[/quote]
,
is divisible by 
for all positive integers 
and 
.
be a polynomial in one variable with integer coefficients. Prove that the number of pairs 
of positive integers such that 
,
be a cyclic quadrilateral. Assume that the points 
are collinear in this order, in such a way that the line 
is tangent to the circle 
,
is tangent to the circle 
.
and 
be the midpoints of segments 
and 
,
,
at point 
,
at point 
.
be nonnegative real numbers such that 
.
.
and determine when the equality holds.
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
,
is chosen on the side 
.
and 
,
and 
are drawn, respectively. The point 
is the circumcenter of 
.
and 
lies on the line 
.
be a point in the plane of triangle 
such that the segments 
,
,
are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to 
is acute.
and 
be a circle with center 
that is tangent to ray 
and ray 
.
be a point on segment 
.
and 
is tangent to 
lie on the plane such that 
is a parallelogram. 
is the orthocenter of 
intersects 
and 
intersects 
The circle with diameter 
intersects the circumcircle of 
again at 
Prove that: The circumcircle of 
passes through the reflection point of 
,
denote the incenter and 
intersects the tangent from 
.
and 
.
.
such that for any positive integer number 
and any arrangement of the numbers from 1 to 
on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than 
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
Y by BatyrKHAN, Adventure10, lian_the_noob12
Cells of 
table are colored with 
colors (each cell is colored with exactly one color). For each color, the total amount of the cells of this color is not less than 
and not greater than 
.
Prove that there exists at least one row or column which contains cells of at least four different colors.
(N. Sedrakyan)
table are colored with 
colors (each cell is colored with exactly one color). For each color, the total amount of the cells of this color is not less than 
and not greater than 
.Prove that there exists at least one row or column which contains cells of at least four different colors.
(N. Sedrakyan)
This post has been edited 1 time. Last edited by Vlados021, Sep 2, 2019, 7:54 AM
be the number of colors, and call them 
For a color 
,
Observe that if we were to prove that 
,
times, then 
denote the number of rows, columns, respectively that 
,
rows and one of the 
columns. This implies by AM-GM that 
As 
,
since the colors appear a combined 
times. We're done.
,
denote the number of colours, appearing in the 
-
denote the number of colours, appearing in the 
.
and let 
denote the number of cells in that colour. Also, let 
denote the number of columns and rows, which the cells of colour 
occupy (i.e. the "contribution" to the sum). Then it's elementary to check that if 
,
,
,
and if 
,
.
and from the third group 
.
which is a slightly stronger bound (and I hope it's correct). And we've proven the problem by the Pigeonhole Principle. Now I also wanted to find if the constant 
is optimal and it is (here is a table 
