Weekly Contest #91

(Ended)

Standings

Q1.

 cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

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


Editorial
Tags: Combinatorics

Q2.

Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$. Find the number of intersections of the graphs of\[y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).\]

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


Editorial
Tags: Algebra

Q3.

Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there had been red vertices is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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


Editorial
Tags: Combinatorics

Q4.

Let $A$, $B$, $C$, and $D$ be points on the hyperbola $\frac{x^2}{20}- \frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi.

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


Editorial
Tags: Geometry