Y by PikaPika999
Determine the lengths of the edges of a right tetrahedron of volume 
so that the sum of its edges' lengths is minumum.
so that the sum of its edges' lengths is minumum.G
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Last Poster
Vertices of a convex polygon if and only if m(S) = f(n)
orl 12
N
by Maximilian113
Source: IMO Shortlist 2000, C3
Let 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](//latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
12 replies
Inequalities
Scientist10 2
N
by arqady
If 
, then prove that the following inequality holds:
![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](//latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
, then prove that the following inequality holds:![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
2 replies
$n$ with $2000$ divisors divides $2^n+1$ (IMO 2000)
Valentin Vornicu 65
N
by ray66
Source: IMO 2000, Problem 5, IMO Shortlist 2000, Problem N3
Does there exist a positive integer such that has exactly 2000 prime divisors and divides 
?
such that has exactly 2000 prime divisors and divides 
?
65 replies
Find the smallest of sum of elements
hlminh 0
Let 
and 
is a subset of 
such that if 
then 
Find the smallest of 
and 
is a subset of 
such that if 
then 
Find the smallest of 
0 replies
Easy IMO 2023 NT
799786 133
N
by Maximilian113
Source: IMO 2023 P1
Determine all composite integers 
that satisfy the following property: if 
, 
, 
, 
are all the positive divisors of 
with 
, then 
divides 
for every 
.
that satisfy the following property: if 
, 
, 
, 
are all the positive divisors of 
with 
, then 
divides 
for every 
.
133 replies
Complicated FE
XAN4 2
N
by cazanova19921
Source: own
Find all solutions for the functional equation 
, in which 
: 
Note: the solution is actually quite obvious -
, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
, in which 
: 
Note: the solution is actually quite obvious -
, but the proof is important.Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
2 replies
Cute diophantine
TestX01 0
Find all sequences of four consecutive integers such that twice their product is perfect square minus nine.
0 replies
\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}
sqing 1
N
by sqing
Source: Own
Let 
and 
Prove that
and 
Prove that
1 reply
Stronger inequality than an old result
KhuongTrang 22
N
by KhuongTrang
Source: own, inspired
Problem. Find the best constant 
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](//latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](http://latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
22 replies
Something nice
KhuongTrang 26
N
by KhuongTrang
Source: own
Problem. Given 
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
26 replies
IMO 2012/5 Mockup
v_Enhance 27
N
by Ilikeminecraft
Source: USA December TST for IMO 2013, Problem 3
Let 
be a scalene triangle with 
, and let 
be the foot of the altitude from 
. Let 
be a point in the interior of the segment 
. Let 
be the point on the segment 
such that 
. Similarly, let 
be the point on the segment 
such that 
. The circumcircle of triangle 
intersects segment 
at a second point 
(other than ). Prove that 
.
be a scalene triangle with 
, and let 
be the foot of the altitude from 
. Let 
be a point in the interior of the segment 
. Let 
be the point on the segment 
such that 
. Similarly, let 
be the point on the segment 
such that 
. The circumcircle of triangle 
intersects segment 
at a second point 
(other than ). Prove that 
.
27 replies
Right tetrahedron of fixed volume and min perimeter
G
H
G
H
Source: Romanian IMO TST 1981, Day 4 P3
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G
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