Difference between revisions of "2003 AMC 12A Problems"
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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 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); | unitsize(2cm); | ||
defaultpen(fontsize(8)+linewidth(0.8)); | defaultpen(fontsize(8)+linewidth(0.8)); | ||
Line 113: | Line 113: | ||
pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); | 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); | draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); | ||
− | label(" | + | label("$A$",A,SE); |
− | label(" | + | label("$B$",B,SW); |
− | label(" | + | label("$C$",C,NW); |
− | label(" | + | label("$D$",D,NE); |
− | label(" | + | label("$K$",K,NNW); |
− | label(" | + | label("$L$",L,E); |
− | label(" | + | label("$M$",M,S); |
− | label(" | + | label("$N$",N,W); |
− | + | </asy> | |
[[2003 AMC 12A Problems/Problem 14|Solution]] | [[2003 AMC 12A Problems/Problem 14|Solution]] |
Revision as of 11:17, 26 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
.
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 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
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 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?
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
are the same set?