maximum profit
by Ecrin_eren, Apr 13, 2025, 2:04 PM
In a meeting attended by 20 businessmen, some of them know each other and do business only with the people they know. The participants are numbered from 1 to 20 according to the order in which they arrived. Let aₖ represent the number of people that person number k knows. (For example, if person 5 knows 9 people, then a₅ = 9.)
If person k knows person n, then the profit that k earns from doing business with n is:
(1 / aₖ) + (1 / aₙ) + (1 / (aₖ × aₙ))
What is the maximum total profit that any participant in this meeting can earn?
If person k knows person n, then the profit that k earns from doing business with n is:
(1 / aₖ) + (1 / aₙ) + (1 / (aₖ × aₙ))
What is the maximum total profit that any participant in this meeting can earn?
satisfied 
Find minimum and maximum of 
and 
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be an acute triangle. Points 
and 
lie on a line in this order and satisfy 
.
and 
be midpoints of 
and 
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is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
be all the positive integers smaller than 
such that![\[gcd(N, c_i + c_{i + 1}) \neq 1\]](http://latex.artofproblemsolving.com/1/0/7/1075290d442e8c1f54237aad8ad07126f455f1f1.png)
for all 
.
of positive integers is called central if for every positive integer 
,
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
be all the positive integers smaller than 
for all 
.
,
,
are concurrent and the point of concurrence is 
,
,
,
be positive real numbers such that 
.![$$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+a^3}{2}}\leqslant a^2+b^2+c^2.$$](http://latex.artofproblemsolving.com/5/a/b/5ab64bd6c09363206f936a4a49a9ac7968218600.png)
of the triangle 
so that 
.
is extended beyond 
to 
so that 
.
,
September 2017
February 2017