be a triangle with 
,
its shortest side. Let 
be the orthocenter of 
be the circle with center 
and radius 
.
be the second point where the line 
meets 
be the second point where 
.
be the intersection point of the lines 
and 
.
is tangent to the circumcircle of the triangle 
.
and circumcircle 
centred at 
,
meets 
;
intersect at 
;
and 
intersect at 
.
of the triangle 
lies on 
are collinear. Prove that 
.
the feet of the altitudes lying on 
respectively. One of the intersection points of the line 
and the circumcircle is 
The lines 
and 
meet at point 
Prove that 
is composite. 
- all divisors of 
are all divisors for some 
(except 
)
be a positive integer, with divisors 
.
is always less than 
,
we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
is tangent to the circle 
at 
is the in-centre of 
,
.
and 
and 
intersect at 
lie on a circle.
to 
.
as points on the parallels through 
to 
are positioned around the sides of 
and 
and 
and 
are the reflections of 
are the intersections of 
and 
respectively with the circumcircle of 
and 
lies on 
.
,
.
in 
respectively and the line through 
in 
respectively. Prove that point 
lie on a circle.
kids in a class. Is it true that for all positive integers 
it is possible to create several clubs each with 3 kids such that any pair of kids are both present in exactly one club?
be an integer. In a configuration of an 
board, each of the 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
lamps 
in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp 
and its neighbours (only one neighbour for 
or 
,
)
Prove that there are infinitely many integers 
for which all the lamps will eventually be off.
Prove that there are infinitely many integers 
Y by
Let 
be polygons, no two of which are similar. Show that there are polygons 
where 
is similar to 
so that for no 
does contain a polygon that's congruent to 
.
Note. Here, the word "contain" means for the construction we have, we cannot select a size for so that is wholly contained in , and so it does not intersect the edges of at all.
be polygons, no two of which are similar. Show that there are polygons 
where 
is similar to 
so that for no 
does contain a polygon that's congruent to 
.Note. Here, the word "contain" means for the construction we have, we cannot select a size for
so that is wholly contained in , and so it does not intersect the edges of at all.This post has been edited 1 time. Last edited by somebodyyouusedtoknow, Apr 26, 2025, 11:41 PM
Reason: Added clarity for the notes
Reason: Added clarity for the notes
is obvious; for fixed 
consider the largest copy of 
inside it and slightly enlarge it to obtain 
.
.
with 
for all 
we seek for appropriate 
.![$i\in [n]:= \{1, \ldots, n\}$](http://latex.artofproblemsolving.com/5/5/7/5572574af6c66362b508aebea9989529caa5811f.png)
define 
as the largest copy of 
inside 
as the smallest copy of ![$i\in [n]$](http://latex.artofproblemsolving.com/e/e/5/ee51c5c5eb09111cc774a00e9970074ad9f9058c.png)
.![$\max_{i\in [n]} R_i^- \subsetneq \min_{i\in [n]}R_i^+$](http://latex.artofproblemsolving.com/5/0/1/501e5e0ead06031f5c49050fed96583c031780da.png)
.
.
