Prove the inequality
by Butterfly, May 20, 2025, 12:36 PM
Inspired by lbh_qys
by sqing, May 20, 2025, 9:13 AM
Let
be real numbers such that
and
Prove that
Equality holds when 
Equality holds when 
Equality holds when 









This post has been edited 2 times. Last edited by sqing, 5 hours ago
Hard Inequality
by danilorj, May 20, 2025, 5:17 AM
Let
with
. Prove that:
with equality if and only if
.


![\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]](http://latex.artofproblemsolving.com/c/b/5/cb551e6d6ce94902b7bcb78f6da210d6c27b96d5.png)

This post has been edited 1 time. Last edited by danilorj, Today at 5:18 AM
Reason: ..
Reason: ..
Interesting inequalities
by sqing, May 20, 2025, 4:24 AM
Inspired by SXJX (12)2022 Q1167
by sqing, May 19, 2025, 4:01 AM
Geometry
by MathsII-enjoy, May 15, 2025, 10:39 AM
Given triangle
inscribed in
with
being the midpoint of
. The tangents at
of
intersect at
. Let
be the projection of
onto
. On the perpendicular bisector of
, take a point
that is not on
and different from M. Circle
intersects
at
. Lines
and
intersect at
. Prove that
is an isosceles triangle.




















Inequality with rational function
by MathMystic33, May 13, 2025, 5:42 PM
Let
be an integer,
a real number, and
be positive real numbers such that
. Prove that:
![\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]](//latex.artofproblemsolving.com/0/8/d/08de0c74a4e36b50a64d17875d3fd93eeb5b52de.png)
When does equality hold?




![\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]](http://latex.artofproblemsolving.com/0/8/d/08de0c74a4e36b50a64d17875d3fd93eeb5b52de.png)
When does equality hold?
Inequality with ^3+b^3+c^3+3abc=6
by bel.jad5, Sep 2, 2018, 12:45 PM
Let
and
. Prove that:
![\[ \frac{a^2+1}{a+1}+\frac{b^2+1}{b+1}+\frac{c^2+1}{c+1} \geq 3\]](//latex.artofproblemsolving.com/7/d/2/7d2e34116452912cd3da2925312ad500bb976f15.png)


![\[ \frac{a^2+1}{a+1}+\frac{b^2+1}{b+1}+\frac{c^2+1}{c+1} \geq 3\]](http://latex.artofproblemsolving.com/7/d/2/7d2e34116452912cd3da2925312ad500bb976f15.png)
This post has been edited 2 times. Last edited by bel.jad5, Sep 2, 2018, 1:01 PM
IMO Shortlist 2014 N4
by hajimbrak, Jul 11, 2015, 9:07 AM
Let
be a given integer. Prove that infinitely many terms of the sequence
, defined by
are odd. (For a real number
,
denotes the largest integer not exceeding
.)
Proposed by Hong Kong


![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](http://latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)



Proposed by Hong Kong
This post has been edited 3 times. Last edited by djmathman, Jul 24, 2015, 8:15 PM
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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