MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1999 AMC 12/AHSME
1999 AMC 12/AHSME
Part of
AMC 12/AHSME
Subcontests
(30)
5
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Alice's Book
The marked price of a book was
30
%
30\%
30%
less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?
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<span class='latex-bold'>(A)</span>\ 25\% \qquad <span class='latex-bold'>(B)</span>\ 30\% \qquad <span class='latex-bold'>(C)</span>\ 35\% \qquad <span class='latex-bold'>(D)</span>\ 60\% \qquad <span class='latex-bold'>(E)</span>\ 65\%
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65%
30
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Diophantine Equation
The number of ordered pairs of integers
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(m,n)
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n
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for which
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n
≥
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mn \ge 0
mn
≥
0
and m^3 \plus{} n^3 \plus{} 99mn \equal{} 33^3 is equal to
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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>\ 33\qquad <span class='latex-bold'>(D)</span>\ 35\qquad <span class='latex-bold'>(E)</span>\ 99
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Tetrahedron with an Inscribed Sphere
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point
P
P
P
is selected at random inside the circumscribed sphere. The probability that
P
P
P
lies inside one of the five small spheres is closest to
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<span class='latex-bold'>(A)</span>\ 0\qquad <span class='latex-bold'>(B)</span>\ 0.1\qquad <span class='latex-bold'>(C)</span>\ 0.2\qquad <span class='latex-bold'>(D)</span>\ 0.3\qquad <span class='latex-bold'>(E)</span>\ 0.4
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28
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Min/Max of Sequence with Conditions
Let
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x_1
x
1
,
x
2
x_2
x
2
,
…
\dots
…
,
x
n
x_n
x
n
be a sequence of integers such that (i) \minus{}1 \le x_i \le 2, for i \equal{} 1,2,3,\dots,n; (ii) x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n \equal{} 19; and (iii) x_1^2 \plus{} x_2^2 \plus{} \cdots \plus{} x_n^2 \equal{} 99. Let
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m
m
and
M
M
M
be the minimal and maximal possible values of x_1^3 \plus{} x_2^3 \plus{} \cdots \plus{} x_n^3, respectively. Then \frac{M}{m} \equal{}
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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>\ 6\qquad <span class='latex-bold'>(E)</span>\ 7
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Trigonometry Relations of a Triangle
In triangle
A
B
C
ABC
A
BC
, 3\sin A \plus{} 4\cos B \equal{} 6 and 4\sin B \plus{} 3\cos A \equal{} 1. Then
∠
C
\angle C
∠
C
in degrees is
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<span class='latex-bold'>(A)</span>\ 30\qquad <span class='latex-bold'>(B)</span>\ 60\qquad <span class='latex-bold'>(C)</span>\ 90\qquad <span class='latex-bold'>(D)</span>\ 120\qquad <span class='latex-bold'>(E)</span>\ 150
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Polygon Perimeter
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length
1
1
1
. The polygons meet at a point
A
A
A
in such a way that the sum of the three interior angles at
A
A
A
is
36
0
∘
360^\circ
36
0
∘
. Thus the three polygons form a new polygon with
A
A
A
as an interior point. What is the largest possible perimeter that this polygon can have?
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<span class='latex-bold'>(A)</span>\ 12\qquad <span class='latex-bold'>(B)</span>\ 14\qquad <span class='latex-bold'>(C)</span>\ 18\qquad <span class='latex-bold'>(D)</span>\ 21\qquad <span class='latex-bold'>(E)</span>\ 24
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Probability of Quadrilateral in a Circle
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords are the sides of a convex quadrilateral?
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1365
<span class='latex-bold'>(A)</span>\ \frac{1}{15}\qquad <span class='latex-bold'>(B)</span>\ \frac{1}{91}\qquad <span class='latex-bold'>(C)</span>\ \frac{1}{273}\qquad <span class='latex-bold'>(D)</span>\ \frac{1}{455}\qquad <span class='latex-bold'>(E)</span>\ \frac{1}{1365}
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23
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Hide problems
Area of a Convex Hexagon
The equiangular convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
has AB \equal{} 1, BC \equal{} 4, CD \equal{} 2, and DE \equal{} 4. The area of the hexagon is
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3
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3
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4
3
<span class='latex-bold'>(A)</span>\ \frac{15}{2}\sqrt{3}\qquad <span class='latex-bold'>(B)</span>\ 9\sqrt{3}\qquad <span class='latex-bold'>(C)</span>\ 16\qquad <span class='latex-bold'>(D)</span>\ \frac{39}{4}\sqrt{3}\qquad <span class='latex-bold'>(E)</span>\ \frac{43}{4}\sqrt{3}
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22
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Absolute Value Graphs
The graphs of y \equal{} \minus{}|x \minus{} a| \plus{} b and y \equal{} |x \minus{} c| \plus{} d intersect at points
(
2
,
5
)
(2,5)
(
2
,
5
)
and
(
8
,
3
)
(8,3)
(
8
,
3
)
. Find a \plus{} c.
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18
<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>\ 13\qquad <span class='latex-bold'>(E)</span>\ 18
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Circumcircle of a Triangle
A circle is circumscribed about a triangle with sides
20
20
20
,
21
21
21
, and
29
29
29
, thus dividing the interior of the circle into four regions. Let
A
A
A
,
B
B
B
, and
C
C
C
be the areas of the non-triangular regions, with
C
C
C
being the largest. Then
(A)
\ A \plus{} B \equal{} C\qquad
(B)
\ A \plus{} B \plus{} 210 \equal{} C\qquad
(C)
\ A^2 \plus{} B^2 \equal{} C^2\qquad \\
(D)
\ 20A \plus{} 21B \equal{} 29C\qquad
(E)
\ \frac{1}{A^2} \plus{} \frac{1}{B^2} \equal{} \frac{1}{C^2}
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1
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Another Sequence Problem
The sequence
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
,
…
\dots
…
satisfies a_1 \equal{} 19, a_9 \equal{} 99, and, for all
n
≥
3
n \ge 3
n
≥
3
,
a
n
a_n
a
n
is the arithmetic mean of the first n \minus{} 1 terms. Find
a
2
a_2
a
2
.
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179
<span class='latex-bold'>(A)</span>\ 29\qquad <span class='latex-bold'>(B)</span>\ 59\qquad <span class='latex-bold'>(C)</span>\ 79\qquad <span class='latex-bold'>(D)</span>\ 99\qquad <span class='latex-bold'>(E)</span>\ 179
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Minimizing Side Length
Consider all triangles
A
B
C
ABC
A
BC
satisfying the following conditions: AB \equal{} AC,
D
D
D
is a point on
A
C
‾
\overline{AC}
A
C
for which
B
D
‾
⊥
A
C
‾
\overline{BD} \perp \overline{AC}
B
D
⊥
A
C
,
A
D
AD
A
D
and
C
D
CD
C
D
are integers, and BD^2 \equal{} 57. Among all such triangles, the smallest possible value of
A
C
AC
A
C
is
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<span class='latex-bold'>(A)</span>\ 9 \qquad <span class='latex-bold'>(B)</span>\ 10 \qquad <span class='latex-bold'>(C)</span>\ 11 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>\ 13
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[asy]defaultpen(linewidth(.8pt)); dotfactor=4;pair B = (0,0); pair C = (5,0); pair A = (2.5,7.5); pair D = foot(B,A,C); dot(A);dot(B);dot(C);dot(D); label("
A
A
A
", A, N);label("
B
B
B
", B, SW);label("
C
C
C
", C, SE);label("
D
D
D
", D, NE); draw(A--B--C--cycle);draw(B--D);[/asy]
18
1
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Zeros of a Function Composition
How many zeros does f(x) \equal{} \cos(\log(x))) have on the interval
0
<
x
<
1
0 < x < 1
0
<
x
<
1
?
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infinitely many
<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>\ 10 \qquad <span class='latex-bold'>(E)</span>\ \text{infinitely many}
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infinitely many
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Remainder of Polynomial Division
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial such that when
P
(
x
)
P(x)
P
(
x
)
is divided by x \minus{} 19, the remainder is
99
99
99
, and when
P
(
x
)
P(x)
P
(
x
)
is divided by x \minus{} 99, the remainder is
19
19
19
. What is the remainder when
P
(
x
)
P(x)
P
(
x
)
is divided by (x \minus{} 19)(x \minus{} 99)?
(A)
\ \minus{}x \plus{} 80 \qquad
(B)
\ x \plus{} 80 \qquad
(C)
\ \minus{}x \plus{} 118 \qquad
(D)
\ x \plus{} 118 \qquad
(E)
\ 0
16
1
Hide problems
Inradius for a Rhombus
What is the radius of a circle inscribed in a rhombus with diagonals of length
10
10
10
and
24
24
24
?
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<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 58/13 \qquad <span class='latex-bold'>(C)</span>\ 60/13 \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ 6
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15
1
Hide problems
Trigonometric Manipulations
Let
x
x
x
be a real number such that \sec x \minus{} \tan x \equal{} 2. Then \sec x \plus{} \tan x \equal{}
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<span class='latex-bold'>(A)</span>\ 0.1 \qquad <span class='latex-bold'>(B)</span>\ 0.2 \qquad <span class='latex-bold'>(C)</span>\ 0.3 \qquad <span class='latex-bold'>(D)</span>\ 0.4 \qquad <span class='latex-bold'>(E)</span>\ 0.5
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14
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Four Singers
Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang
7
7
7
songs, which was more than any other girl, and Mary sang
4
4
4
songs, which was fewer than any other girl. How many songs did these trios sing?
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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>\ 9 \qquad <span class='latex-bold'>(D)</span>\ 10 \qquad <span class='latex-bold'>(E)</span>\ 11
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Real Number Sequence
Define a sequence of real numbers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
,
…
\dots
…
by
a
1
=
1
a_1 = 1
a
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=
1
and
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n
+
1
3
=
99
a
n
3
a_{n + 1}^3 = 99a_n^3
a
n
+
1
3
=
99
a
n
3
for all
n
≥
1
n \ge 1
n
≥
1
. Then
a
100
a_{100}
a
100
equals
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33
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<span class='latex-bold'>(A)</span>\ 33^{33} \qquad <span class='latex-bold'>(B)</span>\ 33^{99} \qquad <span class='latex-bold'>(C)</span>\ 99^{33} \qquad <span class='latex-bold'>(D)</span>\ 99^{99} \qquad <span class='latex-bold'>(E)</span>\ \text{none of these}
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1
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Points of Intersection of Two 4th Degree Polynomials
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions y \equal{} p(x) and y \equal{} q(x), each with leading coefficient
1
1
1
?
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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>\ 8
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Student Locker Numbers
The student locker numbers at Olympic High are numbered consecutively beginning with locker number
1
1
1
. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number
9
9
9
and four centers to label locker number
10
10
10
. If it costs
$
137.94
\$137.94
$137.94
to label all the lockers, how many lockers are there at the school?
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<span class='latex-bold'>(A)</span>\ 2001 \qquad <span class='latex-bold'>(B)</span>\ 2010 \qquad <span class='latex-bold'>(C)</span>\ 2100 \qquad <span class='latex-bold'>(D)</span>\ 2726 \qquad <span class='latex-bold'>(E)</span>\ 6897
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25
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Ahsme 1999 #25
There are unique integers
a
2
,
a
3
,
a
4
,
a
5
,
a
6
,
a
7
a_2, a_3, a_4, a_5, a_6, a_7
a
2
,
a
3
,
a
4
,
a
5
,
a
6
,
a
7
such that \frac {5}{7} \equal{} \frac {a_2}{2!} \plus{} \frac {a_3}{3!} \plus{} \frac {a_4}{4!} \plus{} \frac {a_5}{5!} \plus{} \frac {a_6}{6!} \plus{} \frac {a_7}{7!}, where
0
≤
a
i
<
i
0 \le a_i < i
0
≤
a
i
<
i
for i \equal{} 2,3...,7. Find a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5 \plus{} a_6 \plus{} a_7.
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<span class='latex-bold'>(A)</span>\ 8 \qquad <span class='latex-bold'>(B)</span>\ 9 \qquad <span class='latex-bold'>(C)</span>\ 10 \qquad <span class='latex-bold'>(D)</span>\ 11 \qquad <span class='latex-bold'>(E)</span>\ 12
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Cards in an Envelope
A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false. I. The digit is 1. II. The digit is not 2. III. The digit is 3. IV. The digit is not 4. Which one of the following must necessarily be correct?
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I is true.
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<span class='latex-bold'>(A)</span>\ \text{I is true.} \qquad <span class='latex-bold'>(B)</span>\ \text{I is false.}\qquad <span class='latex-bold'>(C)</span>\ \text{II is true.} \qquad <span class='latex-bold'>(D)</span>\ \text{III is true.} \qquad <span class='latex-bold'>(E)</span>\ \text{IV is false.}
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9
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Odometer Reading
Before Ashley started a three-hour drive, her car’s odometer reading was
29792
29792
29792
, a palindrome. At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of
75
75
75
miles per hour, which of the following was her greatest possible average speed?
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<span class='latex-bold'>(A)</span>\ 33\frac 13 \qquad <span class='latex-bold'>(B)</span>\ 53\frac 13\qquad <span class='latex-bold'>(C)</span>\ 60\frac 23\qquad <span class='latex-bold'>(D)</span>\ 70\frac 13\qquad <span class='latex-bold'>(E)</span>\ 74\frac 13
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Peoples' Ages
At the end of
1994
1994
1994
, Walter was half as old as his grandmother. The sum of the years in which they were born was
3838
3838
3838
. How old will Walter be at the end of
1999
1999
1999
?
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<span class='latex-bold'>(A)</span>\ 48 \qquad <span class='latex-bold'>(B)</span>\ 49\qquad <span class='latex-bold'>(C)</span>\ 53\qquad <span class='latex-bold'>(D)</span>\ 55\qquad <span class='latex-bold'>(E)</span>\ 101
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Sum of the Digits
What is the sum of the digits of the decimal form of the product
2
1999
⋅
5
2001
2^{1999}\cdot 5^{2001}
2
1999
⋅
5
2001
?
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<span class='latex-bold'>(A)</span>\ 2\qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 7\qquad <span class='latex-bold'>(E)</span>\ 10
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Acute Hexagon
What is the largest number of acute angles that a convex hexagon can have?
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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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Prime Numbers
Find the sum of all prime numbers between
1
1
1
and
100
100
100
that are simultaneously
1
1
1
greater than a multiple of
4
4
4
and
1
1
1
less than a multiple of
5
5
5
.
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<span class='latex-bold'>(A)</span>\ 118\qquad <span class='latex-bold'>(B)</span>\ 137\qquad <span class='latex-bold'>(C)</span>\ 158\qquad <span class='latex-bold'>(D)</span>\ 187 \qquad <span class='latex-bold'>(E)</span>\ 245
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Arithmetic Mean
The number halfway between
1
8
\frac {1}{8}
8
1
and
1
10
\displaystyle \frac {1}{10}
10
1
is
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<span class='latex-bold'>(A)</span>\ \frac {1}{80} \qquad <span class='latex-bold'>(B)</span>\ \frac {1}{40} \qquad <span class='latex-bold'>(C)</span>\ \frac {1}{18} \qquad <span class='latex-bold'>(D)</span>\ \frac {1}{9} \qquad <span class='latex-bold'>(E)</span>\ \frac {9}{80}
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80
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Find the False Statement
Which of the following statements is false?
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All equilateral triangles are congruent to each other.
<span class='latex-bold'>(A)</span>\ \text{All equilateral triangles are congruent to each other.}
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All equilateral triangles are congruent to each other.
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All equilateral triangles are convex.
<span class='latex-bold'>(B)</span>\ \text{All equilateral triangles are convex.}
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All equilateral triangles are convex.
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All equilateral triangles are equilangular.
<span class='latex-bold'>(C)</span>\ \text{All equilateral triangles are equilangular.}
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All equilateral triangles are equilangular.
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All equilateral triangles are regular polygons.
<span class='latex-bold'>(D)</span>\ \text{All equilateral triangles are regular polygons.}
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All equilateral triangles are regular polygons.
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All equilateral triangles are similar to each other.
<span class='latex-bold'>(E)</span>\ \text{All equilateral triangles are similar to each other.}
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All equilateral triangles are similar to each other.
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Basic Sequence
1 \minus{} 2 \plus{} 3 \minus{} 4 \plus{} \cdots \minus{} 98 \plus{} 99 \equal{}
(A)
\minus{}\! 50 \qquad
(B)
\minus{}\! 49 \qquad
(C)
\ 0 \qquad
(D)
\ 49 \qquad
(E)
\ 50