Weekly Contest #97

(Ended)

Standings

Q1.

 

Find the number of collections of $16$ distinct subsets of $\{1,2,3,4,5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X \cap Y \not= \emptyset.$

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


Editorial
Tags: Combinatorics

Q2.

            Let $x,y,$ and $z$ be real numbers satisfying the system of equations

s\begin{align*} xy + 4z &= 60 \\ yz + 4x &= 60 \\ zx + 4y &= 60. \end{align*}

                                       Let $S$ be the set of possible values of $x.$ . Find the sum of the squares of the elements of $S.$

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


Editorial
Tags: Algebra

Q3.

                               Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product

\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\]

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


Editorial
Tags: Algebra

Q4.

Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$

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


Editorial
Tags: Trigonometry

Q5.

For each positive integer, $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$

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


Editorial
Tags: Number Theory