# Difference between revisions of "2003 AMC 12A Problems"

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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>. | ||

+ | |||

+ | [asy] | ||

+ | unitsize(2cm); | ||

+ | defaultpen(fontsize(8)+linewidth(0.8)); | ||

+ | pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); | ||

+ | pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); | ||

+ | draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); | ||

+ | label("<math>A</math>",A,SE); | ||

+ | label("<math>B</math>",B,SW); | ||

+ | label("<math>C</math>",C,NW); | ||

+ | label("<math>D</math>",D,NE); | ||

+ | label("<math>K</math>",K,NNW); | ||

+ | label("<math>L</math>",L,E); | ||

+ | label("<math>M</math>",M,S); | ||

+ | label("<math>N</math>",N,W);[/asy] | ||

[[2003 AMC 12A Problems/Problem 14|Solution]] | [[2003 AMC 12A Problems/Problem 14|Solution]] |

## Revision as of 13:19, 7 February 2011

## 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 Leauge 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 Leauge?

## 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 orgin, 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 AKB, BLC, CMD, and DNA are equilateral triangles. If has an area of 16, find the area of .

[asy] unitsize(2cm); defaultpen(fontsize(8)+linewidth(0.8)); pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); label("",A,SE); label("",B,SW); label("",C,NW); label("",D,NE); label("",K,NNW); label("",L,E); label("",M,S); label("",N,W);[/asy]

## 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

## Problem 17

Square has sides of length , and is the midpoint of . A circle with radius and center intersects a circle with raidus 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

## Problem 20

## 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

## 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 are the same set?