Difference between revisions of "2005 AMC 10A Problems"
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− | ==Problem 1== | + | {{AMC10 Problems|year=2005|ab=A}} |
+ | == Problem 1 == | ||
While eating out, Mike and Joe each tipped their server <math>2</math> dollars. Mike tipped <math>10\%</math> of his bill and Joe tipped <math>20\%</math> of his bill. What was the difference, in dollars between their bills? | While eating out, Mike and Joe each tipped their server <math>2</math> dollars. Mike tipped <math>10\%</math> of his bill and Joe tipped <math>20\%</math> of his bill. What was the difference, in dollars between their bills? | ||
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== Problem 2 == | == Problem 2 == | ||
− | For each pair of real numbers <math>a | + | For each pair of real numbers <math>a \neq b</math>, define the [[operation]] <math>\star</math> as |
<math> (a \star b) = \frac{a+b}{a-b} </math>. | <math> (a \star b) = \frac{a+b}{a-b} </math>. | ||
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== Problem 5 == | == Problem 5 == | ||
− | A store normally sells windows at | + | A store normally sells windows at \$100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? |
<math> \mathrm{(A) \ } 100\qquad \mathrm{(B) \ } 200\qquad \mathrm{(C) \ } 300\qquad \mathrm{(D) \ } 400\qquad \mathrm{(E) \ } 500 </math> | <math> \mathrm{(A) \ } 100\qquad \mathrm{(B) \ } 200\qquad \mathrm{(C) \ } 300\qquad \mathrm{(D) \ } 400\qquad \mathrm{(E) \ } 500 </math> | ||
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== Problem 8 == | == Problem 8 == | ||
+ | In the figure, the length of side <math>AB</math> of square <math>ABCD</math> is <math>\sqrt{50}</math> and <math>BE=1</math>. What is the area of the inner square <math>EFGH</math>? | ||
+ | |||
+ | [[File:AMC102005Aq.png]] | ||
+ | |||
+ | <math> \textbf{(A)}\ 25\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 42 </math> | ||
[[2005 AMC 10A Problems/Problem 8|Solution]] | [[2005 AMC 10A Problems/Problem 8|Solution]] | ||
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== Problem 19 == | == Problem 19 == | ||
+ | Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point <math>B</math> from the line on which the bases of the original squares were placed? | ||
+ | |||
+ | <asy> | ||
+ | unitsize(1inch); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
+ | draw((0,0)--((1/3) + 3*(1/2),0)); | ||
+ | fill(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle, rgb(.7,.7,.7)); | ||
+ | draw(((1/6),0)--((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6),(1/2))--cycle); | ||
+ | draw(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle); | ||
+ | draw(((1/6) + 1,0)--((1/6) + 1,(1/2))--((1/6) + (3/2),(1/2))--((1/6) + (3/2),0)--cycle); | ||
+ | draw((2,0)--(2 + (1/3) + (3/2),0)); | ||
+ | draw(((2/3) + (3/2),0)--((2/3) + 2,0)--((2/3) + 2,(1/2))--((2/3) + (3/2),(1/2))--cycle); | ||
+ | draw(((2/3) + (5/2),0)--((2/3) + (5/2),(1/2))--((2/3) + 3,(1/2))--((2/3) + 3,0)--cycle); | ||
+ | label("$B$",((1/6) + (1/2),(1/2)),NW); | ||
+ | label("$B$",((2/3) + 2 + (1/4),(29/30)),NNE); | ||
+ | draw(((1/6) + (1/2),(1/2)+0.05)..(1,.8)..((2/3) + 2 + (1/4)-.05,(29/30)),EndArrow(HookHead,3)); | ||
+ | fill(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle, rgb(.7,.7,.7)); | ||
+ | draw(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle);</asy> | ||
+ | |||
+ | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \frac{3}{2}\qquad\textbf{(D)}\ \sqrt{2}+\frac{1}{2}\qquad\textbf{(E)}\ 2 </math> | ||
[[2005 AMC 10A Problems/Problem 19|Solution]] | [[2005 AMC 10A Problems/Problem 19|Solution]] | ||
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== Problem 23 == | == Problem 23 == | ||
− | <math> | + | Let <math>AB</math> be a diameter of a circle and let <math>C</math> be a point on <math>AB</math> with <math>2\cdot AC=BC</math>. Let <math>D</math> and <math>E</math> be points on the circle such that <math>DC \perp AB</math> and <math>DE</math> is a second diameter. What is the ratio of the area of <math>\triangle DCE</math> to the area of <math>\triangle ABD</math>? |
+ | |||
+ | <asy> | ||
+ | unitsize(2.5cm); | ||
+ | defaultpen(fontsize(10pt)+linewidth(.8pt)); | ||
+ | dotfactor=3; | ||
+ | pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); | ||
+ | pair D=dir(aCos(C.x)), E=(-D.x,-D.y); | ||
+ | draw(A--B--D--cycle); | ||
+ | draw(D--E--C); | ||
+ | draw(unitcircle,white); | ||
+ | drawline(D,C); | ||
+ | dot(O); | ||
+ | clip(unitcircle); | ||
+ | draw(unitcircle); | ||
+ | label("$E$",E,SSE); | ||
+ | label("$B$",B,E); | ||
+ | label("$A$",A,W); | ||
+ | label("$D$",D,NNW); | ||
+ | label("$C$",C,SW); | ||
+ | draw(rightanglemark(D,C,B,2));</asy> | ||
<math> \mathrm{(A) \ } \frac{1}{6} \qquad \mathrm{(B) \ } \frac{1}{4} \qquad \mathrm{(C) \ } \frac{1}{3} \qquad \mathrm{(D) \ } \frac{1}{2} \qquad \mathrm{(E) \ } \frac{2}{3} </math> | <math> \mathrm{(A) \ } \frac{1}{6} \qquad \mathrm{(B) \ } \frac{1}{4} \qquad \mathrm{(C) \ } \frac{1}{3} \qquad \mathrm{(D) \ } \frac{1}{2} \qquad \mathrm{(E) \ } \frac{2}{3} </math> | ||
Line 165: | Line 211: | ||
== Problem 24 == | == Problem 24 == | ||
− | For each positive integer <math> | + | For each positive integer <math> n > 1 </math>, let <math>P(n)</math> denote the greatest prime factor of <math>n</math>. For how many positive integers <math>n</math> is it true that both <math> P(n) = \sqrt{n} </math> and <math> P(n+48) = \sqrt{n+48} </math>? |
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5 </math> | <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5 </math> | ||
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== See also == | == See also == | ||
+ | {{AMC10 box|year=2005|ab=A|before=[[2004 AMC 10B Problems]]|after=[[2005 AMC 10B Problems]]}} | ||
+ | *[[AMC 10 Problems and Solutions]] | ||
* [[AMC Problems and Solutions]] | * [[AMC Problems and Solutions]] | ||
+ | {{MAA Notice}} |
Latest revision as of 21:15, 24 December 2020
2005 AMC 10A (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
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
While eating out, Mike and Joe each tipped their server dollars. Mike tipped of his bill and Joe tipped of his bill. What was the difference, in dollars between their bills?
Problem 2
For each pair of real numbers , define the operation as
.
What is the value of ?
Problem 3
The equations and have the same solution . What is the value of ?
Problem 4
A rectangle with a diagonal of length is twice as long as it is wide. What is the area of the rectangle?
Problem 5
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
Problem 6
The average (mean) of numbers is , and the average of other numbers is . What is the average of all numbers?
Problem 7
Josh and Mike live miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
Problem 8
In the figure, the length of side of square is and . What is the area of the inner square ?
Problem 9
Three tiles are marked and two other tiles are marked . The five tiles are randomly arranged in a row. What is the probability that the arrangement reads ?
Problem 10
There are two values of for which the equation has only one solution for . What is the sum of those values of ?
Problem 11
A wooden cube units on a side is painted red on all six faces and then cut into unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is ?
Problem 12
The figure shown is called a trefoil and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length ?
Problem 13
How many positive integers satisfy the following condition:
?
Problem 14
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
Problem 15
How many positive cubes divide ?
Problem 16
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is . How many two-digit numbers have this property?
Problem 17
In the five-sided star shown, the letters , , , , and are replaced by the numbers , , , , and , although not necessarily in this order. The sums of the numbers at the ends of the line segments , , , , and form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?
Problem 18
Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?
Problem 19
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point from the line on which the bases of the original squares were placed?
Problem 20
An equiangular octagon has four sides of length 1 and four sides of length , arranged so that no two consecutive sides have the same length. What is the area of the octagon?
Problem 21
For how many positive integers does evenly divide ?
Problem 22
Let be the set of the smallest positive multiples of , and let be the set of the smallest positive multiples of . How many elements are common to and ?
Problem 23
Let be a diameter of a circle and let be a point on with . Let and be points on the circle such that and is a second diameter. What is the ratio of the area of to the area of ?
Problem 24
For each positive integer , let denote the greatest prime factor of . For how many positive integers is it true that both and ?
Problem 25
In we have , , and . Points and are on and respectively, with and . What is the ratio of the area of triangle to the area of the quadrilateral ?
See also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2004 AMC 10B Problems |
Followed by 2005 AMC 10B Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.