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Geometry Parallel Proof Problem
CatalanThinker 1
N
by ItzsleepyXD
Source: No source found, just yet, please share if you find it though :)
Let M be the midpoint of the side BC of triangle ABC. The bisector of the exterior angle of point A intersects the side BC in D. Let the circumcircle of triangle ADM intersect the lines AB and AC in E and F respectively. If the midpoint of EF is N, prove that MN || AD.
I have done some constructions, but still did not quite get to the answer, see diagram attached below
I have done some constructions, but still did not quite get to the answer, see diagram attached below
1 reply
.
Let 
.
Kosovo MO 2010 Problem 5
Com10atorics 20
N
by sqing
Source: Kosovo MO 2010 Problem 5
Let 
be positive real numbers such that 
. Prove that
.
be positive real numbers such that 
. Prove that
.
20 replies
Fourth powers and square roots
willwin4sure 39
N
by awesomeming327.
Source: USA TSTST 2020 Problem 4, by Yang Liu
Find all pairs of positive integers 
satisfying the following conditions:
[list]
[*]
divides 
,
[*]
divides 
,
[*]
.
[/list]
Yang Liu
satisfying the following conditions:[list]
[*]
divides 
,[*]
divides 
,[*]
.[/list]
Yang Liu
39 replies
and 
. Prove that
Sum of 1/(a^5(b+2c))^2 at least 1/3 [USA TST 2010 2]
MellowMelon 42
N
by Adywastaken
Let 
be positive reals such that 
. Show that ![\[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]](//latex.artofproblemsolving.com/3/b/d/3bd44cede90261ab949beabd32f8c08411ddc051.png)
be positive reals such that 
. Show that ![\[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]](http://latex.artofproblemsolving.com/3/b/d/3bd44cede90261ab949beabd32f8c08411ddc051.png)
42 replies
Weird function?
ItzsleepyXD 2
N
by ItzsleepyXD
Source: Own
Find all functions 
such that for all 
,
![\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]](//latex.artofproblemsolving.com/d/1/0/d10c01d7d416243614b24f04218d157e8b025ff2.png)
such that for all 
,![\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]](http://latex.artofproblemsolving.com/d/1/0/d10c01d7d416243614b24f04218d157e8b025ff2.png)
2 replies
Almost similar one but more answer lol
ItzsleepyXD 0
Source: Own , Modified
Find all non decreasing functions or non increasing function 
such that for all 
.
such that for all 
.
0 replies
A lot of unexpected answer from non decreasing function
ItzsleepyXD 0
Source: Own
Find all non decreasing function 
such that for all and 
such that 
there exist 
such that 
and 
. Note : 
and 
for all 
.
such that for all and 
such that 
there exist 
such that 
and 
. Note : 
and 
for all 
.
.0 replies
determine the minimum and maximum value of![\[\frac{\sum_{i,j=1}^n|a_i+a_j|}{\sum_{i=1}^n|a_i|}.\]](http://latex.artofproblemsolving.com/6/f/2/6f2427edf9a5e8e8f2e12276d0a841765e8322f1.png)
