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1. For any $k \in\mathbb{Z}^+$ , prove that:-
$\displaystyle{ 2(\sqrt{k+1}-\sqrt{k})<\frac{1}{\sqrt{k}}<2(\sqrt{k}-\sqrt{k-1}) }$
Also compute integral part of $\displaystyle{ \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}} }$.
2. Let $\displaystyle{a_1=1 }$ and $\displaystyle{a_n=n(a_{n-1}+1)}$ for all $\displaystyle{n\ge 2}$ . Define : $\displaystyle{P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)}$ Compute $\displaystyle{\lim_{n\to \infty} P_n}$
3. Let ABCD be a quadrilateral such that the sum of a pair of opposite sides equals the sum of other pair of opposite sides (AB+CD=AD+BC). Prove that the circles inscribed in triangles ABC and ACD are tangent to each other.
4. For a set S we denote its cardinality by |S|. Let $\displaystyle{e_1,e_2,\ldots,e_k }$ be non-negative integers. Let $\displaystyle{A_k}$ (respectively $\displaystyle{B_k}$) be the set of all k-tuples $\displaystyle{(f_1,f_2,\ldots,f_k)}$ of integers such that $\displaystyle{0\leq f_i\leq e_i}$ for all i and $\displaystyle{\sum_{i=1}^k f_i }$ is even (respectively odd). Show that $\displaystyle{|A_k|-|B_k|=0 \textrm{ or } 1}$.
5. Find the point in the closed unit disc $\displaystyle{D={ (x,y) | x^2+y^2\le 1 }}$ at which the function f(x,y)=x+y attains its maximum .
6. Let $a_0=0 $\displaystyle{\int_{a_j}^{a_{j+1}} p(t),dt = 0 \forall 0 \le j\le n-1}$ Show that , for $\displaystyle{0\le j\le n-1}$ , the polynomial p(t) has exactly one root in the interval $\displaystyle{(a_j,a_{j+1})}$
7. Let M be a point in the triangle ABC such that $\displaystyle{\text{area}(ABM)=2 \cdot \text{area}(ACM)}$
Show that the locus of all such points is a straight line.
8. In how many ways can one fill an n*n matrix with +1 and -1 so that the product of the entries in each row and each column equals -1?