by Jackson0423, Jun 6, 2025, 3:59 PM 
be positive integers, and define![\[ n = a + b + c + d + e + f. \]](http://latex.artofproblemsolving.com/9/d/d/9ddfecfa5800050bc15dcaa48d17d12e431fc358.png)
Suppose that 
divides both of the following expressions:![\[
abc + def \quad \text{and} \quad ab + bc + ca - de - ef - fd.
\]](http://latex.artofproblemsolving.com/c/5/8/c5817937f736a7b00c85e1f92b8e561ba26cf8b0.png)
Prove that is a composite number.by Sid-darth-vater, Jun 6, 2025, 3:25 PM 
be the circumcenter and 
the orthocenter of an acute triangle 
Prove that the area of one of the triangles 
and 
is equal to the sum of the areas of the other two.
with any line through the centroid 
and I have no clue as to why that is true. Plz helpby guptaamitu1, Jun 6, 2025, 10:18 AM 
be a triangle with orthocentre . Let 
be the foot of altitudes of 
onto the opposite sides, respectively. Consider 
, the reflection of 
about line 
. Let line 
cut at distinct points 
, and let 
be the orthocenter of 
. Prove that points 
are concyclic.by Rijul saini, Jun 4, 2025, 6:46 PM 
be an integer. We call a pair of integers 
good if ![\[0\leqslant a<k,\hspace{0.2cm} 0<b \hspace{1cm} \text{and} \hspace{1cm} (a+b)^2=ka+b\]](http://latex.artofproblemsolving.com/e/8/d/e8df562e024b8476d69fdaea15338c27bb6c4eb8.png)
Prove that the number of good pairs is a power of 
.by swynca, Apr 27, 2025, 2:03 PM 
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.by Siddharth03, Jun 1, 2024, 6:54 PM 
be an equilateral triangle with incircle . A point on is reflected in the sides of to obtain a new triangle 
. The same point is then reflected over the sides of to obtain another triangle 
. Prove that the circumcircle of is tangent to .by amuthup, Jul 12, 2022, 12:24 PM 
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
by Igor, May 16, 2007, 5:17 PM 
is chosen on the side 
of a triangle with 
in such a way that 
. The incircle of is tangent to 
and at points 
and 
, respectively. Let 
be the incenter of triangle 
. Prove that the line 
intersects the line segment 
at its midpoint.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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such that 
April 2021