by Ashani Dasgupta

The Problem 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. Key Ideas One...
by Ashani Dasgupta

The Problem 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. Key Ideas Set \( \frac{x_2}{x_1} = t \) for all \( x_1, x_2 > 0 \). Do the same...
by Ashani Dasgupta

The Problem 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 \(QOS\). Key...
by Ashani Dasgupta

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

Find all pairs \( (x,y) \) with \(x,y\) real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$ Learn more Suppose that \(PQ\) and \(RS\) are two chords of a circle intersecting at a point \(O\). It is given that \(PO=3...