funny title placeholder
by pikapika007, Mar 21, 2025, 12:10 PM
Let 
be a set of integers with the following properties:
contains all positive integers.
be a set of integers with the following properties:
.- If
and 
, then 
. - If for some
, 
is composite, then all positive divisors of are in .
.
and 
,
.
,
is composite, then all positive divisors of contains all positive integers.This post has been edited 1 time. Last edited by pikapika007, Today at 12:12 PM
Reason: wrong year
Reason: wrong year
usamOOK geometry
by KevinYang2.71, Mar 21, 2025, 12:00 PM
Let 
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.Distributing cupcakes
by KevinYang2.71, Mar 21, 2025, 12:00 PM
Let 
and 
be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .
and 
be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .Scary Binomial Coefficient Sum
by EpicBird08, Mar 21, 2025, 11:59 AM
Determine, with proof, all positive integers 
such that 
is an integer for every positive integer 
such that 
is an integer for every positive integer 
This post has been edited 2 times. Last edited by EpicBird08, Today at 12:06 PM
high tech FE as J1?!
by imagien_bad, Mar 20, 2025, 12:00 PM
Let 
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.
Note: A function is bijective if for every integer 
, there exists exactly one integer 
such that 
.
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.Note: A function
is bijective if for every integer 
, there exists exactly one integer 
such that 
.This post has been edited 1 time. Last edited by imagien_bad, Yesterday at 12:09 PM
On a^4+b^4=c^4+d^4=e^5
by v_Enhance, Apr 29, 2015, 9:13 PM
Let , , , 
, 
be distinct positive integers such that 
. Show that 
is a composite number.
, , , 
, 
be distinct positive integers such that 
. Show that 
is a composite number.
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