3 var inquality
by sqing, Mar 31, 2025, 4:19 AM
Let 
and 
Prove that
and 
Prove that
This post has been edited 1 time. Last edited by sqing, 44 minutes ago
Number theory
by spiderman0, Mar 30, 2025, 4:51 PM
Find all n such that 
is divisibly by 
.
I want a solution that uses order or a solution like “let p be the least prime divisor of n”
is divisibly by 
.I want a solution that uses order or a solution like “let p be the least prime divisor of n”
possible triangle inequality
by sunshine_12, Mar 30, 2025, 2:12 PM
This post has been edited 2 times. Last edited by sunshine_12, Yesterday at 2:41 PM
Inspired by Ruji2018252
by sqing, Mar 30, 2025, 1:09 PM
Let 
be reals such that 
and 
Prove that
be reals such that 
and 
Prove that
This post has been edited 1 time. Last edited by sqing, an hour ago
and 
.
Prove 
Find maximum 
Harmonic Series and Infinite Sequences
by steven_zhang123, Mar 29, 2025, 12:41 AM
Let 
and 
be two infinite sequences of integers. Prove that there exists an infinite sequence of integers 
such that for any positive integer 
, the following holds:
![\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]](//latex.artofproblemsolving.com/2/5/9/259ee32c89befe7a3ee2797e0d5b1df1c3066cf6.png)
and 
be two infinite sequences of integers. Prove that there exists an infinite sequence of integers 
such that for any positive integer 
, the following holds:![\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]](http://latex.artofproblemsolving.com/2/5/9/259ee32c89befe7a3ee2797e0d5b1df1c3066cf6.png)
area of triangle
by QueenArwen, Mar 24, 2025, 12:10 PM
In a triangle 
with right angle 
, the altitude 
is drawn. An arbitrary circle passing through points and 
meets the segments 
, 
and 
for the second time at points 
, 
and 
respectively. Segments 
and 
meet at point 
. What is greater: the area of triangle 
or the sum of areas of triangles 
and 
? (5 marks)
with right angle 
, the altitude 
is drawn. An arbitrary circle passing through points and 
meets the segments 
, 
and 
for the second time at points 
, 
and 
respectively. Segments 
and 
meet at point 
. What is greater: the area of triangle 
or the sum of areas of triangles 
and 
? (5 marks)Attachments:
This post has been edited 1 time. Last edited by QueenArwen, Mar 24, 2025, 12:25 PM
such that for all reals 
Proposed by Aritra12, India
and 
imply that either 
or 
.
implies that 
.
also implies that 
implies that for any 
,
or 
.
,
is injective. 
and 
imply that 
for all 
.
.
such that 
,
gives 
for all such 
.
then gives 
and thus 
.
which means that 
.
for all 
for some 
,
which is impossible. Thus 
in this case.x is rational implies y is rational
by pohoatza, Jun 28, 2007, 7:24 PM
For 
let 
be the number whose 
-th digit after the decimal point is the 
-th digit after the decimal point of 
. Show that if is rational then so is 
.
Proposed by J.P. Grossman, Canada
let 
be the number whose 
-th digit after the decimal point is the 
-th digit after the decimal point of 
. Show that if is rational then so is 
.Proposed by J.P. Grossman, Canada
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