Problem 1
Consider the sequence defined by \(a_1 = 2\), \(a_2 = 3\), and
\[ a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1}, \]
for all integers \(k \geq 1\). Determine all positive integers \(n\) such that
\[ \frac{a_n}{n} \]
is an integer.
Proposed by: Niranjan Balachandran, SS Krishnan, and Prithwijit De
Problem 2
Let \(n\ge 2\) be a positive integer. The integers \(1,2,\dots,n\) are written on a board...
Find all \(n\) for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board.
Proposed by: Rohan Goyal
Problem 3
Euclid has a tool called splitter which can only do the following two types of operations:
- Given three non-collinear marked points \(X,Y,Z\), it can draw the line which forms the interior angle bisector of \(\angle{XYZ}\).
- It can mark the intersection point of two previously drawn non-parallel lines.
Prove that Euclid can use the splitter several times to draw the centre of a circle passing through \(A,B\) and \(C\).
Proposed by: Shankhadeep Ghosh
Problem 4
Let \(n\ge 3\) be a positive integer. Find the largest real number \(t_n\) as a function of \(n\)...
Proposed by: Rohan Goyal and Rijul Saini
Problem 5
Greedy goblin Griphook has a regular 2000-gon, whose every vertex has a single coin...
What is the maximum and minimum number of coins he could have collected?
Proposed by: Pranjal Srivastava and Rohan Goyal
Problem 6
Let \(b \geqslant 2\) be a positive integer. Anu has an infinite collection of notes...
Prove that if there is a payable number, there are infinitely many payable numbers.
Proposed by: Shantanu Nene
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