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Explore the Back-Story(a) Find all prime numbers such that and are also primes.

(b) Determine real numbers such that

If are distinct real numbers such that

find the numerical value of .

and are triangles such that and . If and , without using Trigonometry, prove that .

(a) a, b, c, d are positive real numbers such that . Prove that

(b) In a scalene triangle . The bisectors of the angles and meet the opposite sides in and respectively. Prove that the circle on as diameter passes through the point .

(a) Prove that cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.

(b) segments are marked on a line. Each of these segments intersects at least other segments. Prove that one of these segments intersects all other segments.

If are positive real numbers such that and , find the value

(a) Find all prime numbers such that and are also primes.

(b) Determine real numbers such that

If are distinct real numbers such that

find the numerical value of .

and are triangles such that and . If and , without using Trigonometry, prove that .

(a) a, b, c, d are positive real numbers such that . Prove that

(b) In a scalene triangle . The bisectors of the angles and meet the opposite sides in and respectively. Prove that the circle on as diameter passes through the point .

(a) Prove that cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.

(b) segments are marked on a line. Each of these segments intersects at least other segments. Prove that one of these segments intersects all other segments.

If are positive real numbers such that and , find the value

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