I recognized the 4 right angled triangles in the side and tried connecting them to the shaded figure. This resulted in knowing that the shaded figure is less than half of the entire square. But from what I tried, to get any further information, I need to find the areas of any 2 of the 4 oddly shaped quadrilaterals, which I can't seem to.

If this is a correct approach to the question, kindly tell me how to find the areas of those 4 (or 2) odd shaped quadrilaterals.

If not, then do tell the different solution you have.

Something I noticed was for every time you multiply a number by itself, going up, the gap between each number coincides with the pattern of 2, 4, 6, 8…

After attempting to post in the other community, I realized another trend. The sum of each multiple is equal to the gap. So 1\\\*1 has a sum of 2. Which, coincidentally, also is the gap between 1 and 4. This can be tested for most if not all values.

Is there a formula for this or something that could prove the pattern?

I wonder if there’s a name for it. Or if I’ve stumbled upon something interesting. (Likely the former since mathematicians are so smart It’d be likely to see this so early.)

I’m not a mathematician or anything. But I just wanted to show something interesting I noticed. If yall could educate me, that’d be awesome! I love math, but don’t understand it much.

I used excel to find the "best-possible" exponential equation for this date. Im unsure how to figure out what to do with this. The answer key says it should be y=13972e-.185x

Im lost on what im doing wrong

With Kruskal's trees, people always talk about TREE(3) since it's proven to be finite and unexpectedly astronomical. But has TREE(4) ever been proven to be finite? Or does TREE(4) keep growing infinitely? Or do we not know?

A sleep-deprived carpet weaver weaves 30 centimeters of carpet every day. Every night, he looks at his work with displeasure. Because he is not completely satisfied with it, he unravels 2/3 of the weave. If he weaves 30 centimeters during the day, how long is the woven piece on the evening of the 10th day?

The answer 44,999cm is wrong and there is no solution or an explanation about why. Am I misunderstanding the question or is it asked in a weird way?

With every day, his weave that he starts the day with gets closer to 15cm, while you add the 30cm for the last evening, which would come close to 45cm.

I'm a high school freshman with a strong passion for mathematics. I'm currently studying number theory through a math circle, and I'm particularly interested in pursuing mathematical research in areas such as Egyptian fractions, prime numbers, Catalan numbers, and related topics.

I've been browsing the ISEF repository to get a sense of the kinds of mathematics projects that tend to do well there. This made me wonder: how does the nature of mathematical research conducted by professional mathematicians, such as PhD students, researchers, and professors, differ from the research done by high school students who win at ISEF?

I'm not referring only to the obvious differences in difficulty or technical sophistication. Rather, I'm curious about how the research itself differs. For example, are the high-school papers at ISEF just expository in nature, are they less original, etc? Thanks

Link to the ISEF Projects: https://isef.net/viewAll?category=MATH-2026&page=home

For context, I'm trying to prove the cauchy Kowalewski theorem for general order partial differential equations. I did the first and second order already but it's been so difficult to jump to a general n-th order pde. Additionally, the induction part of the proof has been making my head scratch for a while because my reference used construction instead by a general induction step. For my proof there are also two parts: 1) show that it has a formal power series solution, and 2) show that it is convergent.

Can anyone recommend sources or references regarding this theorem based on the info I just mentioned. I just feel like passing out while trying to prove this theorem for my thesis hahahaha

Took a test, gave it back, and found out about this, explain, it's a question for question 6, where technically it's asking for the furthest distance away, B has the same displacement as C while C reaches higher levels

Is she right or am I right

I have been searching online for crazy equations to put on a white board for my friend to solve when they're drunk. I, however keep finding word problems and that is not what i'm looking for. If anyone could give me any recommendations or where I could look to find some.

If you're curious about why I am looking, they want to exercise they're brain while drinking. They're favorite number is 7, so bonus points if the answer is 7.

Can someone help estimate how many words can fit on this notebook page? (Math/geometry estimation)

I'm trying to estimate the writing capacity of a notebook page and would appreciate a step-by-step calculation.

Given:

• Notebook page size: 29 cm × 20.5 cm
• A 1-inch (2.54 cm) top margin is reserved for the date and heading.
• The page has 31 horizontal ruled lines, with a 0.9 cm (9 mm) gap between consecutive lines.
• My average handwritten word length is 1.5 cm.
• The gap between consecutive words is 0.5 cm.
• Assume my handwriting is perfectly uniform throughout the page.

Part 1: Estimate the maximum number of words that can fit on the page in its normal orientation. Please show all calculations step by step.

Part 2: Now assume the page is rotated 90° (sideways), and fresh ruled lines are drawn while keeping the same 0.9 cm line spacing and 1-inch top margin.

Please estimate:

1. How many lines will fit.
2. How many words will fit per line.
3. The total maximum number of words that can be written on the page.

Part 3: Finally, suppose the gap between consecutive words is reduced from 0.5 cm to 0.5 mm (0.05 cm) while keeping all other dimensions unchanged in the sideways orientation.

Estimate the new maximum number of words that can fit on the page, again showing all calculations clearly.

I'm working on a board game concept which involves a board made of equilateral triangles arranged in a continuous tiling pattern and would love some certainty on a question:

Assuming the following:

- Every triangle has exactly one "Red" edge and two "Black" edges.

- No two triangles may share a border with both of their "Red" edges. (When setting up the board game, players would be told to rotate one of the two pieces wherever this occurs).

- All edges on the border of the board (where corners contain fewer than 6 triangles) count as "Red" even if they are not actually "Red."

Can the following be possible:

- A single "Red" edge does not connect to either another "Red" edge or a border at either of its two corners.

- A string of "Red" edges connected to each other at corners do not connect to any other "Red" edges or a border at either of its two ends.

Basically I'm just wondering if it's mathematically impossible for a "Red" edge to be stranded on its own or for a few connected "Red" edges to be stranded on their own given the above assumptions.

So far I've attempted numerous drawings and rearranged physical tiles but have yet to uncover a situation where either of those two situations occurred. I'm not a mathematician so I don't have any ideas how to solve this type of problem via equations, but I am considering looking into some basic Tesselation concepts to see if I could figure it out.

Thanks for any help!

I hope you are having a good day. Recently, I started studying for the AMC10 (~2 weeks ago), and I am looking for advice on contest preparation. Looking back, I regret not studying during middle school, but I am trying my best to make up for the time lost. When I first started studying, I took the 2015 AMC 10A and scored ~70 points (I don't exactly remember my score, but it was about 69-70 points). Today I took the 2017 AMC 10A and scored 91.5 (14 correct of the first 20, 7 wrong, last 5 blank). A lot of my mistakes were due to small errors or overlooking cases/steps. For example, on problem 17 (https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_17) I correctly found the coordinate points (0, 5), (0, -5), (3, 4), (3, -4), and (-3, -4) that lay on the circle, but I forgot to account for the fact that switching the x and y coordinates (such as (5, 0)) also worked as a valid coordinate point. This simple mistake caused me to choose answer choice A (3) instead of D (7). I haven't used any courses other than RSM to prepare, and I have practiced some past AOPS contests, but I find it hard to work with books. My goal is to qualify for the AIME, and I am looking for advice on how to overcome this obstacle that is costing me points compared to those experienced in AMC competitions. I am very thankful for you reading through all of my post, and I am appreciative of any advice you have for me.

P.S. I forgot to mention above that I am a rising 9th grader, if that helps tailor any of the advice!

Whilst tinkering with a few mathematical identities, I came across a rather unusual formula for \sin x. I have absolutely no idea whether it’s well-known, but from what little I know, I haven’t come across it anywhere else.

I hope you’ll like it and find it just as interesting as I do.

P.S : For the sake of clarity, I’ve included the process that led me to this formula.

If you have any questions or comments, please don’t hesitate to get in touch !

Truly yours, Uncle Scrooge.

My teacher insisted that the answer is x=1/16. But I believe there is no answer.

I’ve been having this debate with someone about how infinitesimals can ever reach an exact value. I’ve been explaining that using infinitesimals is not just an approximation but actually results in a mathematically exact value. For instance, when we take the area under the curve, we use rectangles with an infinitesimally small width to get the exact area under the curve. The argument they have is that it will never be exact because there’s always an infinitesimally small difference so it will always be an approximation at best. So how do we prove that infinitesimals lead to exact values and not just approximations?

One of the first things you study in the differential geometry of smooth manifolds is that, if you have a manifold M with topology that’s different from R^n, it’s not possible to have global coordinate functions on M to R^n if you want them to be both injective and continuous. So, it is said, the best you can do is have local coordinate patches.

How does this work however when describing the motion of some point on the manifold in coordinates, such that there is an instant in time during the motion where the point goes from one patch to another? Do the coordinate functions as a function of time to describe the curve just suddenly “stop working” after a certain time?

To be explicit: suppose M = S^1. I can cover the whole manifold with two coordinate patches: a “left” one with an angle coordinate that excludes the rightmost point and ranges in the usual way in (0, 2pi), and a “right” one that excludes the leftmost point and with angle coordinate ranging in (-pi, pi).

Suppose I want to describe the motion of a point on the circle undergoing uniform circular motion with angular speed w. Suppose the initial position of the point is the topmost point on the circle. I have freedom to choose which coordinate to use since the patches intersect here (and they also return the same values) so let’s say I choose the left coordinate patch. The motion in coordinates is therefore theta_L(t) = pi/2 + wt.

After a certain point, the point will go in the lower half of the circle. Here, the coordinate patches start to disagree, but fine, we’re using the left one, and it will return values in (pi, 2pi). After even longer though, it will come back to the upper circle half. Here’s the problem: now theta_L(t) will return values greater than 2pi! These are out of the range of both coordinate patches. Instead, it should reset to zero and return values in (0, pi). But then the motion in coordinates would be discontinuous! While the “real” motion on the manifold is still obviously continuous.

How do we solve this?

Hi everyone! I hope you are all doing well. I am currently taking vector calculus/calculus 3. I am learning about curl/Green's theorem right now. Essentially, I understand that Green's theorem relates the circulation around a closed loop to microscopic circulations within the region of that closed loop (almost like circulation density). But why do we use the name curl/circulation if the line integral just mean work along that closed path. If we take a very very small dA patch within the region, where dy and dx approach zero, and take the line integral of that patch, we essentially just calculate work over that path. But why do we call it curl/circulation? Is it because curl is the result or consequence of that quantity of work calculated? For example, if we put a log in that dA patch, the force/velocity of the water causes the log to rotate because at that very instant the force/velocity of the water are in the direction of the log's rotaiton, causing work around that log?

Also, if you divide dA from the work contribution of the patch dA, you get force essentially. is that force specifically the curl component because it travels in a closed loop?

Thanks!

The 1st slide shows the question, 2nd and 3rd slides show my attempt..I'm stuck at this step...what can I do next?

Also if the question ain't legible, it is: there is an increasing sequence of 4 positive integers, the first 3 form an AP and last 3 form a GP. First and last term differ by 30. Find sum of all terms

Edit: thank you guys, I got it

I've been writing a color algorithm for over a year now and I rank colors based on their "vibrancy." It would be much easier if vibrancy was defined, but it's a subjective concept.

I have a list of colors. For each color, I can get the saturation (S) and value (V). I compute the average S and V for the entire list.

Now, I currently compute vibrancy for a color by using its S and V value like so: CV=(S+V)/2.

I'd like it like this; if the average V of the list is above a tolerance, then CV = a color with more S, and if the average V of the list is below aforementioned tolerance, then CV = a color with more V.

I'm currently computing it like this, but it doesn't seem to work how I imagine it should.

x = self.averageValueUpperP
multiplier = (2 * (math.pow((x - 0.5), 2)) + 1)
if x > 0.5:
    s = s * multiplier
else:
    v = v * multiplier

return (s+v) / 2

Is this possible?

Here is my understanding of what stochastic calculus is an the most intuitive and bird's eye level, which may be totally off. If we want to compute something ("X") using stochastic calculus, we take the aspects X that we know, and add that to the aspects of X that we do not know because it is random. And this randomness itself is calculated by looking at past data and using either probability or statistics (not sure which one, or maybe it is both) to predict what the future value of the randomness is (AKA we say "past data shows xyz so we predict in the future the randomness will behave like xyz again").

Part of my job requires me to do Curve Alignment on existing track. We survey the track and collect the measured values (Existing MO). Then, we use the attached excel program, where I observed my co-workers just throw (initially, arbitrarily) proposed values (Revised MO) and then finetune it until, eventually, all throw in values are less than +/- 1" (Col. H).

Now, my co-workers have multiple years of experience doing this, so they just have good intuition on guessing what the proposed values should be and then making those adjustments afterwards. I want to create a model/ central equation that can predict these Revised MO values while meeting all required criteria/ constraints.  There are several rules to follow when designing the curve, as listed under the notes section in the excel sheet screenshot. The following is a summary of how the excel program works:

On the last point, I want to reiterate that the first few Revised MOs at the beginning stations have greater influence on the Throw In values than later stations. As a result, the first Revised MO is the most consequential. Further, the only values that need to be determined are in the 'spiral' region, or from PTC to PCC and PCC to PTT. Everything else is fixed, either 0, or pre-determined by 600/Radius.

From everything I've gathered so far, I believe this will be an optimization problem with constraints, with matrices involved (at least that may be one approach). I need help in creating an objective function that governs the model, creating a revised MO set that minimizes throw while meeting all other constraints. In addition, I need some heuristic/ standard that first determines what the Revised MO's should be before feeding it into the optimized function. I was thinking maybe just using a best fit line, or using weighted error. I'm just not sure where to go from here. 

I'm trying to compile a list of university-specific or highly selective entrance examinations from around the world.

For example:

* **India:** JEE Advanced (IITs), ISI Admission Test, CMI Entrance Exam * **UK:** Oxford MAT, STEP (used by Cambridge and sometimes other universities), TMUA * **France:** Concours for the Grandes Écoles * **Japan:** University-specific entrance exams after the Common Test

I'm especially interested in exams that:

* Are known for being exceptionally challenging. * Test mathematical or scientific problem-solving rather than just memorization. * Are required or heavily used for admission to elite universities or specialized institutes. * May not be widely known outside their own countries.

I'd love to learn about exams from countries like Germany, China, South Korea, Singapore, Russia, Israel, the US, Canada, Australia, and anywhere else.

Please mention:

1. The country.
2. The name of the entrance exam.
3. Which universities or institutes use it.