Difference between revisions of "Mock AIME 5 Pre 2005 Problems"
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<math>\zeta_n</math> is defined this way: | <math>\zeta_n</math> is defined this way: | ||
− | <math>\zeta_{n + 1} = (\frac {\zeta_n - i}{\zeta_n + i})^{ - 1}</ | + | <math>\zeta_{n + 1} = (\frac {\zeta_n - i}{\zeta_n + i})^{- 1}</math> |
− | If < | + | If <math>\zeta_0 = i + \frac{1}{121}</math> and 2008th number of the set is in form of (<math>a + bi</math>), find <math>ab</math>. |
[[Mock AIME 5 Pre 2005 Problems/Problem 14|Solution]] | [[Mock AIME 5 Pre 2005 Problems/Problem 14|Solution]] |
Latest revision as of 07:27, 12 October 2022
Contents
Problem 1
Function is given such way that for all
,
If where
and
are relatively prime integers, computer
.
Problem 2
Two 5-digit numbers are called "responsible" if they are:
If two "responsible" numbers are small as possible, what is the sum of the three middle digits of and last two digits on the
? That is,
.
Problem 3
A triangle is called Heronian if its perimeter equals the area of the triangle. Heronian triangle has sides
and
. If the distance from its incenter to the circumcenter is exprssed as
where
is not divisible by squares of any prime, find the remainder when
is divided by
.
Problem 4
Eight boxes are numbered to
. The number of ways you can put 16 identical balls into the boxes such that none of them is empty is expressed as
, where
. What is the remainder when
is divided by
?
Problem 5
There are three rooms in Phil's Motel. One room for one person, one room for three people, and another one room for four people. Let's say Peter and his seven friends came to the Phil's Motel. How many ways are there to house Peter and his seven friends in these rooms?
Problem 6
Larry and his two friends toss a die four times. From all possible outcomes, the probability that outcome has at least one occurrence of 2 is where
and
are relatively prime integers. Find
.
Problem 7
In , X is on
, dividing
. Y is on
, dividing
. V is on
, dividing
. It is found that ray VT intersects
at Z. Find
Problem 8
Let and that:
If with
, then what is
?
Problem 9
On a circle with center , two points,
and
are on a circle and
is outside the circle but in ray XY.
and
. If
where
and
are relatively prime integers, and such that
, find
Problem 10
Given that:
Then what is the smallest integer that is divisible and
?
Problem 11
There are number of ways to represent 10000000 as product of three factors and
number of ways to represent 11390625 as product of three factors. If
, find
.
Problem 12
Let . Find the sum of digits of
.
Problem 13
If the reciprocal of sum of real roots of the following equation can be written in form of where
, find
.
Problem 14
The set contains complex numbers
such that
.
is defined this way:
If and 2008th number of the set is in form of (
), find
.
Problem 15
Call the value of as
. Let the value of
as
. Find the absolute value of
.