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

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  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 \(QOS\).Learn More

  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.Learn more

     

  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. Learn More
  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.

    Hint:

    Isosceles Triangles

  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
    Hint:
    Powers of Pythagorean
  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.

    Hint:

    Divisible by 4