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Regional Math Olympiad 2017

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  1. Let AOB be a given angle less than \( 180^o \) and let P be an interior point of the angular region determined by \( \angle AOB \) . Show, with proof, how to construct, using only ruler and compass, a line segment CD passing through P such that C lies on the ray OA and D lies on the ray OB and CP:PD = 1:2.
  2. Show that the equation $$ a^3 + (a+1)^3 + (a+2)^3 + (a+3)^3 + (a+4)^3 + (a+5)^3 + (a+6)^3 = b^4 + (b+1)^4 $$ has no solutions in integer a, b.

    Discussion by Writabrata Bhattacharya (Associate Faculty – Cheenta)

  3. Let \( P(x) = x^2 + \frac {1}{2} x + b \) and \( Q(x) = x^2 + cx + d \) be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all real roots of P(Q(x)) = 0Discussion: https://www.cheenta.com/forums/topic/rmo-2017-p3-2/
  4. Consider \( n^2 \) unit squares in the xy-plane centered at the point (i, j) with integer coordinates, \( 1 \le i \le n \) , \( 1 \le j \le n \) . It is required to color each unit square in such a way that whenever \( 1 \le i < j  \le n \)  and \( 1 \le k < l \le n \) the thre squares with centers at (i, k), (j, k) , (j, l) have distinct colours. What is the least possible colours needed?
  5. Let \( \Omega \) be a circle with a chord AB which is not a diameter. Let \( \Gamma_1 \) be a circle on one side of AB such that it is tangent to AB at C and internally tangent to \( \Omega \) at D. Likewise let \( \Gamma_2 \) be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to \( \Omega \) at F. Suppose the line DC intersects \( \Omega \) at \( X \neq D \) and the line FE intersects \( \Omega \) at \( Y \neq F \). Prove that XY is a diameter of \( \Omega \)Discussion by Sauvik Mondal (Faculty – Cheenta)
  6. Let x, y, z be real numbers, each greater than 1. Prove that $$ \frac {x+1}{y +1 } + \frac {y+1}{z+1} + \frac {z+1}{x+1} \le\frac {x – 1}{y – 1 } + \frac {y- 1}{z-1} + \frac {z-1}{x-1} $$Discussion by Writabrata Bhattacharya (Associate Faculty – Cheenta)

Problem 2

 

Problem 5

RMO 2017 Problem 5

 

Problem 6

 

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