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1. Find all pairs $$(x,y)$$ with $$x,y$$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$

2. Suppose that $$PQ$$ and $$RS$$ are two chords of a circle intersecting at a point $$O$$. It is given that $$PO=3 \text{cm}$$ and $$SO=4 \text{cm}$$. Moreover, the area of the triangle $$POR$$ is $$7 \text{cm}^2$$. Find the area of the triangle $$QOS$$.
3. Let $$f:\mathbb{R}\to\mathbb{R}$$ be a continuous function such that for all $$x\in\mathbb{R}$$ and for all $$t\geq 0$$, $$f(x)=f(e^tx)$$Show that $$f$$ is a constant function.

4. Let $$f:(0,\infty)\to\mathbb{R}$$ be a continuous function such that for all $$x\in(0,\infty)$$, $$f(2x)=f(x)$$Show that the function $$g$$ defined by the equation $$g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0$$is a constant function.

5. Let $$f:\mathbb{R}\to\mathbb{R}$$ be a differentiable function such that its derivative $$f’$$ is a continuous function. Moreover, assume that for all $$x\in\mathbb{R}$$, $$0\leq \vert f'(x)\vert\leq \frac{1}{2}$$Define a sequence of real numbers $$\{a_n\}_{n\in\mathbb{N}}$$ by :$$a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}$$Prove that there exists a positive real number $$M$$ such that for all $$n\in\mathbb{N}$$, $$\vert a_n\vert \leq M$$

Hint:

The sequence is Cauchy

6. Let, $$a\geq b\geq c >0$$ be real numbers such that for all natural number $$n$$, there exist triangles of side lengths $$a^{n} , b^{n} ,c^{n}$$.
Prove that the triangles are isosceles.

7. Let $$a, b, c$$ are natural numbers such that $$a^{2}+b^{2}=c^{2}$$ and $$c-b=1$$ Prove that
(i) a is odd.
(ii) b is divisible by 4
(iii) $$a^{b}+b^{a}$$ is divisible by c

8. Let $$n\geq 3$$. Let $$A=((a_{ij}))_{1\leq i,j\leq n}$$ be an $$n\times n$$ matrix such that $$a_{ij}\in\{-1,1\}$$ for all $$1\leq i,j\leq n$$. Suppose that $$a_{k1}=1~~\text{for all}~1\leq k\leq n$$and $$~~\sum_{k=1}^n a_{ki}a_{kj}=0~~\text{for all}~i\neq j$$.
Show that n is a multiple of 4.