Weekly Contest #58

(Ended)

Standings

Q1.

A region $S$ in the complex plane is defined by\[S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.\]A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac34 + \frac34i\right)z$ is also in $S$?

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


Editorial
Tags: Algebra Geometry Probability

Q2.

Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\angle DBA = 23^{\circ}$, and $\angle BDC = 46^{\circ}$. The ratio $BC: AD$ is $9: 5$. What is $CD$?

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


Editorial
Tags: Algebra Combinatorics Geometry Probability

Q3.

For how many values of $x$ in $[0,\pi]$ is $\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)$? Note: The functions $\sin^{ - 1} = \arcsin$ and $\cos^{ - 1} = \arccos$ denote inverse trigonometric functions.

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


Editorial
Tags: Algebra Combinatorics

Q4.

How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?

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


Editorial
Tags: Algebra Probability