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## Given a triangle and equation of all its sides, how do I understand if an internal angle is obtuse or acute?

I am given the following question:- In a triangle ABC, if the equation of sides AB, BC, and AC are $2x-y+4 = 0$, $x-2y-1=0$ and $x+3y-3 = 0$ respectively, then what is the tangent of the internal angle a? So I used the following formula, for two lines with slopes $m_1$ and $m_2$, the acute […]

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## Perfect squares between certain divisors of a number

Let $n$ be a positive integer. We will call a divisor $d(<\sqrt{n})$ of $n$ special if there exists no perfect squares between $d$ and $\frac{n}{d}$. Prove that $n$ can have at-most one special divisor. My progress: I boiled down the problem to the following: Suppose $k^2\le a,b,c,d\le (k+1)^2$, then $ab=cd\implies \{a,b\}=\{c,d\}$. But I can’t seem […]

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## Perfect squares between certain divisors of a number

Let $n$ be a positive integer. We will call a divisor $d(<\sqrt{n})$ of $n$ special if there exists no perfect squares between $d$ and $\frac{n}{d}$. Prove that $n$ can have at-most one special divisor. My progress: I boiled down the problem to the following: Suppose $k^2\le a,b,c,d\le (k+1)^2$, then $ab=cd\implies \{a,b\}=\{c,d\}$. But I can’t seem […]

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## Excel Vlookup not letting me press enter

Ok, I have a strange issue. I can’t enter a vlookup in Excel. Pressing enter does nothing and I can still type in the cell. The formula is something basic like =VLOOKUP(A2,B:C,2,FALSE). The formula will only work if I do not include a range, =VLOOKUP(A2,B:C,2,). It won’t let me enter a range. What is also […]

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## Does one element of a field represent some other element of any arbitrary field?

Three elements a,b,c of a field form an arithmetic progression if their successive difference are equal: b-a=c-b. Prove that b=a+c/2. when I start proving it: => b-a+b=c-b+b => b+b-a=c => b+b-a+a=c+a => b+b=c+a in next step, how I write b+b as 2b as it is not specified that b belongs to real number field. for […]

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## conditional probability, how to express P(A,B|C) in term of P(A,C|B)

what i know is, conditional probability: P(X|Y) = P(X,Y) / P(Y) total probability: P(X) = sum Y [P(X|y) P(y)] bayes rule: p(X|Y) = p(Y|X) P(X) Is it possible to express P(A,B|C) in term of P(A,C|B) and joint distribution of two random variable, i.e. P(A,C). P(A,B), P(B,C) Is it possible to express P(A,B|C,D) in term of […]

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## NP oracle queries

The variables $a,b,c \in \{0,1\}$, thus $a^k, b^k, c^k \in \{0,1\}$ I want to pass a query to an oracle that returns the coefficients of each term $(1,a,b,c,ab,ac,bc,abc)$ in the expansion of products such as this one $(1-a+ab)(1-b+bc)(b-bc)$. There could be more variables and more brackets to expand. Do I require a single NP query […]

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## Prove that D is incenter of ABC

Question – AB,AC are tangents from A to a circle touching it at B and C. if D is midpoint of minor arc BC prove that D is incentre of ABC. My try – First I proved that A,D,O are collinear using given condition that D is midpoint of arc BC and tangents from external […]

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## Prove that D is incenter of ABC

Question – AB,AC are tangents from A to a circle touching it at B and C. if D is midpoint of minor arc BC prove that D is incentre of ABC. My try – First I proved that A,D,O are collinear using given condition that D is midpoint of arc BC and tangents from external […]