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Part of 2015 India Regional MathematicaI Olympiad

Problems(5)

2 integer quadratic polynomials, m \ne n, P_1(m) = P_2(n), P_2(m) = P_1(n)

Source: CRMO 2015 region 1 p2

9/30/2018
Let P1(x)=x2+a1x+b1P_1(x) = x^2 + a_1x + b_1 and P2(x)=x2+a2x+b2P_2(x) = x^2 + a_2x + b_2 be two quadratic polynomials with integer coeffcients. Suppose a1≠a2a_1 \ne a_2 and there exist integers m≠nm \ne n such that P1(m)=P2(n),P2(m)=P1(n)P_1(m) = P_2(n), P_2(m) = P_1(n). Prove that a1−a2a_1 - a_2 is even.
algebrapolynomialInteger PolynomialEven
Question 2

Source:

12/6/2015
Let P(x)=x2+ax+bP(x) = x^2 + ax + b be a quadratic polynomial with real coefficients. Suppose there are real numbers s≠t s \neq t such that P(s)=tP(s) = t and P(t)=sP(t) = s. Prove that b−stb-st is a root of x2+ax+b−stx^2 + ax + b - st.
quadraticsalgebrapolynomialRMO 2015
RMO Delhi Q.2

Source:

12/20/2015
2.Let P(x)=x2+ax+bP(x) = x^2 + ax + b be a quadratic polynomial where a, b are real numbers. Suppose P(−1)2P(-1)^2 , P(0)2P(0)^2, P(1)2P(1)^2 is an Arithmetic progression of positive integers. Prove that a, b are integers.
RMO 2015
P(0)^{2},P(1)^{2},P(2)^{2} are integers in a quadratic polynomial

Source: CRMO 2015 region 4 (Karnataka) p2

9/30/2018
Let P(x)=x2+ax+bP(x)=x^{2}+ax+b be a quadratic polynomial where aa is real and b≠2b \neq 2, is rational. Suppose P(0)2,P(1)2,P(2)2P(0)^{2},P(1)^{2},P(2)^{2} are integers, prove that aa and bb are integers.
polynomialIntegersquadratic trinomialalgebra
Finding all three digit numbers satisfying the conditions

Source: RMO (Mumbai Region) 2015 P2

12/6/2015
Determine the number of 3−3-digit numbers in base 1010 having at least one 55 and at most one 33.
combinatoricscombinatorics proposed