# Tagged Questions

Use this tag when you want to determine the thinking that is needed to solve a certain type of problem, as opposed to looking for a specific answer to a question.

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### What is the largest possible area that $PQRS$ could have?

In a $14\times 18$ rectangle $ABCD$, points $P,Q,R$ and $S$ are chosen, one on each side $ABCD$ as pictured. The lengths $AP, PB, BQ, QC, CR, RD, DS$ and $SA$ are all positive integers and $PQRS$ is a ...
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### Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it. So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the ...
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### Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors

Prove that among any 12 consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers ...
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### How to find the average Kendall's distance between 2 rankings

Suppose I have 2 rankings: $1$, $2$, $3$ and $2, 1, 3$ then the Kendall's distance between the two is 1 since there is only one pairwise adjacent switch. My question is, suppose my 2 rankings each ...
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### Probability problem

I created this problem based on the following probability riddle here. You're a king, and you were given two groups of people, and a certain information about them. First group has 2 people. One of ...
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### Fox and Gemstone problem. Weighing stones to compare bags.

There is a problem called Fox and Gemstone at topcoder: https://community.topcoder.com/stat?c=problem_statement&pm=14296 Basically, you have the ability to weigh individual gemstones against ...
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### Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
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### Find the Smallest Value

Find the smallest value of $$a + \frac {1}{(a-b)b}$$ where a>b>0 I found this question in AM-GM inequality problems but I am stuck at this
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### find the number of tuples of positive integers [closed]

find the number of tuples (a,b,c,d) of positive integers \begin{array}{l} {a^3} = {b^2}\\ {c^3} = {d^2}\\ c - a = 64 \end{array} answer should be one of 0 , 1 , 2 , 4
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### proving no real roots exist

Prove that $x^8-x^7+x^2-x+15$ has no real roots. I did it by first assuming it has real roots and then applying Descartes rule of signs. We find that if there are any real roots, they all must be ...
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### number theory problem finding triplets [closed]

Find number of triplets of positive integers satisfying $2^a-5^b\cdot 7^c=1$ Given options are $0 , 1 , 2$ or infinite.
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### How to find the area shared by 4 quadrants inside a square?

I was to find the blue area in this question : As described about how it's a square with 4 quadrants of same radius intertwined with each other, now to find the blue part area I thought about ...
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### Find the solution without “common sense” assumption [closed]

Is it possible to solve the following problem without make a "common sense" assumption? In the year 1887, one person's age was exactly the sum of the digits of his birth's year. What was the person'...
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### Homeomorphism from $S^1\backslash(0,1)$ to $\mathbb{R}$

I am trying to derive a bijection between $S^1\backslash{(0,1)}$ and the real line, but I am stuck on using the most obvious way Let the top point of the circle be $(0,1)$, and the blue line hits ...
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### Polynomial taking irrationals to irrationals

Problem: Find all polynomials from $\mathbb{R}\to \mathbb{R}$ $f$ with integer coefficients taking irrationals to irrationals. My attempt: It is clear that the problem statement is equivalent to ...
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### What is the size of the angle $\angle AMC$? [duplicate]

Suppose we have a triangle $\triangle ABC$ where the size of two angles are given: $\angle B=15^\circ$ and $\angle C=30^\circ$. We draw the median $AM$, so now what is the size of angle $\angle AMC$? ...
### Let $f$ be a differentiable function defined for all real $x$, where $f(x)\ge 0$ for all $x\in[0,a]$
Let $f$ to be a differentiable function defined for all real $x$, where $f(x)\ge 0$ for all $x\in[0,a]$.If $$\int_0^a f(x)\,dx = a,$$ then 2\int_0^{5a}\left(f\left( \frac x 5 \right) + 3 \right)\,...
### Topology: What is a quick way to check whether a subset $D$ is dense in $(X, \mathcal{T})$?
Def $1$: Let $(X, \mathcal{T})$ be a topological space, then $D \subseteq X$ is dense if $\overline {D} = X$ Def $2$: $x \in \overline D$ iff for all \$U \in \mathcal{T}, x \in U \implies D \cap U \...