[/quote]
,
such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
such that 
for all 
?
be a sequence of integers, with the property that every consecutive group of 
'
and 
,![\[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\]](http://latex.artofproblemsolving.com/a/7/b/a7b72c85a43bcba1ebbc642e075c7443ccb6d23c.png)
is always the square of an integer. Prove that the sequence must be constant (all 
be an acute triangle with 
.
be its circumcircle, 
its orthocenter, and 
the foot of the altitude from 
.
be the midpoint of 
.
be the point on 
such that 
and let 
be the point on 
.
,
,
and 
are tangent to each other.
and 
.
(so 
is a function from the positive real numbers) such that![\[ \frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}
\]](http://latex.artofproblemsolving.com/0/4/b/04b955fb570ad384fb7f0e85bf0dacccf4c982a1.png)
satisfying 
and 
of cyclic quadrilateral 
.
,
,
,
such that:
for all 
.
be an integer. Ivan writes the numbers 
each on different cards. He then shuffles these 
cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
?
be six distinct real numbers satisfying the following condition:
,![\[
x_i^6 + (-1)^{i+1} x_i = -x_1 x_2 x_3 x_4 x_5 x_6.
\]](http://latex.artofproblemsolving.com/3/6/f/36f886654cf5f7af956e2475be94acbe62bb8153.png)
.
| 
with 
are positive integers and 
is prime 
)
,
is the midpoint of 
.
is an arbitrary point on 
.
be the intersection of 
and 
.
intersects with the circumcircle of 
,
.
is the circumcenter of triangle 
.
.
Y by ImSh95, Mango247
Sheldon was really annoying Leonard. So to keep him quiet, Leonard decided to do something. He gave Sheldon the following grid
and asked him to transform it to the new grid below
by only applying the following algorithm:
At each step, Sheldon must choose either two rows or two columns.
For two columns 
, if 
are entries in respectively, then we say that 
and 
are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them.
Leonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, "But this is impossible!". Was Sheldon right? Justify.
and asked him to transform it to the new grid below
by only applying the following algorithm:
At each step, Sheldon must choose either two rows or two columns. For two columns 
, if 
are entries in respectively, then we say that 
and 
are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them.Leonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, "But this is impossible!". Was Sheldon right? Justify.
This post has been edited 2 times. Last edited by Severus, Jan 24, 2021, 10:15 PM
denote the entry of the grid on the 
t
t
and 
were chosen then the algorithm transforms them to 
and 
.
matrix 
such that 
,
is the entry on the 
corresponds to the original grid.)
is an invariant for all 
,
.
s
modulo 2, is the only row with five 
s
solutions by me, Distorteddragon1o4 and Sammy27
s
t
and the last row of the next matrix is 
r
