Difference between revisions of "2004 AMC 10B Problems"
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== Problem 20 == | == Problem 20 == | ||
+ | |||
+ | In <math>\triangle ABC</math> points <math>D</math> and <math>E</math> lie on <math>BC</math> and <math>AC</math>, respectively. If <math>AD</math> and <math>BE</math> intersect at <math>T</math> so that <math>AT/DT=3</math> and <math>BT/ET=4</math>, what is <math>CD/BD</math>? | ||
+ | |||
+ | |||
+ | <math> \mathrm{(A) \ } \frac{1}{8} \qquad \mathrm{(B) \ } \frac{2}{9} \qquad \mathrm{(C) \ } \frac{3}{10} \qquad \mathrm{(D) \ } \frac{4}{11} \qquad \mathrm{(E) \ } \frac{5}{12} </math> | ||
+ | |||
+ | |||
+ | <asy> | ||
+ | unitsize(1cm); | ||
+ | defaultpen(0.8); | ||
+ | pair A=(0,0), B=5*dir(60), C=5*(1,0), D=B + (11/15)*(C-B), E = A + (11/16)*(C-A); | ||
+ | draw(A--B--C--cycle); | ||
+ | draw(A--D); | ||
+ | draw(B--E); | ||
+ | pair T=intersectionpoint(A--D,B--E); | ||
+ | label("$A$",A,SW); | ||
+ | label("$B$",B,N); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,NE); | ||
+ | label("$E$",E,S); | ||
+ | label("$T$",T,2*WNW); | ||
+ | </asy> | ||
+ | |||
[[2004 AMC 10B Problems/Problem 20|Solution]] | [[2004 AMC 10B Problems/Problem 20|Solution]] |
Revision as of 17:29, 9 February 2009
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club?
Problem 2
How many two-digit positive integers have at least one 7 as a digit?
Problem 3
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice?
Problem 4
A standard six-sided die is rolled, and P is the product of the five numbers that are visible. What is the largest number that is certain to divide P?
Problem 5
In the expression , the values of
,
,
, and
are
,
,
, and
, although not necessarily in that order. What is the maximum possible value of the result?
Problem 6
Which of the following numbers is a perfect square?
Problem 7
On a trip from the United States to Canada, Isabella took U.S. dollars. At the border she exchanged them all, receiving
Canadian dollars for every
U.S. dollars. After spending
Canadian dollars, she had
Canadian dollars left. What is the sum of the digits of
?
Problem 8
Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
Problem 9
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?
Problem 10
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain?
Problem 11
Two eight-sided dice each have faces numbered 1 through 8. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum?
Problem 12
An annulus is the region between two concentric circles. The concentric circles in the figure have radii and
, with
. Let
be a radius of the larger circle, let
be tangent to the smaller circle at
, and let
be the radius of the larger circle that contains
. Let
,
, and
. What is the area of the annulus?
Problem 13
In the United States, coins have the following thicknesses: penny, mm; nickel,
mm; dime,
mm; quarter,
mm. If a stack of these coins is exactly
mm high, how many coins are in the stack?
Problem 14
A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only of the marbles in the bag are blue. Then yellow marbles are added to the bag until only
of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?
Problem 15
Patty has coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have
cents more. How much are her coins worth?
Problem 16
Three circles of radius are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?
Problem 17
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?
Problem 18
In the right triangle , we have
,
, and
. Points
,
, and
are located on
,
, and
, respectively, so that
,
, and
. What is the ratio of the area of
to that of
?
Problem 19
In the sequence ,
,
,
, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is
. What is the
term in this sequence?
Problem 20
In points
and
lie on
and
, respectively. If
and
intersect at
so that
and
, what is
?
Problem 21
Let ;
;
and
;
;
be two arithmetic progressions. The set
is the union of the first
terms of each sequence. How many distinct numbers are in
?
Problem 22
A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?
Problem 23
Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
Problem 24
In we have
,
, and
. Point
is on the circumscribed circle of the triangle so that
bisects
. What is the value of
?