for all real numbers 
such that 
. Is it possible for 
to have exactly 
distinct real roots?
such that . Is it possible for to have exactly distinct real roots? 2. Let 
denote the sum of divisors of natural number 
, including 
and . For every 
define 
as number of natural numbers 
, for which 
is odd number. Prove that there are infinitely many natural numbers 
, such that 
.
3. Let denote the sum of divisors of natural number , including and . For every define as number of natural numbers , for which is odd number. Prove that there are infinitely many natural numbers , such that . 
and 
be points inside triangle 
satisfying
and 
.a) Prove that feet of perpendiculars from and on the sides of triangle are concyclic.
and on the sides of triangle are concyclic. b) Let 
and 
be feet of perpendiculars from on the lines 
and 
and 
foot of perpendicular from on 
. Let 
be intersection point of 
and . Prove that 
.
and be feet of perpendiculars from on the lines and and foot of perpendicular from on . Let be intersection point of and . Prove that .Κάντε κλικ εδώ, για να κατεβάσετε τα θέματα.
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