by KAME06, Apr 21, 2025, 10:18 PM 
a point in the space and 
the centroid of a tetrahedron 
. Prove that:
by iStud, Apr 21, 2025, 9:51 PM 
be a monic polynomial so that there exists another real coefficients 
that satisfy![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from that satisfies (a), determine whether should have real coefficients or not.by iStud, Apr 21, 2025, 9:47 PM 
with exactly 9 elements that is subset of 
. Prove that there exist two subsets 
and 
that satisfy the following: and are non-empty subsets from , and are equal, and
is an empty subset.by qwerty123456asdfgzxcvb, Apr 21, 2025, 7:23 PM 
be a logarithmic spiral centered at the origin (ie curve satisfying for any point 
on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
on the spiral, the polar of wrt. is tangent to the spiral.by oVlad, Apr 21, 2025, 1:48 PM 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
by oVlad, Apr 21, 2025, 1:36 PM 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
by Jackson0423, Apr 18, 2025, 4:12 PM ![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
by pinetree1, Jun 25, 2019, 5:36 PM 
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and 
such that 
for all integers .by adityaguharoy, Feb 28, 2018, 3:28 PM be an infinite set. And let 
be the collection of all subsets of such that either or 
is finite. Prove that is an algebra on , but not a sigma algebra on . be an . Let be the collection of all subsets of such that either is countable or 
is countable. a 
algebra on ?by 62861, Feb 23, 2017, 5:14 PM 
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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be an arbitrary set. Then a collection 
for each 
the set 
for every finite sequence 
of elements each 
the union 
of elements each 
,
then 
to 
be such a function.
)
be the union 
then 
is an infinite set. Thus, 
April 2021