Difference between revisions of "User:Rowechen"
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[[2006 AIME I Problems/Problem 7|Solution]] | [[2006 AIME I Problems/Problem 7|Solution]] | ||
− | == Problem | + | == Problem 12 == |
− | A | + | In isosceles triangle <math>ABC</math>, <math>A</math> is located at the origin and <math>B</math> is located at (20,0). Point <math>C</math> is in the first quadrant with <math>AC = BC</math> and angle <math>BAC = 75^{\circ}</math>. If triangle <math>ABC</math> is rotated counterclockwise about point <math>A</math> until the image of <math>C</math> lies on the positive <math>y</math>-axis, the area of the region common to the original and the rotated triangle is in the form <math>p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s</math>, where <math>p,q,r,s</math> are integers. Find <math>(p-q+r-s)/2</math>. |
− | [[2007 AIME I Problems/Problem | + | [[2007 AIME I Problems/Problem 12|Solution]] |
− | |||
− | |||
− | [[2006 AIME II Problems/Problem | + | == Problem 13 == |
+ | How many integers <math> N </math> less than 1000 can be written as the sum of <math> j </math> consecutive positive odd integers from exactly 5 values of <math> j\ge 1 </math>? | ||
+ | |||
+ | [[2006 AIME II Problems/Problem 13|Solution]] | ||
== Problem 9== | == Problem 9== |
Revision as of 20:22, 1 June 2020
Here's the AIME compilation I will be doing:
Contents
Problem 7
An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region to the area of shaded region
is 11/5. Find the ratio of shaded region
to the area of shaded region
Problem 12
In isosceles triangle ,
is located at the origin and
is located at (20,0). Point
is in the first quadrant with
and angle
. If triangle
is rotated counterclockwise about point
until the image of
lies on the positive
-axis, the area of the region common to the original and the rotated triangle is in the form
, where
are integers. Find
.
Problem 13
How many integers less than 1000 can be written as the sum of
consecutive positive odd integers from exactly 5 values of
?
Problem 9
The value of the sum
can be expressed in the form
, for some relatively prime positive integers
and
. Compute the value of
.
Problem 8
Determine the remainder obtained when the expression
is divided by
.
Problem 9
Let
where
and
. Determine the remainder obtained when
is divided by
.
Problem 11
A sequence is defined as follows and, for all positive integers
Given that
and
find the remainder when
is divided by 1000.
Problem 10
, and
are positive real numbers such that
Compute the value of
.
Problem 11
,
, and
are complex numbers such that
Let , where
. Determine the value of
.
Problem 12
is a scalene triangle. The circle with diameter
intersects
at
, and
is the foot of the altitude from
.
is the intersection of
and
. Given that
,
, and
, determine the circumradius of
.
Problem 13
Point lies on side
of
so that
bisects
The perpendicular bisector of
intersects the bisectors of
and
in points
and
respectively. Given that
and
the area of
can be written as
where
and
are relatively prime positive integers, and
is a positive integer not divisible by the square of any prime. Find
Problem 15
Let be an acute triangle with circumcircle
and let
be the intersection of the altitudes of
Suppose the tangent to the circumcircle of
at
intersects
at points
and
with
and
The area of
can be written as
where
and
are positive integers, and
is not divisible by the square of any prime. Find
Problem 14
Let be a quadratic polynomial with complex coefficients whose
coefficient is
Suppose the equation
has four distinct solutions,
Find the sum of all possible values of
Problem 13
For each integer , let
be the number of
-element subsets of the vertices of a regular
-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of
such that
.
Problem 15
In triangle , we have
,
, and
. Points
,
, and
are selected
on
,
, and
respectively such that
,
, and
concur at the circumcenter of
. The value of
can be expressed as
where
and
are relatively prime positive integers. Determine
.