[/quote]
,
probability of coming up heads and 
probability of coming up tails. The expected number of flips necessary to first see the sequence 
in that consecutive order can be written as 
for relatively prime positive integers 
,
.
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.
be an acute triangle. Points 
,
lie on a line in this order and satisfy 
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
such for any 
.
satisfy,
Where 
}
be an integer. We are given a balance and 
.
Y by jhu08
The points 
are taken on the sides 
of the square 
respectively such that 
. Let 
be the altitude of triangle 
. Prove that the measure of the angle 
does not depend on the position of the points 
are taken on the sides 
of the square 
respectively such that 
. Let 
be the altitude of triangle 
. Prove that the measure of the angle 
does not depend on the position of the points 
This post has been edited 1 time. Last edited by parmenides51, Aug 21, 2021, 8:21 PM
.
and point 
.
.
.
,
.
.
is a square, and so is 
,
,
,
is a diameter of this circle.
