Solve in gaussian integers
by CHESSR1DER, Mar 21, 2025, 6:45 PM
Solve in gaussian integers.
Graph Theory
by JetFire008, Mar 21, 2025, 4:22 PM
Prove that for any Hamiltonian cycle, if it contain edge 
, then it must not contain edge 
.
, then it must not contain edge 
.
and 
Prove that
.
such that 
.
passing trough 
.
.Assisted perpendicular chasing
by sarjinius, Mar 9, 2025, 3:41 PM
In acute triangle 
with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
(a) Show that
and 
are perpendicular.
(b) Show that
and 
are perpendicular.
with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.(a) Show that
and 
are perpendicular.(b) Show that
and 
are perpendicular.Simple vector geometry existence
by AndreiVila, Mar 8, 2025, 1:31 PM
Let 
be a parallelogram of center . Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.
be a parallelogram of center . Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.a! + b! = 2^{c!}
by parmenides51, Mar 26, 2024, 3:44 PM
Determine all triples 
of positive integers such that
(Walther Janous)
of positive integers such that
(Walther Janous)
CMI Entrance 19#6
by bubu_2001, Nov 1, 2019, 8:11 AM
Compute -![\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}](http://latex.artofproblemsolving.com/8/6/f/86fb48db976012106bd30f74cb6ada8c6e6d7800.png)
For 
define 
Determine the open interval(s) (if any) where 
is decreasing and all the open interval(s) (if any) where is increasing.
Determine all the local minima of (if any) and all the local maxima of (if any) 
This post has been edited 7 times. Last edited by bubu_2001, Nov 16, 2023, 5:52 AM
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