51st Ukrainian National Mathematical Olympiad, 3rd Round

Question

It is known, that x_1, x_2, x_3 are distinct real numbers. a) x_2, x_3 are zeros of the function f_1(x) = x^2 + p_1x + q_1; x_3, x_1 are zeros of the function f_2(x) = x^2 + p_2x + q_2; x_1, x_2 are zeros of the function f_3(x) = x^2 + p_3x + q_3. Does the function f(x) = f_1(x) + f_2(x) + f_3(x) always have zeros? b) x_2, x_3 are zeros of the function f_1(x) = a_1x^2 + b_1x + c_1; x_3, x_1 are zeros of the function f_2(x) = a_2x^2 + b_2x + c_2; x_1, x_2 are zeros of the function f_3(x) = a_3x^2 + b_3x + c_3. Does the function f(x) = f_1(x) + f_2(x) + f_3(x) always have zeros?

Accepted Answer

a) We write our function in the form f(x) = (x - x_2)(x - x_3) + (x - x_1)(x - x_3) + (x - x_1)(x - x_2), WLOG, x_1 < x_2 < x_3. Then f(x_2) = (x_2 - x_1)(x_2 - x_3) < 0, which is equivalent to the existence of the roots of the function. b) Consider the following three functions: f_1(x) = x^2 + x, f_2(x) = x^2 - x, f_3(x) = 1 - x^2, They have the roots 0, -1, 0, 1 and -1, 1 respectively. Their sum f(x) = x^2 + x + x^2 - x + 1 - x^2 = x^2 + 1, does not have zeros.