Question 90
Let $f_1(x) = a_1x^2 + b_1x + c_1$, $f_2(x) = a_2x^2 + b_2x + c_2$, $f_3(x) = a_3x^2 + b_3x + c_3$ be three quadratic functions with real coefficients and $P_1 : y = f_1(x)$, $P_2 : y = f_2(x)$, $P_3 : y = f_3(x)$ their associated parabolas.
a) Give an example where $P_1 \cap P_2 = \{A\}$, $P_2 \cap P_3 = \{B\}$, $P_3 \cap P_1 = \{C\}$, and $A$, $B$, $C$ are the vertices of an equilateral triangle.
b) Prove that if $a_1, a_2, a_3$ are non nil and pairwisely distinct and $P_1 \cap P_2 = \{A\}$, $P_2 \cap P_3 = \{B\}$, $P_3 \cap P_1 = \{C\}$, then $A$, $B$, $C$ coincide.
Vasile Pop
a) Give an example where $P_1 \cap P_2 = \{A\}$, $P_2 \cap P_3 = \{B\}$, $P_3 \cap P_1 = \{C\}$, and $A$, $B$, $C$ are the vertices of an equilateral triangle.
b) Prove that if $a_1, a_2, a_3$ are non nil and pairwisely distinct and $P_1 \cap P_2 = \{A\}$, $P_2 \cap P_3 = \{B\}$, $P_3 \cap P_1 = \{C\}$, then $A$, $B$, $C$ coincide.
Vasile Pop