RMO 2025

(a)Let $n\geq3$ be an integer. Find a configuration of $n$ lines in the plane which has exactly

(i) $n-1$ distinct points of intersection. (ii) $n$ distinct points of intersection

(b) Give configuration of $n$ lines that have exactly $n+1$ distinct points of intersection for (i) $n=8$ and (ii) $n=9$ Proposed by Prof. C.R. Pranesachar

Let $a$ , $b$ , $c$ be distinct non-zero real numbers satisfying $a+\frac{2}{b}= b+\frac{2}{c}=c+\frac{2}{a}$ Determine the value of $|a^2 b+b^2 c+c^2 a|$ Proposed by Prof. B.J. Venkatachala

Let $\Omega$ and $\Gamma$ be circles centred at $O_1$ , $O_2$ respectively. Suppose that they intersect in distinct points $A$ , $B$ . Suppose $O_1$ is outside $\Gamma$ and $O_2$ is outside $\Omega$ . Let $l$ be not passing through $A$ and $B$ that interests $\Omega$ at $P$ and $R$ and $\Gamma$ at $Q$ , $S$ so that $P$ , $Q$ , $R$ , $S$ lie on the line in this order. Furthermore, points $O_1$ , $B$ lie on one side of $l$ and the points $O_2$ , $A$ lie on the other side of $l$ . Given that the points $A$ , $P$ , $Q$ , $O_1$ are concyclic and $B$ , $R$ , $S$ , $O_2$ are concyclic as well, prove that $AQ=BR$ Proposed by Archit Manas and Aadish Jain

Prove that there do not exist positive rational numbers $x$ and $y$ such that $x+y+\frac{1}{x}+\frac{1}{y}= 2025$

Let $ABC$ be an acute-angled triangle with $AB<AC$ , orthocenter $H$ and circumcircle $\Omega$ . Let $M$ be the midpoint of the minor arc $BC$ of $\Omega$ . Suppose the $MH$ is equal to the radius of $\Omega$ . Prove that $\angle BAC=60^\circ$ Proposed by Rijul Saini

Let $p(x)$ be a nonconstant polynomial with integer coefficients, and let $n\geq2$ be an integer such that no term of the sequence $p(0), p(p(0)), p(p(p(0))),...$ is divisible by $n$ . Show that there exist integers $a$ , $b$ such that $0\leq a<b\leq n-1$ and $n$ divides $p(b)-p(a)$ Proposed by Bhavya Tiwari

1. Solve the following system of equations in nonnegative integers $a_1, a_2, \ldots, a_8$ where $a_i \ne 1$ for $i=1, \ldots, 8$ : \begin{align*} a_1 a_2 &= a_3 + a_4 a_3 a_4 &= a_5 + a_6 a_5 a_6 &= a_7 + a_8 a_7 a_8 &= a_1 + a_2 \end{align*}

Let $a, b, c$ be positive real numbers satisfying $abc = 1$ . Prove that $$\frac{2a^2}{a^3+1} + \frac{2b^2}{b^3+1} + \frac{2c^2}{c^3+1} \le a^2 + b^2 + c^2$$

Let $ABCDE$ be a convex pentagon in which $AB=AE$ , $CB=CD$ , and $\angle AED =\angle CDE= 90^\circ$ . Let the internal bisectors of $\angle EAB$ and $\angle DCB$ intersect at $I$ , and let $M$ be the midpoint of $AC$ . Prove that $\angle MIC = \angle AIB$ .

Proposed by - Rijul Saini

A frog is initially at $(0, 0)$ and it reaches $(n, 2)$ , $n \ge 1$ , using the following moves in any order several times:

$R=(1, 0)$ , that is, if the frog is at $(a, b)$ it goes to $(a+1, b)$ ; $U=(0, 1)$ , that is, if the frog is at $(a, b)$ it goes to $(a, b+1)$ ; $D=(1, 1)$ , that is, if the frog is at $(a, b)$ it goes to $(a+1, b+1)$ .

In how many ways can the frog go from $(0, 0)$ to $(n, 2)$ , $n \ge 1$ , using the above steps subject to the condition that steps of the type $UU, DD$ are forbidden? (For example, for $n=3$ , $RDUR, DRD$ are admissible paths, while $DDR, RUURR$ are not.)

Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^\circ$ and $AB < BC < AC$ . Let $M, N$ be the midpoints of $AB, AC$ respectively. Suppose $BE, CF$ are altitudes, with $E$ on $CA$ and $F$ on $AB$ . Let $X$ be the image of $M$ under reflection in the midpoint of $BF$ , and $Y$ be the image of $N$ under reflection in the midpoint of $CE$ . Prove that $XY$ bisects $BC$ .

Proposed by - Rijul Saini

Define the sequence $(a_0, a_1, a_2, \ldots)$ as follows: $a_0 = 49$ and $a_{n} = 10^{2^n} a_{n-1} - 1$ for $n \ge 1$ . Show that $s(a_n^2) = n^2 + n + 7$ for all $n \ge 0$ , where $s(m)$ denotes the sum of digits in base 10 representation of a nonnegative integer $m$ .