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3-variable inequality with min(ab,bc,ca)>=1
mathwizard888 72
N
by math-olympiad-clown
Source: 2016 IMO Shortlist A1
Let 
, 
, 
be positive real numbers such that 
. Prove that ![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](//latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
Proposed by Tigran Margaryan, Armenia
, 
, 
be positive real numbers such that 
. Prove that ![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
Proposed by Tigran Margaryan, Armenia
72 replies
with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
USAMO solutions
ABCD1728 1
N
by ohiorizzler1434
Do USAMO USATSTST and USATST have OFFICIAL solutions? Thanks!
1 reply
Distance between any two points is irrational
orl 21
N
by cursed_tangent1434
Source: IMO 1987, Day 2, Problem 5
Let 
be an integer. Prove that there is a set of 
points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
be an integer. Prove that there is a set of 
points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
21 replies
Symmetric inequality
mrrobotbcmc 5
N
by youochange
Let a,b,c,d belong to positive real numbers such that a+b+c+d=1. Prove that a^3/(b+c)+b^3/(c+d)+c^3/(d+a)+d^3/(a+b)>=1/8
5 replies
Two perpendiculars
jayme 0
Source: Own?
Dear Mathlinkers,
1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. J the center of 1b
7. V the second point of intersection of DJ and 1c.
Prove : CV is perpendicular to BC.
Sincerely
Jean-Louis
1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. J the center of 1b
7. V the second point of intersection of DJ and 1c.
Prove : CV is perpendicular to BC.
Sincerely
Jean-Louis
0 replies
1 viewing
Probably a good lemma
Zavyk09 4
N
by Zavyk09
Source: found when solving exercises
Let 
be a triangle with circumcircle 
. Arbitrary points 
on 
respectively. Circumcircle 
of triangle 
intersects at 
. 
intersects 
at 
. 
cuts and at 
respectively. Construct parallelogram 
. Prove that 
are collinear.
be a triangle with circumcircle 
. Arbitrary points 
on 
respectively. Circumcircle 
of triangle 
intersects at 
. 
intersects 
at 
. 
cuts and at 
respectively. Construct parallelogram 
. Prove that 
are collinear.
4 replies
shade from tub
QueenArwen 1
N
by mikestro
Source: 46th International Tournament of Towns, Senior O-Level P4, Spring 2025
There was a tub on the plane, with its upper base greater that the lower one. The tub was overturned. Prove that the area of its visible shade did decrease. (The tub is a frustum of a right circular cone: its bases are two discs in parallel planes, such that their centers lie on a line perpendicular to these planes. The visible shade is the total shade besides the shade under the tub. Consider the sun rays as parallel.)
1 reply
be real numbers such that 
.
ISI UGB 2025 P4
SomeonecoolLovesMaths 9
N
by nyacide
Source: ISI UGB 2025 P4
Let 
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer such that 
is called the period of 
. Determine the total number of points in 
of period 
.
(Hint :
)
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer such that 
is called the period of 
. Determine the total number of points in 
of period 
.(Hint :
)
9 replies
