Difference between revisions of "User:Fura3334"
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==Problem X== | ==Problem X== | ||
(I haven't decided the problem number yet) | (I haven't decided the problem number yet) | ||
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Let <math>N=109007732774081</math>. Given that <math>N=pq</math> where <math>p</math>, <math>q</math> are distinct primes greater than <math>1000</math>, and that <math>N</math> cannot be expressed as the sum of 2 perfect squares, find the remainder when | Let <math>N=109007732774081</math>. Given that <math>N=pq</math> where <math>p</math>, <math>q</math> are distinct primes greater than <math>1000</math>, and that <math>N</math> cannot be expressed as the sum of 2 perfect squares, find the remainder when | ||
<cmath>\frac{\varphi(N)^2}{2} - (\varphi(N)+1)^6</cmath> | <cmath>\frac{\varphi(N)^2}{2} - (\varphi(N)+1)^6</cmath> | ||
is divided by 144. | is divided by 144. |
Revision as of 06:02, 17 February 2024
IF YOU'RE AN ADMIN, PLS DONT DELETE THIS PAGE, IM WORKING ON SUS MOCK AIME (well, if i haven't edited this page for 2 weeks, you can delete it)
Problem 1
Kube the robot completes a task repeatedly, each time taking minutes. One day, Furaken asks Kube to complete
identical tasks in
hours. If Kube works slower and spends
minutes on each task, it will finish
tasks in exactly
hours. If Kube works faster and spends
minutes on each task, it can finish
tasks in
hours with
minutes to spare. Find
.
Problem 2
Let ,
,
be positive real numbers such that
Find .
Problem 3
Let be an odd prime such that
. Find
.
Problem 4
For triangle , let
be the midpoint of
. Extend
to
such that
. Let
be the point on
such that
, and let
be the point on
such that
. Line
intersects line
at
such that
. Given that
is parallel to
, the maximum possible area of triangle
can be written as
where
and
are relatively prime positive integers. Find
.
Problem 5
Let be a root of the polynomial
where
,
,
,
are integers. Find
.
Problem 6
Fly has a large number of red, yellow, green and blue pearls. Fly is making a necklace consisting of pearls as shown in the diagram. One slot already has a red pearl, and another slot has a green pearl. Find the number of ways to fill the
remaining slots such that any two pearls that are connected directly have different colors.
Problem X
(I haven't decided the problem number yet)
Let . Given that
where
,
are distinct primes greater than
, and that
cannot be expressed as the sum of 2 perfect squares, find the remainder when
is divided by 144.