India National Olympiad 2025 - PROBLEMS

                      

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