I.S.I. B.STAT ENTRANCE SUBJECTIVE 2010

Let $\mathbf{a_1,a_2,\cdots, a_n }$ and $\mathbf{ b_1,b_2,\cdots, b_n }$ be two permutations of the numbers $\mathbf{1,2,\cdots, n }$ . Show that $\mathbf{\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2 }$

Let a,b,c,d be distinct digits such that the product of the 2-digit numbers $\mathbf{\overline{ab}}$ and $\mathbf{\overline{cb}}$ is of the form $\mathbf{\overline{ddd}}$ . Find all possible values of a+b+c+d.

Let $\mathbf{I_1, I_2, I_3}$ be three open intervals of $\mathbf{\mathbb{R}}$ such that none is contained in another. If $\mathbf{I_1\cap I_2 \cap I_3}$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.

A real valued function f is defined on the interval (-1,2). A point $\mathbf{x_0}$ is said to be a fixed point of f if $\mathbf{f(x_0)=x_0}$ . Suppose that f is a differentiable function such that f(0)>0 and f(1)=1. Show that if f’(1)>1, then f has a fixed point in the interval (0,1).

Let A be the set of all functions $\mathbf{f:\mathbb{R} \to \mathbb{R}}$ such that f(xy)=xf(y) for all $\mathbf{x,y \in \mathbb{R}}$ .
(a) If $\mathbf{f \in A}$ then show that f(x+y)=f(x)+f(y) for all x,y $\mathbf{\in \mathbb{R}}$

(b) For $\mathbf{g,h \in A}$ , define a function $\mathbf{g\circ h}$ by $\mathbf{(g \circ h)(x)=g(h(x))}$ for $\mathbf{x \in \mathbb{R}}$ . Prove that $\mathbf{g \circ h}$ is in A and is equal to $\mathbf{h \circ g}$ .

Consider the equation $\mathbf{n^2+(n+1)^4=5(n+2)^3}$
(a) Show that any integer of the form 3m+1 or 3m+2 can not be a solution of this equation.

(b) Does the equation have a solution in positive integers?

Consider a rectangular sheet of paper ABCD such that the lengths of AB and AD are respectively 7 and 3 centimetres. Suppose that B’ and D’ are two points on AB and AD respectively such that if the paper is folded along B’D’ then A falls on A’ on the side DC. Determine the maximum possible area of the triangle AB’D’.

Take r such that $\mathbf{1\le r\le n}$ , and consider all subsets of r elements of the set $\mathbf{\{1,2,\ldots,n\}}$ . Each subset has a smallest element. Let F(n,r) be the arithmetic mean of these smallest elements. Prove that: $\mathbf{F(n,r)={n+1\over r+1}}$ .

Let $\mathbf{f: \mathbb{R}^2 \to \mathbb{R}^2}$ be a function having the following property: For any two points A and B in $\mathbf{\mathbb{R}^2}$ , the distance between A and B is the same as the distance between the points f(A) and f(B).
Denote the unique straight line passing through A and B by l(A,B)

(a) Suppose that C,D are two fixed points in $\mathbf{\mathbb{R}^2}$ . If X is a point on the line l(C,D), then show that f(X) is a point on the line l(f(C),f(D)).

(b) Consider two more point E and F in $\mathbf{\mathbb{R}^2}$ and suppose that l(E,F) intersects l(C,D) at an angle $\mathbf{\alpha}$ . Show that l(f(C),f(D)) intersects l(f(E),f(F)) at an angle \alpha. What happens if the two lines l(C,D) and l(E,F) do not intersect? Justify your answer.

There are 100 people in a queue waiting to enter a hall. The hall has exactly 100 seats numbered from 1 to 100. The first person in the queue enters the hall, chooses any seat and sits there. The n-th person in the queue, where n can be 2, . . . , 100, enters the hall after (n-1)-th person is seated. He sits in seat number n if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which 100 seats can be filled up, provided the 100-th person occupies seat number 100.

I.S.I. B.STAT ENTRANCE SUBJECTIVE 2009

Two train lines intersect each other at a junction at an acute angle $\mathbf{\theta}$ . A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\mathbf{\alpha}$ at a station on the other line. It subtends an angle $\mathbf{\beta (<\alpha)}$ at the same station, when its rear is at the junction. Show that $\mathbf{ \tan\theta=\frac{2\sin\alpha\sin\beta}{\sin(\alpha-\beta)}}$

Let f(x) be a continuous function, whose first and second derivatives are continuous on $\mathbf{[0,2\pi]}$ and $\mathbf{f''(x) \geq 0 }$ for all x in $\mathbf{[0,2\pi]}$ . Show that
$\mathbf{\int_{0}^{2\pi} f(x)\cos x dx \geq 0}$

Let ABC be a right-angled triangle with BC=AC=1. Let P be any point on AB. Draw perpendiculars PQ and PR on AC and BC respectively from P. Define M to be the maximum of the areas of BPR, APQ and PQCR. Find the minimum possible value of M.

A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for $\mathbf{k\geq 2}$ , a sequence is called an arithmetic progression of the k-th order if the differences of the successive terms form an arithmetic progression of the (k-1)-th order.
The numbers
4,6,13,27,50,84
are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the n-th term of this progression.

A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is 60 centimetres and its base has radius 30 centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

Let f(x) be a function satisfying $\mathbf{xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0}$
Show that $\mathbf{f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)}$ where $\mathbf{f^{(n)}(x) }$ denotes the n-th derivative evaluated at x.

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is x, show that the radius of the circumcircle is $\mathbf{\frac{x}{2\sin 36^\circ}}$ .

Find the number of ways in which three numbers can be selected from the set $\mathbf{\{1,2,\cdots ,4n\}}$ , such that the sum of the three selected numbers is divisible by 4.

Consider 6 points located at $\mathbf{P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)}$ . Let R be the region consisting of all points in the plane whose distance from P_0 is smaller than that from any other $\mathbf{P_i, i=1,2,3,4,5}$ . Find the perimeter of the region R.

Let $\mathbf{x_n}$ be the n-th non-square positive integer. Thus $\mathbf{x_1=2, x_2=3, x_3=5, x_4=6}$ , etc. For a positive real number x, denotes the integer closest to it by $\mathbf{\langle x\rangle }$ . If x=m+0.5, where m is an integer, then define $\mathbf{\langle x\rangle=m}$ . For example, $\mathbf{\langle 1.2\rangle =1, \langle 2.8 \rangle =3, \langle 3.5\rangle =3}$ . Show that $\mathbf{x_n=n+\langle \sqrt{n}\rangle}$

I.S.I. B.STAT ENTRANCE SUBJECTIVE 2008

Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

A 40 feet high screen is put on a vertical wall 10 feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

Study the derivatives of the function
$\mathbf{y=\sqrt[3]{x^3-4x}}$
and sketch its graph on the real line.

Suppose P and Q are the centres of two disjoint circles $\mathbf{C_1}$ and $\mathbf{C_2}$ respectively, such that P lies outside $\mathbf{C_2}$ and Q lies outside $\mathbf{C_1}$ . Two tangents are drawn from the point P to the circle $\mathbf{C_2}$ , which intersect the circle $\mathbf{C_1}$ at point A and B. Similarly, two tangents are drawn from the point Q to the circle $\mathbf{C_1}$ , which intersect the circle $\mathbf{C_2}$ at points M and N. Show that AB=MN

Suppose ABC is a triangle with inradius r. The incircle touches the sides BC, CA, and AB at D,E and F respectively. If BD=x, CE=y and AF=z, then show that $\mathbf{r^2=\frac{xyz}{x+y+z}}$

Evaluate: $\mathbf{\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}}$

Consider the equation $\mathbf{x^5+x=10}$ . Show that
(a) the equation has only one real root;
(b) this root lies between 1 and 2;
(c) this root must be irrational.

In how many ways can you divide the set of eight numbers $\mathbf{\{2,3,\cdots,9\}}$ into 4 pairs such that no pair of numbers has $\mathbf{\text{gcd} }$ equal to 2?

Suppose S is the set of all positive integers. For $\mathbf{a,b \in S}$ , define
$\mathbf{a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}}$
For example 8*12=6.
Show that exactly two of the following three properties are satisfied:
(i) If $\mathbf{a,b \in S}$ , then $\mathbf{a*b \in S}$ .
(ii) $\mathbf{(a*b)*c=a*(b*c)}$ for all $\mathbf{a,b,c \in S}$ .
(iii) There exists an element $\mathbf{i \in S}$ such that $\mathbf{a *i =a}$ for all $\mathbf{a \in S}$

Two subsets A and B of the (x,y)-plane are said to be equivalent if there exists a function f: A\to B which is both one-to-one and onto.
(i) Show that any two line segments in the plane are equivalent.
(ii) Show that any two circles in the plane are equivalent.

I.S.I. B.STAT ENTRANCE SUBJECTIVE 2007

Suppose a is a complex number such that $\mathbf{ a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0 }$ If m is a positive integer, find the value of $\mathbf{a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}}$

Use calculus to find the behaviour of the function $\mathbf{ y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty}$ and sketch the graph of the function for $\mathbf{-2\pi \le x \le 2\pi}$ . Show clearly the locations of the maxima, minima and points of inflection in your graph.

Let f(u) be a continuous function and, for any real number u, let [u] denote the greatest integer less than or equal to u. Show that for any x>1, $\mathbf{\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du }$

Show that it is not possible to have a triangle with sides a,b, and c whose medians have length $\mathbf{\frac{2}{3}a, \frac{2}{3}b \text{and} \frac{4}{5}c}$ .

Show that $\mathbf{-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2 }$ for all values of $\mathbf{\theta}$ .

Let $\mathbf{S=\{1,2,\cdots ,n\}}$ where n is an odd integer. Let f be a function defined on $\mathbf{\{(i,j): i\in S, j \in S\}}$ taking values in S such that
(i) $\mathbf{f(s,r)=f(r,s) \text{for all} r,s \in S}$
(ii) $\mathbf{\{f(r,s): s\in S\}=S \text{for all} r\in S}$

Show that $\mathbf{\{f(r,r): r\in S\}=S}$

Consider a prism with triangular base. The total area of the three faces containing a particular vertex A is K. Show that the maximum possible volume of the prism is $\mathbf{\sqrt{\frac{K^3}{54}}}$ and find the height of this largest prism.

The following figure shows a $\mathbf{3^2 \times 3^2 }$ grid divided into $\mathbf{3^2}$ subgrids of size $\mathbf{3 \times 3}$ . This grid has 81 cells, 9 in each subgrid.

Now consider an $\mathbf{n^2 \times n^2}$ grid divided into $\mathbf{n^2}$ subgrids of size $\mathbf{n \times n}$ . Find the number of ways in which you can select n^2 cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.

Let X $\mathbf{\subset \mathbb{R}^2}$ be a set satisfying the following properties:
(i) if $\mathbf{(x_1,y_1)}$ and $\mathbf{(x_2,y_2)}$ are any two distinct elements in X, then
$\mathbf{\text{ either, }\ \ x_1 > x_2 \text{ and } y_1 > y_2\\ \text{ or, } \ \ x_1 < x_2 \text{ and } y_1 < y_2}$
(ii) there are two elements $\mathbf{(a_1,b_1)}$ and $\mathbf{(a_2,b_2)}$ in X such that for any $\mathbf{(x,y) \in X}$ ,
$\mathbf{a_1\le x \le a_2 \text{ and } b_1\le y \le b_2 }$
(iii) if $\mathbf{(x_1,y_1) \text{and} (x_2,y_2)}$ are two elements of X, then for all $\mathbf{\lambda \in [0,1], \left(\lambda x_1+(1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2\right) \in X }$
Show that if $\mathbf{(x,y) \in X}$ , then for some $\mathbf{\lambda \in [0,1], x=\lambda a_1 +(1-\lambda)a_2, y=\lambda b_1 +(1-\lambda)b_2 }$

Let A be a set of positive integers satisfying the following properties:
(i) if m and n belong to A, then m+n belong to A;
(ii) there is no prime number that divides all elements of A.

(a) Suppose $\mathbf{ n_1 \text{and} n_2 }$ are two integers belonging to A such that $\mathbf{n_2-n_1 > 1}$ . Show that you can find two integers $\mathbf{m_1 \text{and} m_2 }$ in A such that $\mathbf{0 < m_2-m_1 < n_2-n_1}$
(b) Hence show that there are two consecutive integers belonging to A.
(c) Let $\mathbf{n_0 \text{and} n_0+1 }$ be two consecutive integers belonging to A. Show that if $\mathbf{n\geq n_0^2 }$ then n belongs to A.