India National Olympiad 2025 - PROBLEMS

                      

Problem 1

Consider the sequence defined by $a_1=2$, $a_2=3$, and

$$a_{2k+1}=2+2a_k,a_{2k+2}=2+a_k+a_{k+1},$$

for all integers $k\ge 1$. 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 $\mathrm{\angle }XYZ$.
• 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 $b⩾2$ 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