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Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 10
2010 AMC 10
2010 AMC 10
Part of
AMC 10
Subcontests
(25)
22
2
Hide problems
Interior Triangles
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
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<span class='latex-bold'>(A)</span>\ 28 \qquad <span class='latex-bold'>(B)</span>\ 56 \qquad <span class='latex-bold'>(C)</span>\ 70 \qquad <span class='latex-bold'>(D)</span>\ 84 \qquad <span class='latex-bold'>(E)</span>\ 140
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Distributing Candy
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
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<span class='latex-bold'>(A)</span>\ 1930\qquad<span class='latex-bold'>(B)</span>\ 1931\qquad<span class='latex-bold'>(C)</span>\ 1932\qquad<span class='latex-bold'>(D)</span>\ 1933\qquad<span class='latex-bold'>(E)</span>\ 1934
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2
Hide problems
Integer-Zero Polynomial
The polynomial x^3\minus{}ax^2\plus{}bx\minus{}2010 has three positive integer zeros. What is the smallest possible value of
a
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?
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<span class='latex-bold'>(A)</span>\ 78 \qquad <span class='latex-bold'>(B)</span>\ 88 \qquad <span class='latex-bold'>(C)</span>\ 98 \qquad <span class='latex-bold'>(D)</span>\ 108 \qquad <span class='latex-bold'>(E)</span>\ 118
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Palindrome divisible by 7
A palindrome between
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is chosen at random. What is the probability that it is divisible by
7
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7?
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?
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<span class='latex-bold'>(A)</span>\ \dfrac{1}{10} \qquad <span class='latex-bold'>(B)</span>\ \dfrac{1}{9} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{1}{7} \qquad <span class='latex-bold'>(D)</span>\ \dfrac{1}{6}\qquad <span class='latex-bold'>(E)</span>\ \dfrac{1}{5}
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2
Hide problems
The Boredest Fly
A fly trapped inside a cubical box with side length
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1
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meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?
(A)
\ 4 \plus{} 4\sqrt2 \qquad
(B)
\ 2 \plus{} 4\sqrt2 \plus{} 2\sqrt3 \qquad
(C)
\ 2 \plus{} 3\sqrt2 \plus{} 3\sqrt3 \qquad
(D)
\ 4\sqrt2 \plus{} 4\sqrt3 \\
(E)
\ 3\sqrt2 \plus{} 5\sqrt3
Two Circles and a Hexagon
Two circles lie outside regular hexagon
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ABCDEF
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\overline{AB}
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. Both are tangent to lines
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. What is the ratio of the area of the second circle to that of the first circle?
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<span class='latex-bold'>(A)</span>\ 18\qquad<span class='latex-bold'>(B)</span>\ 27\qquad<span class='latex-bold'>(C)</span>\ 36\qquad<span class='latex-bold'>(D)</span>\ 81\qquad<span class='latex-bold'>(E)</span>\ 108
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2
Hide problems
Angelina's Driving
Angelina drove at an average rate of
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kph and then stopped
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minutes for gas. After the stop, she drove at an average rate of
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hours including the stop. Which equation could be used to solve for the time
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in hours that she drove before her stop?
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\ 80t\plus{}100(8/3\minus{}t)\equal{}250 \qquad
(B)
\ 80t\equal{}250 \qquad
(C)
\ 100t\equal{}250 \\
(D)
\ 90t\equal{}250 \qquad
(E)
\ 80(8/3\minus{}t)\plus{}100t\equal{}250
Absolute Value Equation
What is the sum of all the solutions of x \equal{} |2x \minus{} |60\minus{}2x\parallel{}?
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<span class='latex-bold'>(A)</span>\ 32\qquad<span class='latex-bold'>(B)</span>\ 60\qquad<span class='latex-bold'>(C)</span>\ 92\qquad<span class='latex-bold'>(D)</span>\ 120\qquad<span class='latex-bold'>(E)</span>\ 124
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2
Hide problems
Interval of Solutions
The length of the interval of solutions of the inequality a\le 2x\plus{}3\le b is
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10
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. What is b\minus{}a?
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<span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 10 \qquad <span class='latex-bold'>(C)</span>\ 15 \qquad <span class='latex-bold'>(D)</span>\ 20 \qquad <span class='latex-bold'>(E)</span>\ 30
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Three Coupons
A shopper plans to purchase an item that has a listed price greater than
$
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\$100
$100
and can use any one of the three coupons. Coupon A gives
15
%
15\%
15%
off the listed price, Coupon B gives
$
30
\$30
$30
the listed price, and Coupon C gives
25
%
25\%
25%
off the amount by which the listed price exceeds
$
100
\$100
$100
.Let
x
x
x
and
y
y
y
be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is y\minus{}x?
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<span class='latex-bold'>(A)</span>\ 50\qquad<span class='latex-bold'>(B)</span>\ 60\qquad<span class='latex-bold'>(C)</span>\ 75\qquad<span class='latex-bold'>(D)</span>\ 80\qquad<span class='latex-bold'>(E)</span>\ 100
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10
2
Hide problems
Marvin's Birthday
Marvin had a birthday on Tuesday, May
27
27
27
in the leap year
2008
2008
2008
. In what year will his birthday next fall on a Saturday?
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<span class='latex-bold'>(A)</span>\ 2011 \qquad <span class='latex-bold'>(B)</span>\ 2012 \qquad <span class='latex-bold'>(C)</span>\ 2013 \qquad <span class='latex-bold'>(D)</span>\ 2015 \qquad <span class='latex-bold'>(E)</span>\ 2017
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Shelby's Scooter
Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?
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<span class='latex-bold'>(A)</span>\ 18\qquad<span class='latex-bold'>(B)</span>\ 21\qquad<span class='latex-bold'>(C)</span>\ 24\qquad<span class='latex-bold'>(D)</span>\ 27\qquad<span class='latex-bold'>(E)</span>\ 30
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8
2
Hide problems
Tony's Timesheets
Tony works
2
2
2
hours a day and is paid
$
0.50
\$0.50
$0.50
per hour for each full year of his age. During a six month period Tony worked
50
50
50
days and earned
$
630
\$630
$630
. How old was Tony at the end of the six month period?
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<span class='latex-bold'>(A)</span>\ 9 \qquad <span class='latex-bold'>(B)</span>\ 11 \qquad <span class='latex-bold'>(C)</span>\ 12 \qquad <span class='latex-bold'>(D)</span>\ 13 \qquad <span class='latex-bold'>(E)</span>\ 14
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Tickets to a School Play
A ticket to a school play costs
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x
dollars, where
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x
x
is a whole number. A group of 9th graders buys tickets costing a total of
$
48
\$48
$48
, and a group of 10th graders buys tickets costing a total of
$
64
\$64
$64
. How many values of
x
x
x
are possible?
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<span class='latex-bold'>(A)</span>\ 1\qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ 3\qquad<span class='latex-bold'>(D)</span>\ 4\qquad<span class='latex-bold'>(E)</span>\ 5
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7
2
Hide problems
Crystal's Runs
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?
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<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ \sqrt2 \qquad <span class='latex-bold'>(C)</span>\ \sqrt3 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ 2\sqrt2
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2
A Triangle and a Rectangle
A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?
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<span class='latex-bold'>(A)</span>\ 16\qquad<span class='latex-bold'>(B)</span>\ 24\qquad<span class='latex-bold'>(C)</span>\ 28\qquad<span class='latex-bold'>(D)</span>\ 32\qquad<span class='latex-bold'>(E)</span>\ 36
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2
Hide problems
Spade Operation
For positive numbers
x
x
x
and
y
y
y
the operation
♠
(
x
,
y
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\spadesuit(x,y)
♠
(
x
,
y
)
is defined as \spadesuit(x,y)\equal{}x\minus{}\frac1yWhat is
♠
(
2
,
♠
(
2
,
2
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)
\spadesuit(2,\spadesuit(2,2))
♠
(
2
,
♠
(
2
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2
))
?
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<span class='latex-bold'>(A)</span>\ \frac23 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ \frac43 \qquad <span class='latex-bold'>(D)</span>\ \frac53 \qquad <span class='latex-bold'>(E)</span>\ 2
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Angles in a Circle
A circle is centered at
O
O
O
,
A
B
‾
\overline{AB}
A
B
is a diameter and
C
C
C
is a point on the circle with \angle COB \equal{} 50^{\circ}. What is the degree measure of
∠
C
A
B
\angle CAB
∠
C
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B
?
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<span class='latex-bold'>(A)</span>\ 20 \qquad<span class='latex-bold'>(B)</span>\ 25 \qquad<span class='latex-bold'>(C)</span>\ 45 \qquad<span class='latex-bold'>(D)</span>\ 50 \qquad<span class='latex-bold'>(E)</span>\ 65
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5
2
Hide problems
Fixed Circumference Circle
The area of a circle whose circumference is
24
π
24\pi
24
π
is
k
π
k\pi
kπ
. What is the value of
k
k
k
?
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<span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 12 \qquad <span class='latex-bold'>(C)</span>\ 24 \qquad <span class='latex-bold'>(D)</span>\ 36 \qquad <span class='latex-bold'>(E)</span>\ 144
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144
Mondays and Wednesdays
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
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<span class='latex-bold'>(A)</span>\ 2\qquad<span class='latex-bold'>(B)</span>\ 3\qquad<span class='latex-bold'>(C)</span>\ 4\qquad<span class='latex-bold'>(D)</span>\ 5\qquad<span class='latex-bold'>(E)</span>\ 6
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4
2
Hide problems
Audiobook CDs
A book that is to be recorded onto compact discs takes
412
412
412
minutes to read aloud. Each disc can hold up to
56
56
56
minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
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<span class='latex-bold'>(A)</span>\ 50.2 \qquad <span class='latex-bold'>(B)</span>\ 51.5 \qquad <span class='latex-bold'>(C)</span>\ 52.4 \qquad <span class='latex-bold'>(D)</span>\ 53.8 \qquad <span class='latex-bold'>(E)</span>\ 55.2
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55.2
A Function
For a real number
x
x
x
, define
♡
(
x
)
\heartsuit (x)
♡
(
x
)
to be the average of
x
x
x
and
x
2
x^2
x
2
. What is \heartsuit(1) \plus{} \heartsuit(2) \plus{}\heartsuit(3)?
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<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 10 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>\ 20
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3
2
Hide problems
Tyrone and Eric's Marbles
Tyrone had
97
97
97
marbles and Eric had
11
11
11
marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
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<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 13 \qquad <span class='latex-bold'>(C)</span>\ 18 \qquad <span class='latex-bold'>(D)</span>\ 25 \qquad <span class='latex-bold'>(E)</span>\ 29
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A Sock Drawer
A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
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<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 8 \qquad <span class='latex-bold'>(E)</span>\ 9
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Squares and Rectangle
Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? [asy]unitsize(8mm); defaultpen(linewidth(.8pt));draw(scale(4)*unitsquare); draw((0,3)--(4,3)); draw((1,3)--(1,4)); draw((2,3)--(2,4)); draw((3,3)--(3,4));[/asy]
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<span class='latex-bold'>(A)</span>\ \frac {5}{4} \qquad <span class='latex-bold'>(B)</span>\ \frac {4}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac {3}{2} \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ 3
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Makayla's Meetings
Makayla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
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<span class='latex-bold'>(A)</span>\ 15 \qquad <span class='latex-bold'>(B)</span>\ 20 \qquad <span class='latex-bold'>(C)</span>\ 25 \qquad <span class='latex-bold'>(D)</span>\ 30 \qquad <span class='latex-bold'>(E)</span>\ 35
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Mary's Books
Mary's top book shelf holds five books with the following widths, in centimeters:
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\frac12
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. What is the average book width, in centimeters?
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Basic Computation
What is 100(100\minus{}3) \minus{} (100 \cdot 100 \minus{} 3)?
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\ \minus{}6 \qquad
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\ 0
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Sequence of Numbers
Jim starts with a positive integer
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n
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numbers: \begin{align*} &55\\ 55-7^2=&\ 6\\ 6-2^2=&\ 2\\ 2-1^2=&\ 1\\ 1-1^2=&\ 0 \end{align*}Let
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be the smallest number for which Jim's sequence has 8 numbers. What is the units digit of
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<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 7 \qquad <span class='latex-bold'>(E)</span>\ 9
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Polynomial with integer coefficients
Let
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be a polynomial with integer coefficients such that P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and} P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.} What is the smallest possible value of
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2
Hide problems
Logic with toads and frogs
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in the swamp, and they make the following statements: Brian: "Mike and I are different species." Chris: "LeRoy is a frog." LeRoy: "Chris is a frog." Mike: "Of the four of us, at least two are toads." How many of these amphibians are frogs?
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<span class='latex-bold'>(A)</span>\ 0\qquad<span class='latex-bold'>(B)</span>\ 1\qquad<span class='latex-bold'>(C)</span>\ 2\qquad<span class='latex-bold'>(D)</span>\ 3\qquad<span class='latex-bold'>(E)</span>\ 4
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A Multiple Choice Test
On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse's total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?
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<span class='latex-bold'>(A)</span>\ 25\qquad<span class='latex-bold'>(B)</span>\ 27\qquad<span class='latex-bold'>(C)</span>\ 29\qquad<span class='latex-bold'>(D)</span>\ 31\qquad<span class='latex-bold'>(E)</span>\ 33
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2
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90! and its last two nonzero digits
The number obtained from the last two nonzero digits of
90
!
90!
90
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is equal to
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n
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. What is
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<span class='latex-bold'>(A)</span>\ 12 \qquad <span class='latex-bold'>(B)</span>\ 32 \qquad <span class='latex-bold'>(C)</span>\ 48 \qquad <span class='latex-bold'>(D)</span>\ 52 \qquad <span class='latex-bold'>(E)</span>\ 68
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Arithmetic and geometric sequence sums
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than
100
100
100
points. What was the total number of points scored by the two teams in the first half?
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<span class='latex-bold'>(A)</span>\ 30 \qquad <span class='latex-bold'>(B)</span>\ 31 \qquad <span class='latex-bold'>(C)</span>\ 32 \qquad <span class='latex-bold'>(D)</span>\ 33 \qquad <span class='latex-bold'>(E)</span>\ 34
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Cube with Holes
A solid cube has side length
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<span class='latex-bold'>(A)</span>\ 7\qquad <span class='latex-bold'>(B)</span>\ 8\qquad <span class='latex-bold'>(C)</span>\ 10\qquad <span class='latex-bold'>(D)</span>\ 12\qquad <span class='latex-bold'>(E)</span>\ 15
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Placings at a Math Contest
Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?
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<span class='latex-bold'>(A)</span>\ 22\qquad<span class='latex-bold'>(B)</span>\ 23\qquad<span class='latex-bold'>(C)</span>\ 24\qquad<span class='latex-bold'>(D)</span>\ 25\qquad<span class='latex-bold'>(E)</span>\ 26
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14
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Hide problems
Triangle with AB = 2*AC
Triangle
A
B
C
ABC
A
BC
has AB \equal{} 2 \cdot AC. Let
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D
and
E
E
E
be on
A
B
‾
\overline{AB}
A
B
and
B
C
‾
\overline{BC}
BC
, respectively, such that \angle{BAE} \equal{} \angle{ACD}. Let
F
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F
be the intersection of segments
A
E
AE
A
E
and
C
D
CD
C
D
, and suppose that
△
C
F
E
\triangle{CFE}
△
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is equilateral. What is
∠
A
C
B
\angle{ACB}
∠
A
CB
?
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<span class='latex-bold'>(A)</span>\ 60^{\circ}\qquad <span class='latex-bold'>(B)</span>\ 75^{\circ}\qquad <span class='latex-bold'>(C)</span>\ 90^{\circ}\qquad <span class='latex-bold'>(D)</span>\ 105^{\circ}\qquad <span class='latex-bold'>(E)</span>\ 120^{\circ}
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Averages
The average of the numbers
1
,
2
,
3
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.
.
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,
98
,
99
1,2,3,...,98,99
1
,
2
,
3
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...
,
98
,
99
, and
x
x
x
is
100
x
100x
100
x
. What is
x
x
x
?
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101
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101
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<span class='latex-bold'>(A)</span>\ \frac{49}{101} \qquad<span class='latex-bold'>(B)</span>\ \frac{50}{101} \qquad<span class='latex-bold'>(C)</span>\ \frac12 \qquad<span class='latex-bold'>(D)</span>\ \frac{51}{101} \qquad<span class='latex-bold'>(E)</span>\ \frac{50}{99}
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16
2
Hide problems
Find the minimum perimeter of the triangle
Nondegenerate
△
A
B
C
\triangle ABC
△
A
BC
has integer side lengths,
B
D
BD
B
D
is an angle bisector, AD \equal{} 3, and DC \equal{} 8. What is the smallest possible value of the perimeter?
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<span class='latex-bold'>(A)</span>\ 30 \qquad <span class='latex-bold'>(B)</span>\ 33 \qquad <span class='latex-bold'>(C)</span>\ 35 \qquad <span class='latex-bold'>(D)</span>\ 36 \qquad <span class='latex-bold'>(E)</span>\ 37
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A Square and a Circle
A square of side length
1
1
1
and a circle of radius
3
/
3
\sqrt3/3
3
/3
share the same center. What is the area inside the circle, but outside the square?
(A)
\ \frac{\pi}3 \minus{} 1 \qquad
(B)
\ \frac{2\pi}{9} \minus{} \frac{\sqrt3}3 \qquad
(C)
\ \frac{\pi}{18} \qquad
(D)
\ \frac14 \qquad
(E)
\ 2\pi/9
12
2
Hide problems
Scale Model
Logan is constructing a scaled model of his town. The city's water tower stands
40
40
40
meters high, and the top portion is a sphere that holds
100
,
000
100,000
100
,
000
liters of water. Logan's miniature water tower holds
0.1
0.1
0.1
liters. How tall, in meters, should Logan make his tower?
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<span class='latex-bold'>(A)</span>\ 0.04\qquad <span class='latex-bold'>(B)</span>\ \frac{0.4}{\pi}\qquad <span class='latex-bold'>(C)</span>\ 0.4\qquad <span class='latex-bold'>(D)</span>\ \frac{4}{\pi}\qquad <span class='latex-bold'>(E)</span>\ 4
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4
Do You Like Math?
At the beginning of the school year,
50
%
50\%
50%
of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and
50
%
50\%
50%
answered "No." At the end of the school year,
70
%
70\%
70%
answered "Yes" and
30
%
30\%
30%
answered "No." Altogether,
x
%
x\%
x
%
of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of
x
x
x
?
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<span class='latex-bold'>(A)</span>\ 0\qquad<span class='latex-bold'>(B)</span>\ 20\qquad<span class='latex-bold'>(C)</span>\ 40\qquad<span class='latex-bold'>(D)</span>\ 60\qquad<span class='latex-bold'>(E)</span>\ 80
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19
2
Hide problems
An equiangular hexagon
Equiangular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
has side lengths AB \equal{} CD \equal{} EF \equal{} 1 and BC \equal{} DE \equal{} FA \equal{} r. The area of
△
A
C
E
\triangle ACE
△
A
CE
is
70
%
70\%
70%
of the area of the hexagon. What is the sum of all possible values of
r
r
r
?
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<span class='latex-bold'>(A)</span>\ \frac {4\sqrt {3}}{3} \qquad <span class='latex-bold'>(B)</span>\ \frac {10}{3} \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ \frac {17}{4} \qquad <span class='latex-bold'>(E)</span>\ 6
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A Circle and an Equilateral Triangle
A circle with center
O
O
O
has area
156
π
156\pi
156
π
. Triangle
A
B
C
ABC
A
BC
is equilateral,
B
C
‾
\overline{BC}
BC
is a chord on the circle, OA \equal{} 4\sqrt3, and point
O
O
O
is outside
△
A
B
C
\triangle ABC
△
A
BC
. What is the side length of
△
A
B
C
\triangle ABC
△
A
BC
?
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<span class='latex-bold'>(A)</span>\ 2\sqrt3 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 4\sqrt3 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 18
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9
2
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Palindromes
A palindrome, such as
83438
83438
83438
, is a number that remains the same when its digits are reversed. The numbers
x
x
x
and x \plus{} 32 are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?
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<span class='latex-bold'>(A)</span>\ 20\qquad <span class='latex-bold'>(B)</span>\ 21\qquad <span class='latex-bold'>(C)</span>\ 22\qquad <span class='latex-bold'>(D)</span>\ 23\qquad <span class='latex-bold'>(E)</span>\ 24
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Lucky Larry
Lucky Larry's teacher asked him to substitute numbers for
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
, and
e
e
e
in the expression a\minus{}(b\minus{}(c\minus{}(d\plus{}e))) and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
were
1
1
1
,
2
2
2
,
3
3
3
, and
4
4
4
, respectively. What number did Larry substitute for
e
e
e
?
(A)
\ \minus{}5\qquad
(B)
\ \minus{}3\qquad
(C)
\ 0\qquad
(D)
\ 3\qquad
(E)
\ 5
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Red marbles in boxes
Each of 2010 boxes in a line contains a single red marble, and for
1
≤
k
≤
2010
1 \le k \le 2010
1
≤
k
≤
2010
, the box in the
k
t
h
kth
k
t
h
position also contains
k
k
k
white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let
P
(
n
)
P(n)
P
(
n
)
be the probability that Isabella stops after drawing exactly
n
n
n
marbles. What is the smallest value of
n
n
n
for which
P
(
n
)
<
1
2010
P(n) < \frac {1}{2010}
P
(
n
)
<
2010
1
?
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<span class='latex-bold'>(A)</span>\ 45 \qquad <span class='latex-bold'>(B)</span>\ 63 \qquad <span class='latex-bold'>(C)</span>\ 64 \qquad <span class='latex-bold'>(D)</span>\ 201 \qquad <span class='latex-bold'>(E)</span>\ 1005
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1005
An Array of Digits
The entries in a
3
×
3
3\times3
3
×
3
array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
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<span class='latex-bold'>(A)</span>\ 18\qquad<span class='latex-bold'>(B)</span>\ 24\qquad<span class='latex-bold'>(C)</span>\ 36\qquad<span class='latex-bold'>(D)</span>\ 42\qquad<span class='latex-bold'>(E)</span>\ 60
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18
2
Hide problems
Bernardo and Silvia counts from sets
Bernardo randomly picks
3
3
3
distinct numbers from the set
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
\{1,2,3,4,5,6,7,8,9\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
and arranges them in descending order to form a
3
3
3
-digit number. Silvia randomly picks
3
3
3
distinct numbers from the set
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
\{1,2,3,4,5,6,7,8\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
and also arranges them in descending order to form a
3
3
3
-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
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<span class='latex-bold'>(A)</span>\ \frac {47}{72}\qquad <span class='latex-bold'>(B)</span>\ \frac {37}{56}\qquad <span class='latex-bold'>(C)</span>\ \frac {2}{3}\qquad <span class='latex-bold'>(D)</span>\ \frac {49}{72}\qquad <span class='latex-bold'>(E)</span>\ \frac {39}{56}
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Probability of being divisible by 3
Positive integers
a
,
b
,
a,b,
a
,
b
,
and
c
c
c
are randomly and independently selected with replacement from the set
{
1
,
2
,
3
,
…
,
2010
}
.
\{ 1,2,3,\dots,2010 \}.
{
1
,
2
,
3
,
…
,
2010
}
.
What is the probability that abc \plus{} ab \plus{} a is divisible by
3
3
3
?
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<span class='latex-bold'>(A)</span>\ \dfrac{1}{3} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{29}{81} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{31}{81} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{11}{27} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{13}{27}
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