[/quote]
,
of positive integers, such that every sum
(with 
)
for all 
;
for all 
?
denotes the greatest common divisor of 
,
.
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
,
,
be positive real numbers such that 
.![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
and 
and 
Prove that![\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]](http://latex.artofproblemsolving.com/0/a/b/0ab4b4e6c28a1ae67788525642661dbcd04d0bd8.png)
and in 
steps and has to reach 
is valid but 
is not valid.
,
lie on a line in this order and satisfy 
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
such for any 
.
satisfy,
Where 
}
be an integer. We are given a balance and 
weights of weight 
.
to satisfy that the equation 
has at least 1 roots 
of the equation![\[n^3=p^2-p-1\]](http://latex.artofproblemsolving.com/f/f/8/ff88093bb0aa93afcfdff3ae25149ec5fbacb61f.png)
is a prime number and 
?
Y by anantmudgal09
A circle centred at 
is tangent to the sides 
, and 
of an acute-angled triangle 
at 
, and 
, respectively. Let 
and 
be the incenters of the quadrilaterals 
and 
, respectively. Let 
be an altitude of triangle . Let the internal angle bisectors of angles 
and 
meet the lines 
and 
at 
and 
, respectively. Prove that is the orthocenter of the triangle 
.
Kolmogorov Cup 2018, Major League, Day 3, Problem 1; A. Zaslavsky
is tangent to the sides 
, and 
of an acute-angled triangle 
at 
, and 
, respectively. Let 
and 
be the incenters of the quadrilaterals 
and 
, respectively. Let 
be an altitude of triangle . Let the internal angle bisectors of angles 
and 
meet the lines 
and 
at 
and 
, respectively. Prove that is the orthocenter of the triangle 
.Kolmogorov Cup 2018, Major League, Day 3, Problem 1; A. Zaslavsky
