Inspired by hlminh
by sqing, Apr 22, 2025, 8:09 AM
Let 
be real numbers such that 
Prove that
Where 
Where 
be real numbers such that 
Prove that
Where 
Where 
and 
.
density over modulo M
by SomeGuy3335, Apr 20, 2025, 4:19 AM
Let 
be a positive integer and let 
be an irrational number. Show that for every integer 
, there exists a positive integer 
such that 
.
be a positive integer and let 
be an irrational number. Show that for every integer 
, there exists a positive integer 
such that 
.This post has been edited 3 times. Last edited by SomeGuy3335, Yesterday at 2:57 PM
JBMO TST Bosnia and Herzegovina 2022 P3
by Motion, May 21, 2022, 9:37 PM
Let 
be an acute triangle. Tangents on the circumscribed circle of triangle at points 
and 
intersect at point 
. Let 
and 
be a foot of the altitudes from onto 
and 
and let be the midpoint of 
. Prove:
A) Prove that is the orthocenter of the triangle 
.
B) Prove that
cuts 
in half.
be an acute triangle. Tangents on the circumscribed circle of triangle at points 
and 
intersect at point 
. Let 
and 
be a foot of the altitudes from onto 
and 
and let be the midpoint of 
. Prove:A) Prove that
is the orthocenter of the triangle 
.B) Prove that
cuts 
in half.Inequality with n-gon sides
by mihaig, Feb 25, 2022, 6:46 AM
If 
are are the lengths of the sides of a 
gon such that
then
![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](//latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
When do we have equality?
(V. Cîrtoaje and L. Giugiuc, 2021)
are are the lengths of the sides of a 
gon such that
then![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](http://latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
When do we have equality?
(V. Cîrtoaje and L. Giugiuc, 2021)
Parity and sets
by betongblander, Mar 18, 2021, 9:17 PM
Let and 
be positive integers with 
. In a group of people, each one or always
speak the truth or always lie. Arnaldo can ask questions for any of these people
provided these questions are of the type: “In set
, what is the parity of people who speak to
true? ”, where is a subset of size 
of the set of people. The answer can only
be “
” or “
”.
a) For which values of and is it possible to determine which people speak the truth and
which people always lie?
b) What is the minimum number of questions required to determine which people
speak the truth and which people always lie, when that number is finite?
and 
be positive integers with 
. In a group of people, each one or alwaysspeak the truth or always lie. Arnaldo can ask questions for any of these people
provided these questions are of the type: “In set
, what is the parity of people who speak totrue? ”, where
is a subset of size 
of the set of people. The answer can onlybe “
” or “
”.a) For which values of
and is it possible to determine which people speak the truth andwhich people always lie?
b) What is the minimum number of questions required to determine which people
speak the truth and which people always lie, when that number is finite?
This post has been edited 14 times. Last edited by betongblander, Jun 7, 2021, 8:08 PM
Diophantine equation !
by ComplexPhi, May 14, 2015, 6:52 AM
Solve in nonnegative integers the following equation :
April 2025
March 2025