Difference between revisions of "2003 AMC 12A Problems"
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+ | {{AMC12 Problems|year=2003|ab=A}} | ||
== Problem 1 == | == Problem 1 == | ||
What is the difference between the sum of the first <math>2003</math> even counting numbers and the sum of the first <math>2003</math> odd counting numbers? | What is the difference between the sum of the first <math>2003</math> even counting numbers and the sum of the first <math>2003</math> odd counting numbers? | ||
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== Problem 2 == | == Problem 2 == | ||
− | Members of the Rockham Soccer | + | Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? |
<math> \mathrm{(A) \ } 77\qquad \mathrm{(B) \ } 91\qquad \mathrm{(C) \ } 143\qquad \mathrm{(D) \ } 182\qquad \mathrm{(E) \ } 286 </math> | <math> \mathrm{(A) \ } 77\qquad \mathrm{(B) \ } 91\qquad \mathrm{(C) \ } 143\qquad \mathrm{(D) \ } 182\qquad \mathrm{(E) \ } 286 </math> | ||
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== Problem 9 == | == Problem 9 == | ||
− | A set <math>S</math> of points in the <math>xy</math>-plane is symmetric about the | + | A set <math>S</math> of points in the <math>xy</math>-plane is symmetric about the origin, both coordinate axes, and the line <math>y=x</math>. If <math>(2,3)</math> is in <math>S</math>, what is the smallest number of points in <math>S</math>? |
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16 </math> | <math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16 </math> | ||
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== Problem 14 == | == Problem 14 == | ||
− | Points <math>K, L, M,</math> and <math>N</math> lie in the plane of the square <math>ABCD</math> such that AKB, BLC, CMD, and DNA are equilateral triangles. If <math>ABCD</math> has an area of 16, find the area of <math>KLMN</math>. | + | Points <math>K, L, M,</math> and <math>N</math> lie in the plane of the square <math>ABCD</math> such that <math>AKB</math>, <math>BLC</math>, <math>CMD</math>, and <math>DNA</math> are equilateral triangles. If <math>ABCD</math> has an area of 16, find the area of <math>KLMN</math>. |
<asy> | <asy> | ||
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label("$N$",N,W); | label("$N$",N,W); | ||
</asy> | </asy> | ||
+ | |||
+ | <math> \textrm{(A)}\ 32\qquad\textrm{(B)}\ 16+16\sqrt{3}\qquad\textrm{(C)}\ 48\qquad\textrm{(D)}\ 32+16\sqrt{3}\qquad\textrm{(E)}\ 64 </math> | ||
[[2003 AMC 12A Problems/Problem 14|Solution]] | [[2003 AMC 12A Problems/Problem 14|Solution]] | ||
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Square <math>ABCD</math> has sides of length <math>4</math>, and <math>M</math> is the midpoint of <math>\overline{CD}</math>. A circle with radius <math>2</math> and center <math>M</math> intersects a circle with radius <math>4</math> and center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance from <math>P</math> to <math>\overline{AD}</math>? | Square <math>ABCD</math> has sides of length <math>4</math>, and <math>M</math> is the midpoint of <math>\overline{CD}</math>. A circle with radius <math>2</math> and center <math>M</math> intersects a circle with radius <math>4</math> and center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance from <math>P</math> to <math>\overline{AD}</math>? | ||
− | + | <asy> | |
+ | pair A,B,C,D,M,P; | ||
+ | D=(0,0); | ||
+ | C=(10,0); | ||
+ | B=(10,10); | ||
+ | A=(0,10); | ||
+ | M=(5,0); | ||
+ | P=(8,4); | ||
+ | dot(M); | ||
+ | dot(P); | ||
+ | draw(A--B--C--D--cycle,linewidth(0.7)); | ||
+ | draw((5,5)..D--C..cycle,linewidth(0.7)); | ||
+ | draw((7.07,2.93)..B--A--D..cycle,linewidth(0.7)); | ||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,NE); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,SW); | ||
+ | label("$M$",M,S); | ||
+ | label("$P$",P,N); | ||
+ | </asy> | ||
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}</math> | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}</math> | ||
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== Problem 19 == | == Problem 19 == | ||
− | A parabola with equation <math>y=ax^2+bx+c</math> is reflected about the <math>x</math>-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of <math>y=f(x)</math> and <math>y=g(x)</math>, respectively. Which of the following describes the graph of <math>y=(f+g)x</math>? | + | A parabola with equation <math>y=ax^2+bx+c</math> is reflected about the <math>x</math>-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of <math>y=f(x)</math> and <math>y=g(x)</math>, respectively. Which of the following describes the graph of <math>y=(f+g)(x)</math>? |
<math> \textbf{(A)}\ \text{a parabola tangent to the }x\text{-axis} </math> | <math> \textbf{(A)}\ \text{a parabola tangent to the }x\text{-axis} </math> | ||
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== Problem 20 == | == Problem 20 == | ||
− | How many 15-letter arrangements of 5 A's, 5 B's, and 5 C's have no A's in the first 5 letters, no B's in the next 5 letters, and no C's in the last 5 letters? | + | How many <math>15</math>-letter arrangements of <math>5</math> A's, <math>5</math> B's, and <math>5</math> C's have no A's in the first <math>5</math> letters, no B's in the next <math>5</math> letters, and no C's in the last <math>5</math> letters? |
− | <math> \ | + | <math> \textrm{(A)}\ \sum_{k=0}^{5}\binom{5}{k}^{3}\qquad\textrm{(B)}\ 3^{5}\cdot 2^{5}\qquad\textrm{(C)}\ 2^{15}\qquad\textrm{(D)}\ \frac{15!}{(5!)^{3}}\qquad\textrm{(E)}\ 3^{15} </math> |
[[2003 AMC 12A Problems/Problem 20|Solution]] | [[2003 AMC 12A Problems/Problem 20|Solution]] | ||
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== Problem 25 == | == Problem 25 == | ||
− | Let <math>f(x)= \sqrt{ax^2+bx} </math>. For how many real values of <math>a</math> is there at least one positive value of <math> b </math> for which the domain of <math>f </math> and the range <math> f </math> are the same set? | + | Let <math>f(x)= \sqrt{ax^2+bx} </math>. For how many real values of <math>a</math> is there at least one positive value of <math> b </math> for which the domain of <math>f </math> and the range of <math> f </math> are the same set? |
<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ \mathrm{infinitely \ many} } </math> | <math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ \mathrm{infinitely \ many} } </math> | ||
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== See also == | == See also == | ||
+ | |||
+ | {{AMC12 box|year=2003|ab=A|before=[[2002 AMC 12B Problems]]|after=[[2003 AMC 12B Problems]]}} | ||
+ | |||
* [[AMC 12]] | * [[AMC 12]] | ||
* [[AMC 12 Problems and Solutions]] | * [[AMC 12 Problems and Solutions]] | ||
* [[2003 AMC 12A]] | * [[2003 AMC 12A]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 22:00, 26 March 2021
2003 AMC 12A (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
What is the difference between the sum of the first even counting numbers and the sum of the first odd counting numbers?
Problem 2
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?
Problem 3
A solid box is cm by cm by cm. A new solid is formed by removing a cube cm on a side from each corner of this box. What percent of the original volume is removed?
Problem 4
It takes Mary minutes to walk uphill km from her home to school, but it takes her only minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?
Problem 5
The sum of the two 5-digit numbers and is . What is ?
Problem 6
Define to be for all real numbers and . Which of the following statements is not true?
for all and
for all and
for all
for all
if
Problem 7
How many non-congruent triangles with perimeter have integer side lengths?
Problem 8
What is the probability that a randomly drawn positive factor of is less than ?
Problem 9
A set of points in the -plane is symmetric about the origin, both coordinate axes, and the line . If is in , what is the smallest number of points in ?
Problem 10
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of , respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?
Problem 11
A square and an equilateral triangle have the same perimeter. Let be the area of the circle circumscribed about the square and the area of the circle circumscribed around the triangle. Find .
Problem 12
Sally has five red cards numbered through and four blue cards numbered through . She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
Problem 13
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?
Problem 14
Points and lie in the plane of the square such that , , , and are equilateral triangles. If has an area of 16, find the area of .
Problem 15
A semicircle of diameter sits at the top of a semicircle of diameter , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
Problem 16
A point P is chosen at random in the interior of equilateral triangle . What is the probability that has a greater area than each of and ?
Problem 17
Square has sides of length , and is the midpoint of . A circle with radius and center intersects a circle with radius and center at points and . What is the distance from to ?
Problem 18
Let be a -digit number, and let and be the quotient and the remainder, respectively, when is divided by . For how many values of is divisible by ?
Problem 19
A parabola with equation is reflected about the -axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of and , respectively. Which of the following describes the graph of ?
Problem 20
How many -letter arrangements of A's, B's, and C's have no A's in the first letters, no B's in the next letters, and no C's in the last letters?
Problem 21
The graph of the polynomial
has five distinct -intercepts, one of which is at . Which of the following coefficients cannot be zero?
Problem 22
Objects and move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object starts at and each of its steps is either right or up, both equally likely. Object starts at and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
Problem 23
How many perfect squares are divisors of the product ?
Problem 24
If what is the largest possible value of
Problem 25
Let . For how many real values of is there at least one positive value of for which the domain of and the range of are the same set?
See also
2003 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2002 AMC 12B Problems |
Followed by 2003 AMC 12B 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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.