k-triangular sets
by navi_09220114, May 19, 2025, 11:52 AM
For an integer 
, we call a set 
of 
points in a plane 
-triangular if no three of them lie on the same line and whenever at most (possibly zero) points are removed from , the convex hull of the resulting set is a non-degenerate triangle. For given positive integer , find all integers such that there exists a -triangular set consisting of 
points.
Note. A set of points in a Euclidean plane is defined to be convex if it contains the line segments connecting each pair of its points. The convex hull of a shape is the smallest convex set that contains it.
, we call a set 
of 
points in a plane 
-triangular if no three of them lie on the same line and whenever at most (possibly zero) points are removed from , the convex hull of the resulting set is a non-degenerate triangle. For given positive integer , find all integers such that there exists a -triangular set consisting of 
points.Note. A set of points in a Euclidean plane is defined to be convex if it contains the line segments connecting each pair of its points. The convex hull of a shape is the smallest convex set that contains it.
This post has been edited 1 time. Last edited by navi_09220114, 2 hours ago
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](http://latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.
denotes the number of positive divisors of the positive integer 
.
with real coefficients of degree 
such that the following two conditions hold:
(they are not necessarily distinct);
are 
.
is monic since the coefficient of 
is one.
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to 
,
,
,
lie on a semicircle 
in this order with diameter 
,
.
and 
lie on segments 
and 
respectively such that 
and 
.
passes through 
.
,
are concurrent.
around a circle so that the sum of every 
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
.
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when 
.
What is 
be a scalene triangle, and let 
.
and 
are similar and quadrilaterals 
and 
are cyclic. Let lines 
and 
meet at 
and 
meet at 
.
and 
are parallel.
be an interior point of a triangle 
meet the sides 
in 
respectively. Show that ![\[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}. \]](http://latex.artofproblemsolving.com/7/5/9/759482bace109eb619aaea692625f7854faa9fec.png)
April 2021
February 2021