DottedCalculator's blog

By DottedCalculator, 7 months ago, In English

In this post, I want to go over each of my problems. I got this idea from antontrygubO_o and flamestorm.

Number Problem Statement Source Comments
001 In right triangle $$$ABC$$$, we have $$$\angle B=90^{\circ}$$$. If $$$AC=18$$$ and the length $$$x$$$ of the altitude from $$$B$$$ to $$$AC$$$ satisfies

then the maximum possible value of the area of $ABC$ can be expressed in the form $$$\frac{a+b\sqrt c}d$$$ for integers $$$a$$$, $$$b$$$, $$$c$$$, and $$$d$$$, where $$$\gcd(a,b,d)=1$$$, $$$c$$$ is squarefree, and $$$d$$$ is positive. Compute $$$a+b+c+7d$$$.

CNCM PoTD 07/03/2021 The answer is not $$$665$$$.
002 Let $$$n$$$ be a positive integer such that $$$n=a^{665}m$$$, where $$$a$$$ is as large as possible. Then, define $$$\psi(n)=\mu(m)$$$, where $$$\mu(n)=0$$$ if $$$p^2\mid n$$$ for some $$$p$$$, and $$$\mu(n)=(-1)^k$$$ if $$$n$$$ is squarefree and has $$$k$$$ prime factors. Find
$$$\sum_{i=1}^n\psi(i)\left\lfloor\frac ni\right\rfloor$$$

in terms of $n$.

CNCM PoTD 11/05/2021 Classic summation swap with $$$665$$$.
003 Bob has infinitely many matches of equal length and wants to make as many equilateral triangles as possible. He uses one square yard of tape for each point two matches intersect. What is the smallest number of square inches of tape he needs to create at least seven equilateral triangles? CNCM PoTD 11/14/2021 Think outside the box and don't get trolled.
004 Let $$$ABC$$$ be a triangle such that $$$AB=7$$$, $$$BC=8$$$, and $$$CA=9$$$. There exists a unique point $$$X$$$ such that $$$XB=XC$$$ and $$$XA$$$ is tangent to the circumcircle of $$$ABC$$$. If $$$XA=\frac ab$$$, where $$$a$$$ and $$$b$$$ are coprime positive integers, find $$$a+b$$$. OMMC 2022/11 Nice length chasing problem with right triangles, but there are also a lot of bash solutions.
005 Alex writes down some distinct integers on a blackboard. For each pair of integers, he writes the positive difference of those on a piece of paper. Find the sum of all $$$n\leq2022$$$ such that it is possible for the numbers on the paper to contain all the positive integers between $$$1$$$ and $$$n$$$, inclusive exactly once. OMMC Tiebreaker 2022/2 Annoying grinding problem.
006 Compute the number of solutions in positive integers to the equation $$$ab+bc+ca=p^{2k}$$$ satisfying $$$a\leq b\leq c$$$ where $$$p$$$ is a prime and $$$k$$$ is a positive integer. ELSMO Revenge 2022/1 The solution may or may not exist.
007 Let $$$ABC$$$ be a triangle. Let $$$\omega_A$$$ be the ellipse with foci $$$B$$$ and $$$C$$$ passing through $$$A$$$, and define $$$\omega_B$$$ and $$$\omega_C$$$ similarly. Let $$$A_1$$$ and $$$A_2$$$ be the intersections of $$$\omega_B$$$ and $$$\omega_C$$$, and define $$$B_1$$$, $$$B_2$$$, $$$C_1$$$, and $$$C_2$$$ similarly. Show that $$$A_1A_2$$$, $$$B_1B_2$$$, and $$$C_1C_2$$$ are concurrent and find the point of concurrency. OTIS Suggestion Also Sharygin 2023/23

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