1. Let 
be points on the sides 
respectively of a triangle 
such that 
and 
Show that 
is equilateral.
2. Call a natural number
faithful if there exist natural numbers 
such that 
and 
and 
Show that all but a finite number of natural numbers are faithful.
Find the sum of all natural numbers which are not faithful.
3. Let
and 
be two polynomials with integral coefficients such that 
is a prime and 
and 
Suppose that there exists a rational number 
such that 
Prove that 
4. Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
5. Let
be a cyclic quadrilateral inscribed in a circle 
Let 
be the midpoints of arcs 
of 
respectively. Suppose that 
Show that 
are all concurrent.
6. Find all functions
satisfying
For all
where 
denotes the set of all real numbers.
be points on the sides respectively of a triangle such that and Show that is equilateral. 2. Call a natural number
faithful if there exist natural numbers such that and and Show that all but a finite number of natural numbers are faithful. Find the sum of all natural numbers which are not faithful. 3. Let
and be two polynomials with integral coefficients such that is a prime and and Suppose that there exists a rational number such that Prove that 4. Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
5. Let
be a cyclic quadrilateral inscribed in a circle Let be the midpoints of arcs of respectively. Suppose that Show that are all concurrent. 6. Find all functions
satisfyingFor all
where denotes the set of all real numbers.
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