by Maths_VC, May 27, 2025, 7:49 PM 
satisfy 
. Determine the minimal and the maximal value of the expression 
by Maths_VC, May 27, 2025, 7:38 PM 
be a random point on the smaller arc 
of the circumcircle of square 
, and let 
be the intersection point of segments 
and 
. The feet of the tangents from point 
to the circumcircle of the triangle 
are 
and 
, where 
is the center of the square. Prove that points 
, 
, and lie on a single circle.by M11100111001Y1R, May 27, 2025, 7:20 AM 
is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number 
, there exists a lucky number such that 
.by OgnjenTesic, May 22, 2025, 4:06 PM 
such that:
is strictly increasing,
such that 
for all 
,
, there exists 
such that![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](http://latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinovićby lminsl, Jul 16, 2019, 12:11 PM 
, point 
lies on side 
and point 
lies on side . Let and be points on segments 
and 
, respectively, such that 
is parallel to . Let 
be a point on line 
, such that lies strictly between and , and 
. Similarly, let 
be the point on line 
, such that lies strictly between and , and 
.
, and are concyclic.by gobathegreat, Sep 17, 2018, 10:29 AM are positive integers, and smallest side is equal to 
. Determine the area of triangle if 
, where 
, 
and 
are lengths of altitudes in triangle from vertices , 
and , respectively.by sqing, Sep 2, 2018, 7:41 AM 
and 
be positive real numbers satisfying 
Prove that
by MithsApprentice, Sep 30, 2005, 8:10 PM be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides and , respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex is denoted by . Prove that 
.by MithsApprentice, Sep 27, 2005, 8:01 PM be a triangle. A circle passing through and intersects segments and at and 
, respectively. Lines and 
intersect at 
, while lines 
and 
intersect at . Prove that 
if and only if 
.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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,
be positive integers such that 
is a multiple of 
for all positive integers 
.
.
April 2021