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ISI Entrance Paper 2016 – from Indian Statistical Institute’s B.Stat, B.Math Entrance

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 $$\lim_{n \to \infty} a_n =1$$.