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INMO 2017

  1. In the given figure, \(ABCD\) is a square sheet of paper. It is folded along \(E F\) such that \(A\) goes to a point \(A’\) different from Band \(C\), on the side \(BC\) and \(D\) goes to \(D’\) . The line \(A’ D’\) cuts \(C D\) in \(G\). Show that the inradius of the triangle \(GC A’\) is the sum of the inradii of the triangles \(GD’F\) and \(A’ BE\).
  2. Suppose \(n \ge 0\) is an integer and all the roots of \( x^3 + ax + 4 -(2 \times {2016^n})\) = 0 are integers. Find all possible values of \(\alpha\).
  3. Find the number of triples \((x, a, b)\) where \(x\) is a real number and a, b belong to the set \({{1,2,3,4,5,6,7,8,9}}\) such that
    \(x^2 – a \{x\} + b = 0\)
    where \(\{x\}\) denotes the fractional part of the real number \(x\).    (For example \(\{1.1\}\) = 0.1 =\(\{-0.9\}\) ).
  4. Let \(ABCDE\) be a convex pentagon in which \({\angle A} ={\angle B} ={\angle C} ={\angle D}\) =\(120^{\circ}\) and side lengths are five consecutive integers in some order. Find all possible values of  \(AB + BC + CD\).
  5. Let \(ABC\) be a triangle with \(\angle A =90^{\circ}\) and \(AB < AC\). Let \(AD\) be the altitude from \(A\) on to BC. Let \(P, Q\) and I denote respectively the incentres of triangles \(ABD, ACD\) and \(ABC\). Prove that \(AI\) is prependicular to \(PQ\) and \(AI = PQ\).
  6. Let \(n \ge 1\) be an integer and consider the sum
    \( \displaystyle{ x ={\sum_{k \ge 0}} \dbinom{n}{2k} 2{^{n-2k}} {3^k} \\  = \dbinom{n}{0} 2^n + \dbinom{n}{2} 2{^{n-2}} . 3 + \dbinom{n}{4} 2{^{n-4}} . 3^2 + \dots }\)
    Show that \(2x – 1, 2x, 2x+1  \) from the sides of a triangle whose area and inradius are also integers.

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