Difference between revisions of "2003 AMC 12B Problems"
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== Problem 24 == | == Problem 24 == | ||
| + | Positive integers <math>a,b,</math> and <math>c</math> are chosen so that <math>a<b<c</math>, and the system of [[equation]]s | ||
| + | <center><math>2x + y = 2003 \quad</math> and <math>\quad y = |x-a| + |x-b| + |x-c|</math></center> | ||
| + | has exactly one solution. What is the minimum value of <math>c</math>? | ||
| + | |||
| + | <math>\mathrm{(A)}\ 668 | ||
| + | \qquad\mathrm{(B)}\ 669 | ||
| + | \qquad\mathrm{(C)}\ 1002 | ||
| + | \qquad\mathrm{(D)}\ 2003 | ||
| + | \qquad\mathrm{(E)}\ 2004</math> | ||
[[2003 AMC 12B Problems/Problem 24|Solution]] | [[2003 AMC 12B Problems/Problem 24|Solution]] | ||
Revision as of 18:06, 1 February 2010
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
Which of the following is the same as
![\[\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}\]](http://latex.artofproblemsolving.com/6/d/8/6d8019823e919fab14bd496a543fe92645572845.png)
?
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Let 
be a 5-digit number, and let q and r be the quotient and remainder, respectively, when is divided by 100. For how many values of is 
divisible by 11?
Problem 19
Let 
be the set of permutations of the sequence 
for which the first term is not 
. A permutation is chosen randomly from . The probability that the second term is 
, in lowest terms, is 
. What is 
?
Problem 20
Problem 21
An object moves 
cm in a straight line from 
to 
, turns at an angle 
, measured in radians and chosen at random from the interval 
, and moves 
cm in a straight line to 
. What is the probability that 
?
Problem 22
Let 
be a rhombus with 
and 
. Let 
be a point on 
, and let 
and 
be the feet of the perpendiculars from to 
and 
, respectively. Which of the following is closest to the minimum possible value of 
?
![[asy] size(200); defaultpen(0.6); pair O = (15*15/17,8*15/17), C = (17,0), D = (0,0), P = (25.6,19.2), Q = (25.6, 18.5); pair A = 2*O-C, B = 2*O-D; pair P = (A+O)/2, Q=(B+O)/2, N=(A+B)/2; draw(A--B--C--D--cycle); draw(A--O--B--O--C--O--D); draw(P--N--Q); label("\(A\)",A,WNW); label("\(B\)",B,ESE); label("\(C\)",C,ESE); label("\(D\)",D,SW); label("\(P\)",P,SSW); label("\(Q\)",Q,SSE); label("\(N\)",N,NNE); [/asy]](http://latex.artofproblemsolving.com/c/2/2/c22d4d20155faaf8bd0cf51f26a5795d14b32322.png)
Problem 23
Problem 24
Positive integers 
and 
are chosen so that 
, and the system of equations
and 
has exactly one solution. What is the minimum value of ?
Problem 25
Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distance between the points are less than the radius of the circle?
