MathDB

1

Part of 1994 All-Russian Olympiad

Problems(3)

|P_1(x)|+|P_2(x)|=|P_3(x)| has at most 8 real roots, if P_i quadratics

Source: All Russian MO 1994 ARO

7/29/2018
Let be given three quadratic polynomials: P1(x)=x2+p1x+q1,P2(x)=x2+p2x+q2,P3(x)=x2+p3x+q3P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3. Prove that the equation ∣P1(x)∣+∣P2(x)∣=∣P3(x)∣|P_1(x)|+|P_2(x)| = |P_3(x)| has at most eight real roots.
algebrapolynomialquadratics
if (x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1, then x+y = 0.

Source: All Russian MO 1994 ARO IX P1

7/29/2018
Prove that if (x+x2+1)(y+y2+1)=1(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1, then x+y=0x+y = 0.
algebraradicalequation
Simple and nice problem

Source:

3/11/2018
a,ba,b are natural numbers such that a+1b\frac{a+1}{b} and b+1a\frac{b+1}{a} are integers.Let d=GCD(a;b)d=GCD(a;b).Prove that d2≤a+bd^2\le a+b
number theory