MathDB

1

Part of 2010 Postal Coaching

Problems(5)

A bound on a Ratio of Areas

Source:

10/21/2010
Let γ,Γ\gamma,\Gamma be two concentric circles with radii r,Rr,R with r<Rr<R. Let ABCDABCD be a cyclic quadrilateral inscribed in γ\gamma. If AB→\overrightarrow{AB} denotes the Ray starting from AA and extending indefinitely in B′sB's direction then Let AB→,BC→,CD→,DA→\overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CD} , \overrightarrow{DA} meet Γ\Gamma at the points C1,D1,A1,B1C_1,D_1,A_1,B_1 respectively. Prove that [A1B1C1D1][ABCD]≥R2r2\frac{[A_1B_1C_1D_1]}{[ABCD]} \ge \frac{R^2}{r^2} where [.][.] denotes area.
ratiogeometrycyclic quadrilateralgeometry unsolved
Easy Squares

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12/9/2010
In a family there are four children of different ages, each age being a positive integer not less than 22 and not greater than 1616. A year ago the square of the age of the eldest child was equal to the sum of the squares of the ages of the remaining children. One year from now the sum of the squares of the youngest and the oldest will be equal to the sum of the squares of the other two. How old is each child?
inequalitiesnumber theory unsolvednumber theory
Determine all possible values...

Source:

12/9/2010
Let A,B,C,DA, B, C, D be four distinct points in the plane such that the length of the six line segments AB,AC,AD,BC,BD,CDAB, AC, AD, BC, BD, CD form a 22-element set a,b{a, b}. If a>ba > b, determine all the possible values of ab\frac ab.
pigeonhole principleratiogeometry unsolvedgeometry
Maximum Value of a Polynomial

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10/22/2010
A polynomial P(x)P (x) with real coefficients and of degree n≥3n \ge 3 has nn real roots x1<x2<⋯<xnx_1 <x_2 < \cdots < x_n such that x2−x1<x3−x2<⋯<xn−xn−1x_2 - x_1 < x_3 - x_2 < \cdots < x_n - x_{n-1} Prove that the maximum value of ∣P(x)∣|P (x)| on the interval [x1,xn][x_1 , x_n ] is attained in the interval [xn−1,xn][x_{n-1} , x_n ].
algebrapolynomialalgebra unsolved
Very Hard

Source:

12/9/2010
Does there exist an increasing sequence of positive integers a1,a2,⋯a_1 , a_2 ,\cdots with the following two properties?
(i) Every positive integer nn can be uniquely expressed in the form n=aj−ain = a_j - a_i ,
(ii) akk3\frac{a_k}{k^3} is bounded.
number theory unsolvednumber theory