2 var inquality
by sqing, Apr 3, 2025, 5:16 AM
Let 
and 
Prove that
and 
Prove that
such that 
.
GCD of x^2-y, y^2-z and z^2-x
by EmersonSoriano, Apr 2, 2025, 9:38 PM
Find all positive integers 
that can be written in the form
where 
are pairwise coprime positive integers such that 
, 
, and 
.
that can be written in the form
where 
are pairwise coprime positive integers such that 
, 
, and 
.This post has been edited 3 times. Last edited by EmersonSoriano, 5 hours ago
Reason: change
Reason: change
and 
: Prove that :
such that 
for reals 
.Functional Equation from strings to a set of integers
by Jia_Le_Kong, Jul 29, 2023, 3:40 AM
Let 
be the set of all non-empty sub-strings of "Socrates".
Thus, "Soc", "crates"
but "teS", "sorat", "setarcoS" 
.
Define the sum of two strings
and 
as follows: take the string with the first term that appears first in "Socrates" (if both have the same first term, take the shorter one) and WLOG let it be . Then the sum of these two strings will be the string formed by all the letters between the first term of and the last term of . For instance, "Socr" + "te" 
"Socrate" and "cra" + "ates" "crates".
Find all functions
such that
for all
.
Bonus. What if we change "Socrates" to a string assumed to be of an arbitrary natural number length, with distinct letters (or symbols, if we need more)? The range
is still the same.
be the set of all non-empty sub-strings of "Socrates".Thus, "Soc", "crates"
but "teS", "sorat", "setarcoS" 
.Define the sum of two strings
and 
as follows: take the string with the first term that appears first in "Socrates" (if both have the same first term, take the shorter one) and WLOG let it be . Then the sum of these two strings will be the string formed by all the letters between the first term of and the last term of . For instance, "Socr" + "te" 
"Socrate" and "cra" + "ates" "crates".Find all functions
such that
for all
.Bonus. What if we change "Socrates" to a string assumed to be of an arbitrary natural number length, with distinct letters (or symbols, if we need more)? The range
is still the same.This post has been edited 1 time. Last edited by Jia_Le_Kong, Jul 29, 2023, 3:40 AM
Reason: typo
Reason: typo
Another FE
by Ankoganit, Jul 22, 2016, 9:04 AM
Find all functions 
such that 
for all reals 
.
such that 
for all reals 
.This post has been edited 1 time. Last edited by Ankoganit, Jul 22, 2016, 9:04 AM
Perpendicular following tangent circles
by buzzychaoz, Mar 21, 2016, 5:53 AM
The diagonals of a cyclic quadrilateral 
intersect at 
, and there exist a circle 
tangent to the extensions of 
at 
respectively. Circle 
passes through points 
, and is externally tangent to circle at . Prove that 
.
intersect at 
, and there exist a circle 
tangent to the extensions of 
at 
respectively. Circle 
passes through points 
, and is externally tangent to circle at . Prove that 
.Attachments:
Sequence of numbers in form of a^2+b^2
by TheOverlord, May 11, 2015, 2:14 PM
Let 
be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like
which 
.
be the sequence of all natural numbers which are sum of squares of two natural numbers.Prove that there exists infinite natural numbers like
which 
.IMO ShortList 2001, combinatorics problem 2
by orl, Sep 30, 2004, 5:23 PM
Let 
be an odd integer greater than 1 and let 
be integers. For each permutation 
of 
, define 
. Prove that there exist permutations 
of such that 
is a divisor of 
.
be an odd integer greater than 1 and let 
be integers. For each permutation 
of 
, define 
. Prove that there exist permutations 
of such that 
is a divisor of 
.Attachments:
This post has been edited 2 times. Last edited by orl, Oct 25, 2004, 12:13 AM
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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