AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
, for 
in at least 
different ways.
be positive real numbers satisfying 
. Prove that![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
+1 w
be real numbers such that 
. Prove that
Let be real numbers such that 
. Prove that
and 
, we define 
as follows:
where 
are all the positive divisors of . For example,
.
Find all positive integers such that 
.
Find all positive integers such that 
.
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
. Let 
and 
. Prove that the points 
, 
, and are concyclic. and are not the diagnols)
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
. Prove that there exists a positive integer satisfying 
.
?
and colors are given. We will say that a set of 
points on a plane is 
, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of the least integer such that: in every set of 
points of a plane, no three of which are collinear, consisting of points of every color there exists a subset.
, let 
be the set of all positive integers not exceeding that are relatively prime to . Consider the polynomial![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](http://latex.artofproblemsolving.com/3/a/c/3ac60cb70baf4dfcf297e9af87480a8ee34ba70c.png)
and a polynomial 
with integer coefficients such that![\[
P_n(x) = (x^{r_n} + 1) Q_n(x).
\]](http://latex.artofproblemsolving.com/7/0/e/70e19a9d3e0a7e97abc7aa08d3f23126c14ea7d0.png)
such that 
is irreducible in ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
be reals such that 
and 
.
,
,
be the sides of a triangle, and 
its area. Prove:![\[ a^{2} + b^{2} + c^{2}\geq 4S \sqrt {3}
\]](http://latex.artofproblemsolving.com/2/d/1/2d1e7e3cedeb2ef8b7b1960f5650a33d25b3eec4.png)
.
.
;
,
,
,
and find that 
.
. Because 
the inequality reduces to the following:
.
.
with area 
,
When do we have equality?
Substituting 
and 
,
which is equivalent to 
Equality when 
!
SXTB,(3)2022,Q2651![\[ a^2+b^2+c^2\ge 4\sqrt{4-sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}}S.\]](http://latex.artofproblemsolving.com/1/d/6/1d6dd9e1e497f29722d7149579875f8609160ce1.png)
Ionescu, early 1900
or 
.
so we need to prove 
or 
which is true.
,
is a convex function. so we get
as desired
be the semi-perimeter and 
be the area of the triangle respectively .
![$\frac{(s-a)+(s-b)+(s-c)}{3}\geq \sqrt[3]{(s-a)(s-b)(s-c)}\implies s\geq 3 \sqrt[3]{\frac{{\triangle}^2}{s}} \implies s^4\geq 27{\triangle}^2$](http://latex.artofproblemsolving.com/8/1/b/81b7cc9628adbe6821e226b549a06dbfd17b2c44.png)
Let 
,
,
well known lemma 
,
,
.
where 
.
![$16S^2=[(b+c+a)(b+c-a)(a+c-b)(a-(c-b)]$](http://latex.artofproblemsolving.com/9/d/3/9d312c0417af4b8b4306cbb57988853ae54fbb55.png)
![$48S^2=3[(b+c)^2-a^2)(a^2-(b-c)^2]$](http://latex.artofproblemsolving.com/e/2/2/e2283646504d0ee4a35f4016aa70087b0ebaedbc.png)
![$\sqrt{48S^2}=\sqrt{3[(b+c)^2-a^2)(a^2-(b-c)^2]}$](http://latex.artofproblemsolving.com/3/6/4/364fb697b859b631612497297068c2ffd074be45.png)
![$a^2+b^2+c^2\geq\sqrt{48S^2} \implies a^2+b^2+c^2\geq\sqrt{3[(b+c)^2-a^2)(a^2-(b-c)^2]}$](http://latex.artofproblemsolving.com/d/4/c/d4cd5e10619aa4eed08b674e90d97920e036744f.png)
![$a^2+b^2+c^2\geq\sqrt{3[(b+c)^2-a^2)(a^2-(b-c)^2]} \iff$](http://latex.artofproblemsolving.com/f/0/9/f096621f9eb1eb5ab3fb27fd3e92c320665bcc33.png)
![$(a^2+b^2+c^2)^2\geq3[(b+c)^2-a^2)(a^2-(b-c)^2] \iff$](http://latex.artofproblemsolving.com/4/e/8/4e8a20daaccef9c5464c1c0dc662195353483905.png)
![$(a^2+b^2+c^2)^2\geq3[(b+c)(b-c)]^2+[(b+c)a]^2-a^4+[(a(b-c)^2)] \iff$](http://latex.artofproblemsolving.com/2/e/e/2ee041eedc2e54901abf15a2bab5966b1c34b027.png)
![$(a^2+b^2+c^2)^2\geq3[(b^2-c^2)^2+(ab+ac)^2-a^4+(ab-ac)^2] \iff$](http://latex.artofproblemsolving.com/e/3/c/e3c5a09818d78e4f64be3389b2b7df0253dd3194.png)
![$(b^2-c^2)^2+2[(-a^2(a^2-b^2)+c^2(a^2-b^2)]\le0 \iff$](http://latex.artofproblemsolving.com/d/7/4/d74a5025b8f7787970ca9d26d0158294d3e1938d.png)
so if we let 
be the greater side of the triangle we'll have that 
but 
.
and this is true because since they're sides of a triangle then 
so 
and this finishes the first part.
in other words when 
or 
but in both cases we get 
.
.
s.t. 
and 
.