Difference between revisions of "2004 AMC 10A Problems"
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== Problem 20 == | == Problem 20 == | ||
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+ | [[Point]]s <math>\mathrm {E}</math> and <math>\mathrm {F}</math> are located on [[square]] <math>\mathrm {ABCD}</math> so that <math>\triangle \mathrm {BEF}</math> is [[equilateral]]. What is the [[ratio]] of the [[area]] of <math>\triangle \mathrm {DEF}</math> to that of <math>\triangle \mathrm {ABE}</math>? | ||
[[2004 AMC 10A Problems/Problem 20|Solution]] | [[2004 AMC 10A Problems/Problem 20|Solution]] |
Revision as of 08:50, 24 September 2007
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
You and five friends need to raise dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?
Problem 2
For any three real numbers ,
, and
, with
, the operation
is defined by:
What is
?
Problem 3
Alicia earns 20 dollars per hour, of which is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
Problem 4
What is the value of if
?
Problem 5
Problem 6
Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
Problem 7
A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack?
Problem 8
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players ,
, and
start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game?
Problem 9
Problem 10
Coin is flipped three times and coin
is flipped four times. What is the probability that the number of heads obtained from flipping the two fair foins is the same?
Problem 11
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by without altering the volume, by what percent must the height be decreased?
Problem 12
Henry's Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A costomer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?
Problem 13
At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?
Problem 14
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more quarter, the average would be 21 cents. How many dimes does she have in her purse?
Problem 15
Given that and
, what is the largest possible value of (x+y)/x?
Problem 16
Problem 17
Problem 18
A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the secon term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
Problem 19
Problem 20
Points and
are located on square
so that
is equilateral. What is the ratio of the area of
to that of
?
Problem 21
Problem 22
Problem 23
Problem 24
Let , be a sequence with the following properties.
- (i)
, and
- (ii)
for any positive integer
.
What is the value of ?
Problem 25
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?