[/quote]
,
,
,
are pairwise distinct real numbers satisfying 
Prove that
Let 
Prove that
chessboard such that no two pawns attack each other?
and 
be two positive integers. Prove that the integer![\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]](http://latex.artofproblemsolving.com/1/4/5/14509b500811314c0c363da994e6de875ba468ba.png)
is not a square. (Here 
denotes the least integer greater than or equal to 
be a triangle with incenter 
,
be a point on line 
satisfying 
.
at 
.
on 
and 
on 
analogously, using the excircles opposite 
and 
,
is cyclic, then 
is tangent to the circumcircle of 
.
and 
Prove that
Let 
Prove that
and incenter 
be the midpoint of 
,
are selected on sides 
,
,
.
intersects 
other than 
and 
meet on 
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
such that 
is a perfect square.
Y by cubres
An insect starts from 
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.
*Too many edits, my brain had gone to a trip
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.*Too many edits, my brain had gone to a trip
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This post has been edited 4 times. Last edited by Rohit-2006, Apr 16, 2025, 2:23 AM
