The Art of Problem solving
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The Art of Problem solving
par methodiX Mar 17 Aoû - 22:18
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13 March 2007
|
are multiples of 
? 
is equal to 
, where 
is a positive real number and 
. Given that the imaginary parts of 
and 
are equal, find 
years. Find 
to the corresponding Celsius temperature 
is 
. An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer. For how many integer Fahrenheit temperatures 
with 
does the original temperature equal the final temperature? 
denotes the greatest integer that is less than or equal to x, and 
denotes the least integer that is greater than or equal to x. 
is cubic. What is the largest value of 
for which the polynomials 
and 
are both factors of 
with right angle 
and 
. Its legs 
and 
are extended beyond 
and 
. Points 
and 
lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center 
, where 
and 
are relatively prime positive integers. Find 
. 
grid shown, 
of the 
be the number of shadings with this property. Find the remainder when 
. 
denote the unique positive integer 
. For example, 
and 
. If 
, find the remainder when S is divided by 1000. 
. Point 
and 
. If 
is rotated counterclockwise about point 
where 
, 
are integers. Find 
. 
and vertex 
has eight edges of length 4. A plane passes through the midpoints of 
, 
, and 
. The plane's intersection with the pyramid has an area that can be expressed as 
. Find 
, 
, and for 
, 
. Find the largest integer less than or equal to 
. 
and 
and 
, respectively, with 
and 
. Point 
such that 
. The area of triangle 
is 
. The two possible values of the length of side 
, where 
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Re: The Art of Problem solving
par methodiX Mar 17 Aoû - 22:18
| II | ||||||||||||||||||||||||||||||||||||||||||||
28 March 2007
|
. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 
. 
where 
, 
are positive integers, 
. 
, and points 
and 
. Find 
. 
is drawn on graph paper with each square representing one unit in each direction. How many of the 
by 
satisfies 
if 
is odd, and 
is 
let 
For a certain integer 
there are exactly 
positive integers 
such that 
and 
for all 
such that 
for 
and 
Points 
and 
The inscribed circle of triangle 
is tangent to 
at point 
and the inscribed circle of triangle 
Find 
be a set with six elements. Let 
be the set of all subsets of 
Subsets 
is 
where 
, 
(The set 
) 
and rolls along the surface toward the smaller tube, which has radius 
from where it starts. The distance 
where 
and 
consists entirely of integral powers of 
Given that 
and 
With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a 
or a 
? 
be a polynomial with real coefficients such that 
and for all 
Find 
and 
with the same radius are drawn in the interior of triangle 
is tangent to sides 
, 
to 
, 
, and 
is externally tangent to 
and 
the radius of 
, where 
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