Inspired by lbh_qys
by sqing, May 20, 2025, 1:16 PM
Let 
be real numbers such that 
and 
Prove that
Where 
be real numbers such that 
and 
Prove that
Where 
be positive real numbers. Prove 
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 
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:
![\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]](//latex.artofproblemsolving.com/c/b/5/cb551e6d6ce94902b7bcb78f6da210d6c27b96d5.png)
with equality if and only if 
.
with 
. Prove that:![\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]](http://latex.artofproblemsolving.com/c/b/5/cb551e6d6ce94902b7bcb78f6da210d6c27b96d5.png)
with equality if and only if 
.This post has been edited 1 time. Last edited by danilorj, Today at 5:18 AM
Reason: ..
Reason: ..
.
.
Where 
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.
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?
be an integer, 
a real number, and 
be positive real numbers such that 
. Prove that:![\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]](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)
and 
. Prove that:![\[ \frac{a^2+1}{a+1}+\frac{b^2+1}{b+1}+\frac{c^2+1}{c+1} \geq 3\]](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 ![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](//latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)
are odd. (For a real number 
, 
denotes the largest integer not exceeding .)
Proposed by Hong Kong
be a given integer. Prove that infinitely many terms of the sequence 
, defined by ![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](http://latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)
are odd. (For a real number 
, 
denotes the largest integer not exceeding .)Proposed by Hong Kong
This post has been edited 3 times. Last edited by djmathman, Jul 24, 2015, 8:15 PM
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