MathDB

Problem 1

Part of 2001 Mongolian Mathematical Olympiad

Problems(2)

Recurrence relation, maximum of x1

Source: Mongolia MO 2001 Grade 10 P1

4/12/2021
Suppose that a sequence x1,x2,…,x2001x_1,x_2,\ldots,x_{2001} of positive real numbers satisfies 3xn+12=7xnxn+1−3xn+1−2xn2+xn and x37=x2001.3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.Find the maximum possible value of x1x_1.
Sequencerecurrence relationalgebra
polynomial FE, prove existence in p(x)p(4-x)=p(x(4-x))

Source: Mongolia MO 2001 Teachers P1

4/12/2021
Prove that for every positive integer nn there exists a polynomial p(x)p(x) of degree nn with real coefficients, having nn distinct real roots and satisfying p(x)p(4−x)=p(x(4−x))p(x)p(4-x)=p(x(4-x))
functional equationPolynomials