#### ISI Entrance Paper – from Indian Statistical Institute’s B.Stat Entrance 2005

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- Let a,b and c be the sides of a right angled triangle. Let be the smallest angle of this triangle. If and are also the sides of a right angled triangle then show that
- Let for all real x. Sketch the graph of f(x). What is the minimum value of f(x)?
- Let f be a function defined on satisfying for all i for all k such that i <k
- Find all real solutions of the equation .
- Consider an acute angled triangle PQR such that C,I and O are the circumcentre, incentre and orthocentre respectively. Suppose and , measured in degrees, are respectively. Show that >
- Let f be a function defined on as follows: . Let h be a function defined for all as . Suppose that g(x)=f(h(x)) for all . Show that h is a strictly increasing function. Show that there exists a real number such that g is strictly decreasing in the interval and strictly increasing in the interval .
- For integers , Let and denote the following sets:
- given that for all i
- given that and for all i
- given that for all i
- Define a one-one onto map from onto .
- Define a one-one onto map from onto .
- Find the number of elements of the sets and

- A function f(n) is defined on the set of positive integers is said to be multiplicative if f(mn)=f(m)f(n) whenever m and n have no common factors greater than 1. Are the following functions multiplicative? Justify your answer.
- where k is the number of distinct primes which divide n.
- …., 1 otherwise

- Suppose that to every point of the plane a colour, either red or blue, is associated.
- Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points A,B and C of the same colour such that B is the midpoint of AC.
- Show that there must be an equilateral triangle with all vertices of the same colour.

- Let ABC be a triangle. Take n point lying on the side AB (different from A and B) and connect all of them by straight lines to the vertex C. Similarly, take n points on the side AC and connect them to B. Into how many regions is the triangle ABC partitioned by these lines? Further, take n points on the side BC also and join them with A. Assume that no three straight lines meet at a point other than A,B and C. Into how many regions is the triangle ABC partitioned now?

doubt question no. 4and 5

We have posted the requested discussions. Find them here: https://cheenta.com/2015/05/07/solutions-to-an-equation-b-stat-2005-subjective-problem-4-solution/

https://cheenta.com/2015/05/07/geometric-inequality-i-s-i-b-stat-2005-problem-5-solution/

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Do you please help me giving solutions of I.S.I. B.STAT ENTRANCE SUBJECTIVE PAPERS of 2

005, 2006,2007,2008,2009,2010,2011,2012