Try the followings:

Consider a prism with a 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 $\displaystyle \sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.

Consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$. 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 ach row and one from each column.

If $a_1, \cdots, a_7 \in (1,13)$ are not necessarily distinct reals, show that we can choose three of them such that they are lengths of the sides of a triangle.

Show that there cannot exist a non-constant polynomial $P(x) \in \mathbb{Z}[x]$ such that $P(n)$ is prime for all positive integers $n$.

**(Calculus)**Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose that $f$ also satisfies $f(x) \geq 0$, $f'(x) \leq 0$ and $f''(x) \leq f(x)$, for all $x \geq 0$. Show that $f'(0) \geq - \sqrt{2}$.

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## Comments

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There must be a value of $x \in(0,1)$ satisfying $f'(x_{0}) = \dfrac{f(1) - 1}{1-0}$

For this $x$ , $(f'(x_{0}))^2 = (1 - f(1))^2$

Now,

$f''(x) \leq f(x) \Rightarrow f'(x) f''(x) \geq f'(x) f(x)$

Hence,

$d((f'(x))^2) \geq d((f(x))^2)$

Integrate in limits 0 to $x_{0}$ to get:

$(f'(x_{0}))^2 - (f'(0))^2 \geq (f(x_{0}))^2 - 1$

Hence, $(f'(0))^2 \leq 1 +(1-f(1))^2 - (f(x_{0}))^2$

Clearly, $(f'(0))^2 \leq 2$ ,or $f'(0) \geq - \sqrt{2}$

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Good!

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hey paramjeet i cant get it can u explain me in detail

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What needs explanation? And it's Paramj

it. :)Log in to reply

Problem 3 is essentially USAMO 2012 Problem 1 in disguise. Can you see why?

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