Difference between revisions of "1996 AIME Problems"
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== Problem 13 == | == Problem 13 == | ||
+ | In triangle <math>ABC</math>, <math>AB=\sqrt{30}</math>, <math>AC=\sqrt{6}</math>, and <math>BC=\sqrt{15}</math>. There is a point <math>D</math> for which <math>\overline{AD}</math> bisects <math>\overline{BC}</math>, and <math>\angle ADB</math> is a right angle. The ratio | ||
+ | |||
+ | <cmath>\dfrac{Area(\triangle ADB)}{Area(\triangle ABC)}</cmath> | ||
+ | |||
+ | can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math> | ||
[[1996 AIME Problems/Problem 13|Solution]] | [[1996 AIME Problems/Problem 13|Solution]] |
Revision as of 16:22, 24 September 2007
Contents
Problem 1
Problem 2
For each real number , let
denote the greatest integer that does not exceed x. For how man positive integers
is it true that
and that
is a positive even integer?
Problem 3
Find the smallest positive integer for which the expansion of
, after like terms have been collected, has at least 1996 terms.
Problem 4
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of a shadow, which does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed
.
Problem 5
Suppose that the roots of are
,
, and
, and that the roots of
are
,
, and
. Find
.
Problem 6
In a five-team tournament, each team plays one game with every other team. Each team has a chance of winning any game it plays. (There are no ties.) Let
be the probability that the tournament will product neither an undefeated team nor a winless team, where
and
are relatively prime integers. Find
.
Problem 7
Two squares of a checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible?
Problem 8
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers with
is the harmonic mean of
and
equal to
?
Problem 9
A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?
Problem 10
Find the smallest positive integer solution to .
Problem 11
Let be the product of the roots of
that have an imaginary part, and suppose that
, where
and
. Find
.
Problem 12
For each permutation of the integers
, form the sum
.
The average value of all such sums can be written in the form , where
and
are relatively prime positive integers. Find
.
Problem 13
In triangle ,
,
, and
. There is a point
for which
bisects
, and
is a right angle. The ratio
can be written in the form , where
and
are relatively prime positive integers. Find
Problem 14
A rectangular solid is made by gluing together
cubes. An internal diagonal of this solid passes through the interiors of how many of the
cubes?