Weekly Contest #88

(Ended)

Standings

Q1.

When the roots of the polynomial

\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot \cdot (x-10)^{10}\]

are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Algebra

Q2.

You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Combinatorics

Q3.

Let $\mathcal{B}$ be the set of rectangular boxes with surface area $54$ and volume $23$. Let $r$ be the radius of the smallest sphere that can contain each of the rectangular boxes that are elements of $\mathcal{B}$. The value of $r^2$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Geometry

Q4.

Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.

Answer must be a floating-point or integer value and precision error less than 10^-6 is allowed.


Editorial
Tags: Combinatorics Number Theory