Found problems: 5
2023 India National Olympiad, 6
Euclid has a tool called [i]cyclos[/i] which allows him to do the following:
[list]
[*] Given three non-collinear marked points, draw the circle passing through them.
[*] Given two marked points, draw the circle with them as endpoints of a diameter.
[*] Mark any intersection points of two drawn circles or mark a new point on a drawn circle.
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Show that given two marked points, Euclid can draw a circle centered at one of them and passing through the other, using only the cyclos.
[i]Proposed by Rohan Goyal, Anant Mudgal, and Daniel Hu[/i]
2023 India National Olympiad, 3
Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying:
[list]
[*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;
[*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$.
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[i]Proposed by Sutanay Bhattacharya[/i]
2023 India National Olympiad, 5
Euler marks $n$ different points in the Euclidean plane. For each pair of marked points, Gauss writes down the number $\lfloor \log_2 d \rfloor$ where $d$ is the distance between the two points. Prove that Gauss writes down less than $2n$ distinct values.
[i]Note:[/i] For any $d>0$, $\lfloor \log_2 d\rfloor$ is the unique integer $k$ such that $2^k\le d<2^{k+1}$.
[i]Proposed by Pranjal Srivastava[/i]
2023 India National Olympiad, 1
Let $S$ be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs $(x,y)$ in $S\times S$ so that the product $xy$ is a perfect square. Prove that one can find at least four distinct elements in $S$ so that none of their pairwise products is a perfect square.
[i]Note:[/i] As an example, if $S=\{1,2,4\}$, there are exactly five such ordered pairs: $(1,1)$, $(1,4)$, $(2,2)$, $(4,1)$, and $(4,4)$.
[i]Proposed by Sutanay Bhattacharya[/i]
2023 India National Olympiad, 2
Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$:
$$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real.
[i]Proposed by Prithwijit De[/i]