I.S.I. Entrance 2016 (Subjective Paper)

  1. In a sports tournament of \(n\) players, each pair of players plays against each other exactly one match and there are no draws. Show that the players can be arranged in an order \(P_1,P_2, …. , P_n\) such that \(P_i\) defeats \(P_{i+1}\) for all \(1 \le i \le n-1\).
  2. Consider the polynomial \(ax^3+bx^2+cx+d\) where \(a,b,c,d\) are integers such that \(ad\) is odd and \(bc\) is even. Prove that not all of its roots are rational.
  3. If \(P(x)=x^n+a_1x^{n-1}+…+a_{n-1}\) be a polynomial with real coefficients and \(a_1^2<a_2\) then prove that not all roots of \(P(x)\) are real.
  4. Given a square \(ABCD\) with two consecutive vertices, say \(A\) and \(B\) on the positive \(x\)-axis and positive \(y\)-axis respectively. Suppose the other vertice \(C\) lying in the first quadrant has coordinates \((u , v)\). Then find the area of the square \(ABCD\) in terms of \(u\) and \(v\).
  5. Prove that there exists a right angle triangle with rational sides and area \(d\) if and only if \(x^2,y^2\) and \(z^2\) are squares of rational numbers and are in Arithmetic Progression
    Here \(d\) is an integer.
  6. Suppose in a triangle \(\triangle ABC\), \(A\) , \(B\) , \(C\) are the three angles and \(a\) , \(b\) , \(c\) are the lengths of the sides opposite to the angles respectively. Then prove that if \(sin(A-B)= \frac{a}{a+b}\sin A \cos B – \frac{b}{a+b}\sin B \cos A\) then the triangle \(\triangle ABC\) is isoscelos.
  7. \(f\) is a differentiable function such that \(f(f(x))=x\) where \(x \in [0,1]\).Also \(f(0)=1\).Find the value of
    $$\int_0^1(x-f(x))^{2016}dx$$
  8. Suppose that \((a_n)_{n\geq 1}\) is a sequence of real numbers satisfying \(a_{n+1} = \frac{3a_n}{2+a_n}\).
    1. Suppose \(0 < a_1 <1\), then prove that the sequence \(a_n\) is increasing and hence show that \(\lim_{n \to \infty} a_n =1\).
    2.  Suppose \( a_1 >1\), then prove that the sequence \(a_n\) is decreasing and hence show that \(\lim_{n \to \infty} a_n =1\).

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