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Also known as night-shining clouds, noctilucent clouds are Earth's highest clouds and are found in the mesosphere, roughly 50 miles above the surface (just for comparison, most are found in the troposphere, under 10 miles). They are mostly observed at high latitudes and during astronomical twilight (1.5-2 hrs before sunrise or after sunset). For information, you can visit the wiki site here: https://en.wikipedia.org/wiki/Noctilucent_cloud#:~:text=Noctilucent%20clouds%2C%20or%20night%20shining,%22night%20shining%22%20in%20Latin.

If you could only eat one thing for the rest of your life, what would it be?

These pouch like protrusions composed of ice are typically found at the bottom of cumulonimbus clouds and can stretch hundreds of miles in every direction. They are typically harbingers of violent storm weather. For more info on these, you can check out the wiki page here: https://en.wikipedia.org/wiki/Mammatus_cloud

Hi guys, Just wanted to share with you these awesome clouds, which occur in the stratosphere at polar latitudes during the wintertime. For more information, you can visit the wiki page here: https://en.wikipedia.org/wiki/Polar_stratospheric_cloud

For the first Math Trick of the Week (MTW), which will be uploaded every 23:30 Wednesday GMT time (as near that time as I can make), is going to be on... SFFT. WHAT DOES SFFT STAND FOR? SFFT stands for: Simon's Favorite Factoring Trick, which was popularized by AoPS (Art of Porblem Solving), in one of their videos. WHAT IS SFFT? To explain it, let us look at an example: "Solve for all integer pairs (x,y) in this: xy+x+y=0". At first glance, this problem seems hard. Then, we might observe this solution: (0,0). But, is that the only one? If we can find another, how can we prove there are only these pairs? Method 1 (Works in general, but slow) xy+y=-x (x+1)y=-x y=-x/(x+1) So, when is y an integer given x is an integer? Well, after a bit of deliberation, it is clear that only x=0 and x=-2 work, so the pairs are (0, 0) and (-2, -2). (Took me 1:34 mins using this method) Method 2 (Fast, SFFT) xy+y=-x (x+1)y=-x (x+1)y-1=-(x+1) (x+1)(y+1)=1 So, (x+1, y+1)=(1, 1) or (-1, -1). So, (x,y)=(0, 0) or (-2, -2) What is SFFT in general? axy+bx+cy=d xy+(b/a)x+(c/a)y=d/a (y+b/a)x+(c/a)y=d/a The next step, is to make y+b/a. (y+b/a)x+(c/a)y+bc/(a^2)=d/a+bc/(a^2) (y+b/a)x+(y+b/a)(c/a)=d/a+bc/(a^2)=(da+bc)/(a^2) (y+b/a)(x+c/a)=(da+bc)/(a^2) (ay+b)(ax+c)=da+bc. You now find factors of the right hand side, and then you can find (x, y). Conclusion SFFT is a powerful trick that solves diophantine equations of the form axy+bx+cy=d. I hope you understand this trick now. THE END

Hi guys, If any of you guys have experience with number theory/proofs, I'd really appreciate your help with some of my math camp problems! They are so hard, I'm not even sure where to start

Here are the rules: I have chosen five from the SAs, As, and Cs, and withdraws will be accepted if wanted. I will pair them with members with similar ratings ASAP. I will use the ratings at the time i am writing this. ALSO, NO COLLABORATION. 14 days daily chess for each game @LLLhk (985) BLACK will be paired with @COURT12CHAMPION (983) @Hyper_Speed20 (1399) BLACK will be paired with @Ashok_Chess_Novice (1396) @ChessSensasian (1216) BLACK will be paired with @Anu_2107 (1206) @cutelittle (1112) WHITE will be paired with @Kaidranzer112 (1136) @virtuousabyss29 (1719) WHITE will be paired with @DrOwaisKhan (1720)

In this forum you are allowed to talk about topics related to mathematics and science.

first puzzle: ldebnur

For example, the first person might write: Once For example, the second person might write: Once upon. So, start from Once upon.

If you would like to be promoted, please put down here why you want to be promoted and how you will improve the club.

PLEASE TRY IT YOURSELF AND DO NOT ANALYSE Day 1: (Mate)

Please write down suggestions for the club rules.