,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
be the midpoint of side 
in 
,
is such that the quadrilateral 
is cyclic and the circumcircle of 
is tangent to the line 
.
is the second common point of the line 
and the circumcircle of 
.
of positive integers such that 
is divisible by 
.
different coloured marbles on it and another one is empty
The altitudes 
, intersect
.
and 
o
.
b
intersects 
,
intersects 
at 
lie on the same circle.
intersects the circle 
The line 
.
are collinear.
such that 
is an exact 5th power, 
is an exact 6th power, and 
is an exact 7th power.
denote the set of natural numbers and let 
denote 
ordered pairs in 
.
for 
whose product is a perfect cube.
. Prove that
have real roots and real coefficients with 
.
?
,
.
have roots 
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
such that 
is a perfect cube.
digits are distinct?
Solve for 
,
,
,
.
be the point on 
,
.
is a point such that 
is a rectangle, compute the area of 
.
has a sidelength of 
,
,
,
,
,
,
are length 
.
.
satisfies for all integers 
If 
and 
,
.
children have black hair, all but 
have red hair, and all but 
can be partitioned into n Akash subsets.
:
.
Let 
.
Y by ImSh95
A unit square is removed from the corner of an 
grid, where 
. Prove that the remainder can be covered by copies of the figures of 
or 
unit squares depicted in the drawing below.
![[asy]
import geometry;
draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle);
draw((-3.5,1)--(-2.5,1)--(-2.5,0));
draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle);
draw((1.5,0)--(1.5,1));
draw((2.5,0)--(2.5,1));
draw((0.5,1)--(1.5,1));
draw((0.5,2)--(1.5,2));
[/asy]](//latex.artofproblemsolving.com/5/7/8/578baca60367441fe9b771689f1a2e91c0ccf984.png)
Note: Every square must be covered once and figures must not go over the bounds of the grid.
grid, where 
. Prove that the remainder can be covered by copies of the figures of 
or 
unit squares depicted in the drawing below.![[asy]
import geometry;
draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle);
draw((-3.5,1)--(-2.5,1)--(-2.5,0));
draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle);
draw((1.5,0)--(1.5,1));
draw((2.5,0)--(2.5,1));
draw((0.5,1)--(1.5,1));
draw((0.5,2)--(1.5,2));
[/asy]](http://latex.artofproblemsolving.com/5/7/8/578baca60367441fe9b771689f1a2e91c0ccf984.png)
Note: Every square must be covered once and figures must not go over the bounds of the grid.
This post has been edited 2 times. Last edited by Rukevwe, May 6, 2022, 2:54 PM
rectangular, so 
3-unit figure can make 
rectangular.
have been proved. Then when 
,
.
exists, 
with a unit removed from a corner can be made as below.
or 
,
.
is trivial.
:
:
:
and 
.
only needs one case: the top-right 
can be reduced to 
we split into three cases, also note that one can build a 
then build a big L shape on the bottom-left and use 
it's the same method but start with a middle L shape
then it's the same method but start with the 
