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== Problem 9 == | == Problem 9 == | ||
− | Let <math>ABCDEF</math> be a regular hexagon. Let <math>G</math>, <math>H</math>, <math>I</math>, <math>J</math>, <math>K</math>, and <math>L</math> be the midpoints of sides <math>AB</math>, <math>BC</math>, <math>CD</math>, <math>DE</math>, <math>EF</math>, and <math>AF</math>, respectively. The segments <math>\ | + | Let <math>ABCDEF</math> be a regular hexagon. Let <math>G</math>, <math>H</math>, <math>I</math>, <math>J</math>, <math>K</math>, and <math>L</math> be the midpoints of sides <math>AB</math>, <math>BC</math>, <math>CD</math>, <math>DE</math>, <math>EF</math>, and <math>AF</math>, respectively. The segments <math>\overline{AH}</math>, <math>\overline{BI}</math>, <math>\overline{CJ}</math>, <math>\overline{DK}</math>, <math>\overline{EL}</math>, and <math>\overline{FG}</math> bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of <math>ABCDEF</math> be expressed as a fraction <math>\frac {m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. |
[[2010 AIME II Problems/Problem 9|Solution]] | [[2010 AIME II Problems/Problem 9|Solution]] |
Revision as of 01:33, 1 March 2015
2010 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
[hide]Problem 1
Let be the greatest integer multiple of
all of whose digits are even and no two of whose digits are the same. Find the remainder when
is divided by
.
Problem 2
A point is chosen at random in the interior of a unit square
. Let
denote the distance from
to the closest side of
. The probability that
is equal to
, where
and
are relatively prime positive integers. Find
.
Problem 3
Let be the product of all factors
(not necessarily distinct) where
and
are integers satisfying
. Find the greatest positive integer
such that
divides
.
Problem 4
Dave arrives at an airport which has twelve gates arranged in a straight line with exactly feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks
feet or less to the new gate be a fraction
, where
and
are relatively prime positive integers. Find
.
Problem 5
Positive numbers ,
, and
satisfy
and
. Find
.
Problem 6
Find the smallest positive integer with the property that the polynomial
can be written as a product of two nonconstant polynomials with integer coefficients.
Problem 7
Let , where
,
, and
are real. There exists a complex number
such that the three roots of
are
,
, and
, where
. Find
.
Problem 8
Let be the number of ordered pairs of nonempty sets
and
that have the following properties:
-
,
-
,
- The number of elements of
is not an element of
,
- The number of elements of
is not an element of
.
Find .
Problem 9
Let be a regular hexagon. Let
,
,
,
,
, and
be the midpoints of sides
,
,
,
,
, and
, respectively. The segments
,
,
,
,
, and
bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of
be expressed as a fraction
where
and
are relatively prime positive integers. Find
.
Problem 10
Find the number of second-degree polynomials with integer coefficients and integer zeros for which
.
Problem 11
Define a T-grid to be a matrix which satisfies the following two properties:
- Exactly five of the entries are
's, and the remaining four entries are
's.
- Among the eight rows, columns, and long diagonals (the long diagonals are
and
, no more than one of the eight has all three entries equal.
Find the number of distinct T-grids.
Problem 12
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is . Find the minimum possible value of their common perimeter.
Problem 13
The cards in a deck are numbered
. Alex, Blair, Corey, and Dylan each pick a card from the deck randomly and without replacement. The two people with lower numbered cards form a team, and the two people with higher numbered cards form another team. Let
be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards
and
, and Dylan picks the other of these two cards. The minimum value of
for which
can be written as
, where
and
are relatively prime positive integers. Find
.
Problem 14
Triangle with right angle at
,
and
. Point
on $\overbar{AB}$ (Error compiling LaTeX. Unknown error_msg) is chosen such that
and
. The ratio
can be represented in the form
, where
,
,
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 15
In triangle ,
,
, and
. Points
and
lie on
with
and
. Points
and
lie on
with
and
. Let
be the point, other than
, of intersection of the circumcircles of
and
. Ray
meets
at
. The ratio
can be written in the form
, where
and
are relatively prime positive integers. Find
.
See also
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.