3 questions from 2021
Consider a triangle \( \Delta \) whose two sides lie on the \( x \)-axis and the line \( x + y + 1 = 0 \). If the orthocenter of \( \Delta \) is \( (1, 1) \), then the equation of the circle passing through the vertices of the triangle \( \Delta \) is:
Let \( M = \{(x,y)\in \mathbb{R}\times \mathbb{R} : x^2 + y^2 \le r^2\} \), where \( r>0 \). Consider the geometric progression \( a_n=\frac{1}{2^{\,n-1}},\; n=1,2,3,\ldots \). Let \( S_0=0 \) and, for \( n\ge 1 \), let \( S_n \) denote the sum of the first \( n \) terms of this progression. For \( n\ge 1 \), let \( C_n \) denote the circle with center \( (S_{n-1},0) \) and radius \( a_n \), and \( D_n \) denote the circle with center \( (S_{n-1},S_{n-1}) \) and radius \( a_n \).
Consider \( M \) with \( r=\frac{1025}{513} \). Let \( k \) be the number of all those circles \( C_n \) that are inside \( M \). Let \( l \) be the maximum possible number of circles among these \( k \) circles such that no two circles intersect. Then
Let \( M = \{(x,y)\in \mathbb{R}\times \mathbb{R} : x^2 + y^2 \le r^2\} \), where \( r>0 \). Consider the geometric progression \( a_n=\frac{1}{2^{\,n-1}},\; n=1,2,3,\ldots \). Let \( S_0=0 \) and, for \( n\ge 1 \), let \( S_n \) denote the sum of the first \( n \) terms of this progression. For \( n\ge 1 \), let \( C_n \) denote the circle with center \( (S_{n-1},0) \) and radius \( a_n \) and \( D_n \) denote the circle with center \( (S_{n-1},S_{n-1}) \) and radius \( a_n \).
Consider \( M \) with \( r=\dfrac{(2^{100}-1)\sqrt{2}}{2^{198}} \). The number of all those circles \( D_n \) that are inside \( M \) is:
