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Inspired by hlminh
sqing 1
N
by sqing
Source: Own
Let 
be real numbers such that 
Prove that
Where 
Where 
be real numbers such that 
Prove that
Where 
Where 
1 reply
Two very hard parallel
jayme 3
N
by jayme
Source: own inspired by EGMO
Dear Mathlinkers,
1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.
Prove : UV is parallel to PM.
Sincerely
Jean-Louis
1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.
Prove : UV is parallel to PM.
Sincerely
Jean-Louis
3 replies
Inequality with n-gon sides
mihaig 3
N
by mihaig
Source: VL
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)
3 replies
Advanced topics in Inequalities
va2010 23
N
by Novmath
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
23 replies
JBMO TST Bosnia and Herzegovina 2022 P3
Motion 7
N
by cafer2861
Source: JBMO TST Bosnia and Herzegovina 2022
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.
7 replies
and 
.
density over modulo M
SomeGuy3335 3
N
by ja.
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 
.
3 replies
Diophantine equation !
ComplexPhi 5
N
by aops.c.c.
Source: Romania JBMO TST 2015 Day 1 Problem 4
Solve in nonnegative integers the following equation :
5 replies
Combo problem
soryn 0
The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.
0 replies
Parity and sets
betongblander 7
N
by ihategeo_1969
Source: Brazil National Olympiad 2020 5 Level 3
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?
7 replies
