[/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 Adventure10
The numbers 
are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers 
from the board, replaces them by their sum 
and writes 
to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all 
numbers written in the notebook. Let 
and 
be the maximum and the minimum possible sums that Ann san obtain.
Find the value of
.
(I. Voronovich)
are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers 
from the board, replaces them by their sum 
and writes 
to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all 
numbers written in the notebook. Let 
and 
be the maximum and the minimum possible sums that Ann san obtain.Find the value of
.(I. Voronovich)
This post has been edited 1 time. Last edited by Vlados021, Sep 2, 2019, 8:30 AM
.
be the last two remaining numbers on the board, then Ann's sum is simply:
and 
it's clear that 
or 
Hence, our answer is:
