PartialMathematician

numbers that end in 2:        20

numbers that have tens digit 2:           20 

numbers that have hundreds digit 2:      0

numbers overcounted:           2              (the numbers we overcounted were 322 and 422)

Answer = 20+20-2 = 38.

- PM

And good catch, guest! 

I hope the above solution helps. 

Hope this helps, 

- PM

Lol I thought you were smart enough to understand... The parallelogram is a rhombus.

rhom - bus

/ˈrämbəs/

noun

a parallelogram with opposite equal acute angles, opposite equal obtuse angles, and four equal sides.

any parallelogram with equal sides, including a square.

The largest possible length of the shortest side is 5. 

Well, if you got the correct answer, good for you. 

2) Think about it. The question says to find the "largest possible length of the shortest side". Well if you go too big, the "shortest side" becomes the "longest side". What if there was no "shortest side" and "longest side"?

- PM

1) For this problem, you would use a formula similar to the Pythag theorem. The pythag theorem says a^2 + b^2 = c^2 for right triangles, so if a triangle is acute, a^2 + b^2 > c^2. Likewise for obtuse triangles, a^2 + b^2 < c^2.

Use this and brute force to find the answer.

Hope this helps,

- PM

Yeah, I've seen it happen to me too. One vote might not seem like much, but the troll searches up the person's profile and repeatedly thumbs down each post. Usually I have 0-2 thumbs ups for most of my math responses, but the troll keeps on disliking my posts every hour. I have noticed that the same person can downvote or upvote the same post; he or she just has to wait a couple of hours before the program forgets that the user has already voted. That's what happened when I had around 850 one day, and when I checked later in the evening, I was down to 700. I don't know why this person is doing it, and honestly I think it's a bit dissapointing for the user to find out that they have lost 50 status points in a few minutes. 

(And I think the troll doesn't have anything to do in his/her life, so he/she just feasts on other people's likes)

- PM

Oh, I see what you are saying. I thought the first condition with x+1 if floor x is prime was saying: If floor x is prime, then p(x) = floor x + 1.

That explains a lot. But again, the number can be infinitely close to 8, such as 7.99999..., but the number can never be 8, so the answer range is \(\boxed{[3,9)}\).

I tried that for guest, and I am pretty sure it is incorrect. I think p(x) can equal 8 because if x = 7, p(x) = 7+1 = 8. 

You are a guest, so it does not matter. If you have a real accout, then downvoting is truly dissappointing. (*hint hint* i am continuously donvoted by someone)

Vieta's formulas. 

Say we have a polynomial \(ax^2 + bx + c = 0\) with solutions X_1 and X_2.

Vieta's formulas prove that...

a) X_1 + X_2 = -b/a.

b) (X_1) * (X_2) = c/a.

- PM

There is a tag at the bottom left corner of the page that says "Online Users". If there are some profile pictures there, that means some experienced (or people that use the website a lot) users are online.

The radical in \(\dfrac{\sqrt{28}}{14}\) can be simplified to \(2 \sqrt{7}\). \(\dfrac{2 \sqrt{7}}{14}\) simplifies into \(\boxed{\dfrac{\sqrt{7}}{7}}\), the answer. 

Hope this helps,

- PM

Assuming that the numberator is the same as the denominator, the answer is \(\boxed{1}\). 

Treat \(\sqrt{45 + \sqrt{1}} + \sqrt{45 + \sqrt{2}} ... + \sqrt{45 + \sqrt{2024}}\) as X. We will have \(\dfrac{X}{X}\), which simplifies into 1. 

Hope this helps, 

- PM

But anyways, think of it this way. The whole trip takes her X minutes, from Home to Friend's House. Let's call the distance between the Post Office and Friend's House \(Y\). To walk (2+Y) miles, it takes her X minutes. 

If she can walk a mile in Z minutes, she can walk the distance from Home to Post Office in 2Z minutes. The time she takes to walk from Post Office to Friend's House is X - 2Z, which we will call A. To find the distance between PO and FH, we have (A minutes) * (1 mile / Z minutes). The minutes cancel out, giving us A miles / Z. Putting any values for A and Z will give us the answer for the distance from PO to FH. 

Hope this helps,

- PM

Umm...how long does it take Georgia to walk a whole trip? I don't think the Latex registered for the number of minutes.