Here, you will find all the questions of ISI Entrance Paper 2008 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1 :
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. Suppose
$$
f(x)=\frac{1}{t} \int_{0}^{t}(f(x+y)-f(y)) d y
$$
for all $x \in \mathbb{R}$ and all $t>0$. Then show that there exists a constant $c$ such that $f(x)=c x$ for all $x$.
Problem 2 :
Suppose that $P(x)$ is a polynomial with real coefficients such that for some positive real numbers $c, d$ and for all natural numbers $n,$ we have
$$
c|n|^{3} \leq|P(n)| \leq d|n|^{3}
$$
Prove that $P(x)$ has a real zero.
Problem 3 :
Let $z$ be a complex number such that $z, z^{2}, z^{3}$ are collinear in the complex plane. Show that $z$ is a real number.
Problem 4 :
Let $a_{1}, \ldots, a_{n}$ be integers. Show that there exists integers $k$ and $r$ such that the sum
$$
a_{k}+a_{k+1}+\cdots+a_{k+r}
$$
is divisible by $n$.
Problem 5 :
If a polynomial $P$ with integer coefficients has three distinct integer zeroes, then show that $P(n) \neq 1$ for any integer $n$.
Problem 6 :
Let $\left(\begin{array}{l}n \\ k\end{array}\right)$ denote the binomial coefficient $\frac{n !}{k !(n-k) !},$ and $F_{m}$ be the $m$ th Fibonacci number given by $F_{1}=F_{2}=1$ and $F_{m+2}=F_{m}+F_{m+1}$ for all $m \geq 1$. Show that
$$
\sum\left(\begin{array}{l}
n \\
k
\end{array}\right)=F_{m+1} \quad \text { for all } m \geq 1
$$
Here, the above sum is over all pairs of integers $n \geq k \geq 0$ with $n+k=m$.
Problem 7 :
Let,
$$
C=\{(i, j) \mid i, j \text { integers such that } 0 \leq i, j \leq 24\}
$$
How many squares can be formed in the plane all of whose vertices are in $C$ and whose sides are parallel to the $x$ -axis and $y$ -axis?
Problem 8 :
Let,
$$
a^{2}+b^{2}=1, \quad c^{2}+d^{2}=1, \quad a c+b d=0
$$.
Prove that,
$$
a^{2}+c^{2}=1, \quad b^{2}+d^{2}=1, \quad a b+c d=0
$$.
Problem 9 :
For $n \geq 3,$ determine all real solutions of the system of $n$ equations:
$x_{1}+x_{2}+\cdots+x_{n-1}=\frac{1}{x_{n}}$
$\ldots \ldots \ldots$
$x_{1}+x_{2}+\cdots+x_{i-1}+x_{i+1}+\cdots+x_{n}=\frac{1}{x_{i}}$
$\ldots \ldots \ldots$
$x_{2}+\cdots+x_{n-1}+x_{n}=\frac{1}{x_{1}}$
Problem 10 :
If $p$ is a prime number and $a>1$ is a natural number, then show that the greatest common divisor of the two numbers $a-1$ and $\frac{a^{p}-1}{a-1}$ is either $1$ or $p$.
Here, you will find all the questions of ISI Entrance Paper 2008 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1 :
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. Suppose
$$
f(x)=\frac{1}{t} \int_{0}^{t}(f(x+y)-f(y)) d y
$$
for all $x \in \mathbb{R}$ and all $t>0$. Then show that there exists a constant $c$ such that $f(x)=c x$ for all $x$.
Problem 2 :
Suppose that $P(x)$ is a polynomial with real coefficients such that for some positive real numbers $c, d$ and for all natural numbers $n,$ we have
$$
c|n|^{3} \leq|P(n)| \leq d|n|^{3}
$$
Prove that $P(x)$ has a real zero.
Problem 3 :
Let $z$ be a complex number such that $z, z^{2}, z^{3}$ are collinear in the complex plane. Show that $z$ is a real number.
Problem 4 :
Let $a_{1}, \ldots, a_{n}$ be integers. Show that there exists integers $k$ and $r$ such that the sum
$$
a_{k}+a_{k+1}+\cdots+a_{k+r}
$$
is divisible by $n$.
Problem 5 :
If a polynomial $P$ with integer coefficients has three distinct integer zeroes, then show that $P(n) \neq 1$ for any integer $n$.
Problem 6 :
Let $\left(\begin{array}{l}n \\ k\end{array}\right)$ denote the binomial coefficient $\frac{n !}{k !(n-k) !},$ and $F_{m}$ be the $m$ th Fibonacci number given by $F_{1}=F_{2}=1$ and $F_{m+2}=F_{m}+F_{m+1}$ for all $m \geq 1$. Show that
$$
\sum\left(\begin{array}{l}
n \\
k
\end{array}\right)=F_{m+1} \quad \text { for all } m \geq 1
$$
Here, the above sum is over all pairs of integers $n \geq k \geq 0$ with $n+k=m$.
Problem 7 :
Let,
$$
C=\{(i, j) \mid i, j \text { integers such that } 0 \leq i, j \leq 24\}
$$
How many squares can be formed in the plane all of whose vertices are in $C$ and whose sides are parallel to the $x$ -axis and $y$ -axis?
Problem 8 :
Let,
$$
a^{2}+b^{2}=1, \quad c^{2}+d^{2}=1, \quad a c+b d=0
$$.
Prove that,
$$
a^{2}+c^{2}=1, \quad b^{2}+d^{2}=1, \quad a b+c d=0
$$.
Problem 9 :
For $n \geq 3,$ determine all real solutions of the system of $n$ equations:
$x_{1}+x_{2}+\cdots+x_{n-1}=\frac{1}{x_{n}}$
$\ldots \ldots \ldots$
$x_{1}+x_{2}+\cdots+x_{i-1}+x_{i+1}+\cdots+x_{n}=\frac{1}{x_{i}}$
$\ldots \ldots \ldots$
$x_{2}+\cdots+x_{n-1}+x_{n}=\frac{1}{x_{1}}$
Problem 10 :
If $p$ is a prime number and $a>1$ is a natural number, then show that the greatest common divisor of the two numbers $a-1$ and $\frac{a^{p}-1}{a-1}$ is either $1$ or $p$.
