Misa-Misa's blog

By Misa-Misa, history, 9 months ago,

1

Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

2

A plane contains $40$ lines, no $2$ of which are parallel. Suppose there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.

Solution

3

The sum of all positive integers $m$ for which $\tfrac{13!}{m}$ is a perfect square can be written as $2^{a}3^{b}5^{c}7^{d}11^{e}13^{f}$, where $a, b, c, d, e,$ and $f$ are positive integers. Find $a+b+c+d+e+f$.

Solution

4

There exists a unique positive integer $a$ for which the sum $[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor]$ is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$.

(Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)

5

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$.

Sources

What is Misa-Math
• +32

 » 9 months ago, # |   -9 Auto comment: topic has been updated by Misa-Misa (previous revision, new revision, compare).
 » 9 months ago, # |   +54 When posting problems you did not create, you should credit the original source. (The first four problems come from the 2023 AIME I and the last comes from the 2023 USAJMO.)
•  » » 9 months ago, # ^ |   +24 Ok i will do that, I starting practising some maths after reading your blog. Thought it would be good idea to post problems here, for others and my own reference too.
•  » » » 9 months ago, # ^ |   +1 Makes sense, good luck with your training!
 » 9 months ago, # |   -8 Hi