3

Part of 2013 Czech-Polish-Slovak Match

Problems(2)

x^2-rx and x^3-rx are rationals implies x is rational

Source: Czech - Polish - Slovak Match 2013: P3

For each rational number

r

consider the statement: If

x

is a real number such that

x^2-rx

x^3-rx

are both rational, then

x

is also rational.
(a) Prove the claim for

r \ge \frac43

r \le 0

p,q

be different odd primes such that

3p <4q

. Prove that the claim for

r=\frac{p}q

quadraticsalgebrapolynomialalgebra proposed

circumcircle passes through the midpoint

Source: Czech-Polish-Slovak Match 2013 day 2 P3

{ABC}

be a triangle inscribed in a circle. Point

{P}

is the center of the arc

{BAC}

. The circle with the diameter

{CP}

intersects the angle bisector of angle

{\angle BAC}

{K, L}

{(|AK| <|AL|)}

{M}

is the reflection of

{L}

with respect to line

{BC}

. Prove that the circumcircle of the triangle

{BKM}

passes through the center of the segment

{BC}

geometrycircumcirclereflectionarc midpoint