by mihaig, May 15, 2025, 5:33 AM 
be real numbers such that 
Prove that if 
is a positive constant, then
is not always trueby Miquel-point, May 14, 2025, 6:15 PM 
, 
, 
and 
there exists infinitely many positive integers 
for which 
and 
are not relatively primes.by a_507_bc, Jul 29, 2023, 12:15 PM 
in which 
and 
. Let 
be a point on the side 
and 
a point on the line 
such that 
. Prove that 
.by microsoft_office_word, Apr 12, 2018, 11:02 AM 
or 
tile.
be an integer. Dominoes are placed on an 
board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some 
such that each row and each column has a value of 
. Prove that a balanced configuration exists for every , and find the minimum number of dominoes needed in such a configuration.by ABCDE, Jul 7, 2016, 7:52 PM 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.by v_Enhance, Jul 23, 2013, 1:32 AM be a cyclic quadrilateral inscribed in circle 
whose diagonals meet at . Lines 
and 
meet at . Segment 
intersects at 
. Lines 
and meet at 
, and lines 
and meet at 
. Prove that 
and concur with the tangent to at ."Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu
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Prove that
Let 
Prove that
.
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)