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Recent questions tagged isi2021-pcb-mathematics

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2
ISI 2021 | PCB Mathematics | Question: 2
Consider three real numbers $a \geq b \geq c>0$. If $\left(a^{x}-b^{x}-c^{x}\right)(x-2)>0$ for any rational number $x \neq 2$, show that $a, b$ and $c$ can be the lengths of the three sides of a triangle $A B C;$ $A B C$ is a right-angled triangle.
Consider three real numbers $a \geq b \geq c>0$. If $\left(a^{x}-b^{x}-c^{x}\right)(x-2)>0$ for any rational number $x \neq 2$, show that$a, b$ and $c$ can be the lengths...
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4
ISI 2021 | PCB Mathematics | Question: 4
In the figure below, there are three circles touching each other externally and also touching the line below. Let $r_{1}, r_{2}$ and $r_{3}$ be the radii of the three circles as shown in the figure. If $r_{1}=25$ and $r_{3}=9$, then find $r_{2}$.
In the figure below, there are three circles touching each other externally and also touching the line below. Let $r_{1}, r_{2}$ and $r_{3}$ be the radii of the three cir...
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ISI 2021 | PCB Mathematics | Question: 6
Let $p$ be an odd prime and let $n=(p-1)(p+1)$. Show that $p$ divides $n 2^{n}+1$. Show that there are infinitely many integers $m$ such that $p$ divides $m 2^{m}+1$.
Let $p$ be an odd prime and let $n=(p-1)(p+1)$.Show that $p$ divides $n 2^{n}+1$.Show that there are infinitely many integers $m$ such that $p$ divides $m 2^{m}+1$.
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7
ISI 2021 | PCB Mathematics | Question: 7
Let $G$ be a cubic graph, that is, every vertex has degree exactly $3$. Prove that the number of vertices of $G$ cannot be $101$. Prove that if $G$ contains $100$ vertices, then it contains a bipartite subgraph that has at least $75$ edges.
Let $G$ be a cubic graph, that is, every vertex has degree exactly $3$.Prove that the number of vertices of $G$ cannot be $101$.Prove that if $G$ contains $100$ vertices,...
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10
ISI 2021 | PCB Mathematics | Question: 10
Let $a>0$ and $x_{1}>0$. Define $x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)$ for all $n \in \mathbb{N}$. Show that $x_{n}>\sqrt{a}$ for all $n \geq 2;$ the sequence $\left\{x_{n}: n \geq 1\right\}$ converges to $\sqrt{a}.$
Let $a>0$ and $x_{1}>0$. Define $x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)$ for all $n \in \mathbb{N}$. Show that$x_{n}>\sqrt{a}$ for all $n \geq 2;$the seque...
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