Let $latex A_1$, $latex A_2$, $latex A_3$, $latex \ldots$, $latex A_{20}$ be a $latex 20$-sided polygon $latex P$ in the plane. Suppose all of the side lengths of $latex P$ are $latex 1$, the interior angle at $latex A_i$ measures…

There is no submitter. Hint: For any positive integer $latex m$ and prime $latex p$, let $latex v_p(m)$ be the exponent of $latex p$ in the prime factorization of $latex m$. Then $latex v(ab) = v_p(a) + v_p(b)$ for any…

For all positive integers $latex n$, find the value $latex \displaystyle\frac{1}{n}\sum_{k=1}^n \frac{k\cdot k!\cdot \binom{n}{k}}{n^k}$. Due day is April 19.

Find all positive integers $latex n$ that have distinct positive divisors $latex d_1$, $latex d_2$,$latex \ldots$, $latex d_k$, where $latex k > 1$, that are in arithmetic progression and $latex n=d_1 +d_2 + \cdots +d_k.$ Note that $latex d_1$, $latex…

Results: 17-045 박민철 A 17-063 송민학 A, delay 17-071 어세훈 A+ 15-033 남동현 C I would like to thank 박민철, who informed me of the error in grading.

The positive integers $latex a_1$, $latex a_2$, $latex \ldots$, $latex a_9$ are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $latex n$ and the product of any two…

A point $latex P$ in the interior of a convex polyhedron in Euclidean space is called a pivot point of the polyhedron if every line through $latex P$ contains exactly $latex 0$ or $latex 2$ vertices of the polyhedron. Determine,…

A sequence $latex a_1,\ a_2,\ \ldots,\ a_n$ is called almost-increasing if $latex a_i\le a_{i+1}+8$ for all $latex i$. For example, the sequence $latex 160,\ 170,\ 165,\ 180,\ 175$ is almost-increasing. (a) How many different rearrangement of the sequence $latex 140,\…