2023 SSMO Tiebreaker Round Problems

Problem 1

From the phrase "Summer Solstice", how many ways are there to make a 4 letter "word" such that the second and third letters aren't spaces and the first letter is capital. Note: You can only use a letter twice if it appears in "Summer Solstice" twice.

Solution

Problem 2

Let $P(x) = x^3 + 3ax^2 + 3bx + (a+b)$ be a real polynomial with nonnegative and nonzero real roots $p, q, r$ . Suppose that $(p + 1)^3 + (q + 1)^3 + (r+1)^3 + 3P(-1) = 0.$ If $P(1) = a_1+b_1\sqrt{c_1},$ for squarefree $c_1,$ find $a_1+b_1+c_1$ .

Solution

Problem 3

For $n\geq4,$ let $a_n$ be the maximum possible value of $P(n+1)$ given that $P(x)$ is a $n$ degree monic polynomial that satisfies $P(i)\in\{1,2,3\dots,n\}$ for $1\leq i \leq n.$ If $\frac{m}{n} = \sum_{n=4}^{\infty}\frac{a_n-n!}{3^n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

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2023 SSMO Tiebreaker Round Problems

Problem 1

Problem 2

Problem 3