full/1991-state-countdown-round 123/123