Difference between revisions of "2011 AIME II Problems"
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== Problem 15 == | == Problem 15 == | ||
Let <math>P(x) = x^2 - 3x - 9</math>. A real number <math>x</math> is chosen at random from the interval <math>5 \le x \le 15</math>. The probability that <math>\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}</math> is equal to <math>\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}</math> , where <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, and <math>e</math> are positive integers, and none of <math>a</math>, <math>b</math>, or <math>c</math> is divisible by the square of a prime. Find <math>a + b + c + d + e</math>. | Let <math>P(x) = x^2 - 3x - 9</math>. A real number <math>x</math> is chosen at random from the interval <math>5 \le x \le 15</math>. The probability that <math>\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}</math> is equal to <math>\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}</math> , where <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, and <math>e</math> are positive integers, and none of <math>a</math>, <math>b</math>, or <math>c</math> is divisible by the square of a prime. Find <math>a + b + c + d + e</math>. | ||
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[[2011 AIME II Problems/Problem 15|Solution]] | [[2011 AIME II Problems/Problem 15|Solution]] |
Revision as of 12:05, 1 April 2011
2011 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Gary purchased a large bevarage, but only drank m/n of it, where m and n are relatively prime positive integers. If he had purchased half as much and drank twice as much, he would have wasted only 2/9 as much bevarage. Find m+n.
Problem 2
On square ABCD, point E lies on side AD and point F lies on side BC, so that BE=EF=FD=30. Find the area of the square ABCD.
Problem 3
The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.
Problem 4
In triangle ABC, AB=(20/11)AC. The angle bisector of angle A intersects BC at point D, and point M is the midpoint of AD. Let P be the point of intersection of AC and the line BM. The ratio of CP to PA can be expresses in the form m/n, where m and n are relatively prime positive integers. Find m+n.
Problem 5
The sum of the first 2011 terms of a geometric sequence is 200. The sum of the first 4022 terms is 380. Find the sum of the first 6033 terms.
Problem 6
Define an ordered quadruple (a, b, c, d) as interesting if , and a+d>b+c. How many interesting ordered quadruples are there?
Problem 7
Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let be the maximum number of red marbles for which such an arrangement is possible, and let
be the number of ways he can arrange the
marbles to satisfy the requirement. Find the remainder when
is divided by
.
Problem 8
Let ,
,
,
,
be the 12 zeroes of the polynomial
. For each
, let
be one of
or
. Then the maximum possible value of the real part of
can be written as
, where
and
are positive integers. Find
.
Problem 9
Let ,
,
,
be nonnegative real numbers such that
, and
. Let
and
be positive relatively prime integers such that
is the maximum possible value of
. Find
.
Problem 10
A circle with center has radius 25. Chord
of length 30 and chord
of length 14 intersect at point
. The distance between the midpoints of the two chords is 12. The quantity
can be represented as
, where
and
are relatively prime positive integers. Find the remainder when
is divided by 1000.
Problem 11
Let be the
matrix with entries as follows: for
,
; for
,
; all other entries in
are zero. Let
be the determinant of matrix
. Then
can be represented as
, where
and
are relatively prime positive integers. Find
.
Note: The determinant of the
matrix
is
, and the determinant of the
matrix
; for
, the determinant of an
matrix with first row or first column
is equal to
, where
is the determinant of the
matrix formed by eliminating the row and column containing
.
Problem 12
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be , where
and
are relatively prime positive integers. Find
.
Problem 13
Point lies on the diagonal
of square
with $AP \gt CP$ (Error compiling LaTeX. Unknown error_msg). Let
and
be the circumcenters of triangles
and
, respectively. Given that
and
, then
, where
and
are positive integers. Find
.
Problem 14
There are permutations
of
such that for
,
divides
for all integers
with
. Find the remainder when
is divided by 1000.
Problem 15
Let . A real number
is chosen at random from the interval
. The probability that
is equal to
, where
,
,
,
, and
are positive integers, and none of
,
, or
is divisible by the square of a prime. Find
.