RMM 2026

Let $n$ be a positive integer. Alice draws a unit area triangle on the board. Then she draws additional triangles by performing $n$ moves in a row. On each move, she chooses a drawn triangle $\Delta$ with no marked points in its interior, marks a point $P$ in its interior, and draws three smaller triangles by joining $P$ to each vertex of $\Delta$ with a segment.

Once these $n$ moves have been performed, Bob chooses three distinct drawn triangles $\Delta_1, \Delta_2,$ and $\Delta_3$ which contain no marked points in their interiors, such that $\Delta_2$ shares one side with $\Delta_1$ and another with $\Delta_3$. In terms of $n$, determine the largest possible constant $c$ such that Bob can guarantee that the sum of the areas of $\Delta_1, \Delta_2,$ and $\Delta_3$ is at least $c$, regardless of Alice’s choices.

Let $p \geq 11$ be a prime. Suppose that, if $a$ and $b$ are integers such that $1 \leq a < b \leq p-3,$ then $b! - a!$ is not divisible by $p$. Prove that $p-5$ is divisible by 8.

Let $S$ be a finite subset of $\mathbb{R}^3$. Prove that there exist three polynomials $P(x,y,z), Q(x,y,z)$ and $R(x,y,z)$ with real coefficients, such that a triple of real numbers $(a,b,c)$ is in $S$ if and only if the system of equations

$$P(x,y,z) = a,$$

$$Q(x,y,z) = b,$$

$$R(x,y,z) = c,$$

does not have a solution in real numbers $x, y,$ and $z$.

For any positive integer $m$, let $\varphi(m)$ be the number of positive integers less than or equal to $m$ and coprime to $m$. Define $\varphi_0(m) = m$ and, for each positive integer $k$, $\varphi_k(m) = \varphi(\varphi_{k-1}(m))$. For any integer $n \geq 3$, prove that

$$\varphi_0(2^n - 3) \cdot \varphi_1(2^n - 3) \cdot \varphi_2(2^n - 3) \cdot \ldots \cdot \varphi_n(2^n - 3)$$

has at most $n$ distinct prime divisors.

Let $ABC$ be a triangle with $AB < AC$, let $O$ be its circumcentre and let $XYZT$ be a parallelogram inside triangle $ABC$ such that

$$\angle AXB = \angle AZC, \angle AZB = \angle AXC,$$

$$\angle AYB = \angle ATC, \angle ATB = \angle AYC.$$

Prove that the diagonals $XZ$ and $YT$ of the parallelogram intersect on the circumcircle of $BOC$.

Let $k > 1$ be an integer, and let $S$ denote the set of all $(k+1)$-tuples of integers $X = (x_1, \ldots, x_{k+1})$ such that $1 \leq x_1 < \ldots < x_{k+1} \leq k^2 + 1$. If $\sigma$ is a permutation of the numbers $1, 2, \ldots, k^2 + 1$, say that an element $X$ of $S$ is $\sigma$-nice if the sequence $\sigma(x_1), \sigma(x_2), \ldots, \sigma(x_{k+1})$ is monotone. Prove that

$$\min_{1 \leq i \leq k} \left\lfloor \frac{x_i}{i} \right\rfloor + \min_{2 \leq i \leq k+1} \left\lfloor \frac{k^2 + 2 - x_i}{k + 2 - i} \right\rfloor \geq k + 1$$

if and only if there exists a permutation $\sigma$ such that $X$ is the unique $\sigma$-nice tuple in $S$.