[/quote]
,
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
and 
be positive integers with 
.
,
is a subset of 
such that every element of 
.
![\[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \]](http://latex.artofproblemsolving.com/0/9/d/09d80fe5c25c5c6ea57c545b68c7e11b319cbada.png)
with integer coefficients 
,
,
,
and 
such that ![\[ f(a)=f(b)=f(c)=f(d)=5, \]](http://latex.artofproblemsolving.com/8/2/d/82d0c5d0c7a6dd48802ad8fbe04b211278c1fa58.png)
show that there is no integer 
such that 
.
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
such that 
divides by 
s
n
r
t
hats, numbered from 
hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
be an acute triangle with circumcircle 
.
be the midpoint of 
and let 
be the midpoint of 
.
be the foot of the altitude from 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
be the incentre of acute-angled triangle 
,
,
respectively. Let line 
intersect the circumcircle of the triangle at 
and 
,
lies between 
and 
.
is internally tangent to 
at 
and 
be points on line 
is the midpoint of 
and 
is the midpoint of 
.
and 
meet at 
and intersect 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
such that points 
lie on the line 
is tangent to the segments 
and 
and to the circle 
is tangent to the segments 
and 
and to the circle 
and 
intersect at point 
.
be the set of all positive integers. Find all functions 
such that 
divides 
for all positive integers 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
are given. Losyash likes some of the cells of the 
checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of 
,
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)
.
.
