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2025 SSMO Relay Round 2 Problems

Problem 1

Let $(p,q,r)$ be the real solution to following system: \begin{align*} pq &= 20 \\ qr &= 25 \\ r+p &= 9. \end{align*} Compute $p^2+q^2+r^2$.

Solution

Problem 2

Let $T = TNYWR.$ Let $x_1,x_2,\dots, x_{T}$ be an increasing sequence of positive integers such that for every positive integer $1\le n \le T,$ the sum $x_1+x_2+\cdots + x_n$ is a multiple of $n$. Find the smallest possible value of $x_{T} - x_1$.

Solution

Problem 3

Let $T = TNYWR.$ Define a \textit{multiplicative partition} of a positive integer $n$ as the value of a product $a_1a_2\cdots a_k,$ where $a_1+a_2+\cdots + a_k = n$ and every $a_i$ is a positive integer. Let $f(x)$ denote the maximal possible value of a multiplicative partition of $x$. If the sum of all possible values of $\frac{f\left(x+\left\lfloor\sqrt{T}\right\rfloor\right)}{f(x)}$ for integers $x>1$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Solution

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