MathDB

5

Part of 2009 Postal Coaching

Problems(6)

P_n(x) = x_{n-1}x_{n+1} - x_n^2, x_{n+2} = xx_{n+1} + nx_n

Source: Indian Postal Coaching 2009 set 1 p5

5/26/2020
Define a sequence <xn><x_n> by x1=1,x2=x,xn+2=xxn+1+nxn,n≥1x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1. Consider the polynomial Pn(x)=xn−1xn+1−xn2P_n(x) = x_{n-1}x_{n+1} - x_n^2, for each n≥2n \ge 2. Prove or disprove that the coefficients of Pn(x)P_n(x) are all non-negative, except for the constant term when nn is odd.
polynomialSequencealgebra
2(P(d))^2 + 2P(ab + bc + ca) = (P(a + b + c))^2 , when a^2 + b^2 + c^2 = 2d^2

Source: Indian Postal Coaching 2009 set 2 p5

5/26/2020
Find all real polynomials P(x)P(x) such that for every four distinct natural numbers a,b,c,da, b, c, d such that a2+b2+c2=2d2a^2 + b^2 + c^2 = 2d^2 with gcd(a,b,c,d)=1gcd(a, b, c, d) = 1 the following equality holds: 2(P(d))2+2P(ab+bc+ca)=(P(a+b+c))22(P(d))^2 + 2P(ab + bc + ca) = (P(a + b + c))^2 .
algebrapolynomial
f(n) &lt; 3\sqrt{n} and f(n) &gt; k if n &gt; k! + k, L(n, 1) &lt; L(n, 2) &lt; ... &lt; L(n, k)

Source: Indian Postal Coaching 2009 set 3 p5

5/26/2020
For positive integers n,kn, k with 1≤k≤n1 \le k \le n, define L(n,k)=Lcm (n,n−1,n−2,...,n−k+1)L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1) Let f(n)f(n) be the largest value of kk such that L(n,1)<L(n,2)<...<L(n,k)L(n, 1) < L(n, 2) < ... < L(n, k). Prove that f(n)<3nf(n) < 3\sqrt{n} and f(n)>kf(n) > k if n>k!+kn > k! + k.
number theoryleast common multipleLCMinequalities
circumcircle of AKP passes through fixed point independent choiced D,P

Source: Indian Postal Coaching 2009 set 4 p5

5/26/2020
A point DD is chosen in the interior of the side BCBC of an acute triangle ABCABC, and another point PP in the interior of the segment ADAD, but not lying on the median through CC. This median (through CC) intersects the circumcircle of a triangle CPDCPD at K(≠C)K(\ne C). Prove that the circumcircle of triangle AKPAKP always passes through a fixed point M(≠A)M(\ne A) independent of the choices of the points DD and P.P.
geometryFixed pointfixedcircumcircle
OE x d = r^2, incircle related

Source: Indian Postal Coaching 2009 set 5 p5

5/26/2020
Let ABCDABCD be a quadrilateral that has an incircle with centre OO and radius rr. Let P=AB∩CDP = AB \cap CD, Q=AD∩BCQ = AD \cap BC, E=AC∩BDE = AC \cap BD. Show that OE⋅d=r2OE \cdot d = r^2, where dd is the distance of OO from PQPQ.
geometryincircledistance
PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}

Source: Indian Postal Coaching 2009 set 6 p5

5/26/2020
Let PP be an interior point of a circle and A1,A2...,A10A_1,A_2...,A_{10} be points on the circle such that ∠A1PA2=∠A2PA3=...=∠A10PA1=36o\angle A_1PA_2 = \angle A_2PA_3 = ... = \angle A_{10}PA_1 = 36^o. Prove that PA1+PA3+PA5+PA7+PA9=PA2+PA4+PA6+PA8+PA10PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}.
equal anglesgeometryanglesSum