# I.S.I. B.STAT ENTRANCE SUBJECTIVE 2005

1. Let a,b and c be the sides of a right angled triangle. Let $$\displaystyle{\theta }$$ be the smallest angle of this triangle. If $$\displaystyle{ \frac{1}{a}, \frac{1}{b} }$$ and $$\displaystyle{ \frac{1}{c} }$$ are also the sides of a right angled triangle then show that $$\displaystyle{ \sin\theta=\frac{\sqrt{5}-1}{2}}$$
2. Let $$\displaystyle{f(x)=\int_0^1 |t-x|t , dt }$$ for all real x. Sketch the graph of f(x). What is the minimum value of f(x)?
3. Let f be a function defined on $$\displaystyle{ {(i,j): i,j \in \mathbb{N}} }$$ satisfying $$\displaystyle{ f(i,i+1)=\frac{1}{3} }$$ for all i $$\displaystyle{ f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j) }$$ for all k such that i <k
4. Find all real solutions of the equation $$\displaystyle{ \sin^{5}x+\cos^{3}x=1 }$$.
1. Discussion
5. Consider an acute angled triangle PQR such that C,I and O are the circumcentre, incentre and orthocentre respectively. Suppose $$\displaystyle{ \angle QCR, \angle QIR }$$ and $$\displaystyle{ \angle QOR }$$, measured in degrees, are $$\displaystyle{ \alpha, \beta and \gamma }$$ respectively. Show that $$\displaystyle{ \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} }$$ > $$\displaystyle{ \frac{1}{45} }$$
1. Discussion
6. Let f be a function defined on $$\displaystyle{ (0, \infty ) }$$ as follows: $$\displaystyle{ f(x)=x+\frac1x }$$ . Let h be a function defined for all $$\displaystyle{ x \in (0,1) }$$ as $$\displaystyle{h(x)=\frac{x^4}{(1-x)^6} }$$. Suppose that g(x)=f(h(x)) for all $$\displaystyle{x \in (0,1)}$$. Show that h is a strictly increasing function.
1. Show that there exists a real number $$\displaystyle{x_0 \in (0,1)}$$ such that g is strictly decreasing in the interval $$\displaystyle{ (0,x_0] }$$ and strictly increasing in the interval $$\displaystyle{[x_0,1)}$$.
7. For integers $$\displaystyle{ m,n\geq 1 }$$, Let $$\displaystyle{ A_{m,n} , B_{m,n} }$$ and $$\displaystyle{ C_{m,n}}$$ denote the following sets:
$$\displaystyle{A_{m,n}={(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n}}$$ given that $$\displaystyle{\alpha _i \in \mathbb{Z}}$$ for all i
$$\displaystyle{B_{m,n}={(\alpha _1,\alpha _2,\ldots ,\alpha _m) \colon \alpha _1+\alpha _2+\ldots + \alpha _m=n}}$$ given that $$\displaystyle{\alpha _i \geq 0}$$ and $$\displaystyle{\alpha_ i \in \mathbb{Z}}$$ for all i
$$\displaystyle{C_{m,n}={(\alpha _1,\alpha _2,\ldots,\alpha _m)\colon 1\leq \alpha _1 \alpha_2 \ldots; \alpha_m \leq n}}$$ given that $$\displaystyle{\alpha _i \in \mathbb{Z}}$$ for all i

1. Define a one-one onto map from $$\displaystyle{A_{m,n}}$$ onto $$\displaystyle{B_{m+1,n-1}}$$.
2. Define a one-one onto map from $$\displaystyle{A_{m,n}}$$ onto $$\displaystyle{C_{m,n+m-1}}$$.
3. Find the number of elements of the sets $$\displaystyle{A_{m,n}}$$ and $$\displaystyle{B_{m,n}}$$.
8. 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.
1. $$\displaystyle{ g(n)=5^k }$$ where k is the number of distinct primes which divide n.
2. $$\displaystyle{ h(n)= 0} \text{if} n \text{is divisible by} k^2 \text{for some integer} k>1$$ …., 1 otherwise
9. Suppose that to every point of the plane a colour, either red or blue, is associated.
1. 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.
2. Show that there must be an equilateral triangle with all vertices of the same colour.
10. 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?

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