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ISI UGB 2025
Entrepreneur   6
N Yesterday at 3:15 PM by oty
Source: ISI UGB 2025
1.)
Suppose $f:\mathbb R\to\mathbb R$ is differentiable and $|f'(x)|<\frac 12\;\forall\;x\in\mathbb R.$ Show that for some $x_0\in\mathbb R,f(x_0)=x_0.$

3.)
Suppose $f:[0,1]\to\mathbb R$ is differentiable with $f(0)=0.$ If $|f'(x)|\le f(x)\;\forall\;x\in[0,1],$ then show that $f(x)=0\;\forall\;x.$

4.)
Let $S^1=\{z\in\mathbb C:|z|=1\}$ be the unit circle in the complex plane. Let $f:S^1\to S^1$ be the map given by $f(z)=z^2.$ We define $f^{(1)}:=f$ and $f^{(k+1)}=f\circ f^{(k)}$ for $k\ge 1.$ The smallest positive integer $n$ such that $f^n(z)=z$ is called period of $z.$ Determine the total number of points $S^1$ of period $2025.$

6.)
Let $\mathbb N$ denote the set of natural numbers, and let $(a_i,b_i), 1\le i\le 9,$ be nine distinct tuples in $\mathbb N\times\mathbb N.$ Show that there are $3$ distinct elements in the set $\{2^{a_i}3^{b_i}:1\le i\le 9\}$ whose product is a perfect cube.

8.)
Let $n\ge 2$ and let $a_1\le a_2\le\cdots\le a_n$ be positive integers such that $$\sum_{i=1}^n a_i=\prod_{i=1}^n a_i.$$Prove that $$\sum_{i=1}^n a_i\le 2n$$and determine when equality holds.
6 replies
Entrepreneur
May 27, 2025
oty
Yesterday at 3:15 PM
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