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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Prove that in each year, the $13$ th day of some month occurs on a Friday.

**Problem 2:**

In the accompanying figure, $y=f(x)$ is the graph of a one-to-one continuous function $f$. At each point $P$ on the graph of $y=2 x^{2}$, assume that the areas $O A P$ and $O B P$ are equal. Here $P A \cdot P B$ are the horizontal and vertical segments. Determine the function $f$.

**Problem 3:**

Show that. for any positive integer $n,$ the sum of $8 n+4$ consecutive positive integers cannot be a perfect square.

**Problem 4:**

If $a, b, c \in(0,1)$ satisfy $a+b+c=2,$ prove that

$\frac{a b c}{(1-a)(1-b)(1-c)} \geq 8$.

**Problem 5:**

Let $a_{1}>a_{2}>\cdots>a_{r}$ be positive real numbers. Compute $\lim _{n \rightarrow \infty}\left(a_{1}^{n}+a_{2}^{n}+\cdots+a_{r}^{n}\right)^{1 / n}$.

**Problem 6:**

Let each of the vertices of a regular $9$-gon (polygon of $9$ equal sides and equal angles) be coloured black or white.

(a) Show that there are two adjacent vertices of the same colour.

(b) Show there are 3 vertices of the same colour forming an isosceles triangle.

**Problem 7:**

Let $a, b, c$ be real numbers and. assume that all the roots of $x^{3}+a x^{2}+b x+c=0$ have the same absolute value, Show that $a=0$ if, and only if, $b=0$.

**Problem 8:**

I et $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}$ exists, and is finite. Prove that $f^{\prime}(0)=0$.

**Problem 9:**

Let $f(x)$ be a polynomial with integer coefficients. Assume that 3 divides the value f(n) for each integer $n$. Prove that when $f(x)$ is divided by $x^{3}-x$ the remainder is of the form $3 r(x)$, where $r(x)$ is a polynomial with integer coefficients.

**Problem 10:**

Consider a regular heptagon (polygon of 7 equal sides and equal angles) ABCDEFG.

(a) Prove $\frac{1}{\sin \frac{\pi}{7}}=\frac{1}{\sin \frac{2 \pi}{7}}+\frac{1}{\sin \frac{3 \pi}{7}}$.

(b) Using (a) or otherwise, show that $\frac{1}{A G}=\frac{1}{A F}+\frac{1}{A E}$. (See the figure appearing in the next page.)

Here, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Prove that in each year, the $13$ th day of some month occurs on a Friday.

**Problem 2:**

In the accompanying figure, $y=f(x)$ is the graph of a one-to-one continuous function $f$. At each point $P$ on the graph of $y=2 x^{2}$, assume that the areas $O A P$ and $O B P$ are equal. Here $P A \cdot P B$ are the horizontal and vertical segments. Determine the function $f$.

**Problem 3:**

Show that. for any positive integer $n,$ the sum of $8 n+4$ consecutive positive integers cannot be a perfect square.

**Problem 4:**

If $a, b, c \in(0,1)$ satisfy $a+b+c=2,$ prove that

$\frac{a b c}{(1-a)(1-b)(1-c)} \geq 8$.

**Problem 5:**

Let $a_{1}>a_{2}>\cdots>a_{r}$ be positive real numbers. Compute $\lim _{n \rightarrow \infty}\left(a_{1}^{n}+a_{2}^{n}+\cdots+a_{r}^{n}\right)^{1 / n}$.

**Problem 6:**

Let each of the vertices of a regular $9$-gon (polygon of $9$ equal sides and equal angles) be coloured black or white.

(a) Show that there are two adjacent vertices of the same colour.

(b) Show there are 3 vertices of the same colour forming an isosceles triangle.

**Problem 7:**

Let $a, b, c$ be real numbers and. assume that all the roots of $x^{3}+a x^{2}+b x+c=0$ have the same absolute value, Show that $a=0$ if, and only if, $b=0$.

**Problem 8:**

I et $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}$ exists, and is finite. Prove that $f^{\prime}(0)=0$.

**Problem 9:**

Let $f(x)$ be a polynomial with integer coefficients. Assume that 3 divides the value f(n) for each integer $n$. Prove that when $f(x)$ is divided by $x^{3}-x$ the remainder is of the form $3 r(x)$, where $r(x)$ is a polynomial with integer coefficients.

**Problem 10:**

Consider a regular heptagon (polygon of 7 equal sides and equal angles) ABCDEFG.

(a) Prove $\frac{1}{\sin \frac{\pi}{7}}=\frac{1}{\sin \frac{2 \pi}{7}}+\frac{1}{\sin \frac{3 \pi}{7}}$.

(b) Using (a) or otherwise, show that $\frac{1}{A G}=\frac{1}{A F}+\frac{1}{A E}$. (See the figure appearing in the next page.)

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