10x = 9.999...

In reality, though, the only thing this equation is telling us is that 9.999... = 9.999..., and by dividing .999... ends up with the equation 1 = 1. I'm not sure if they were just confused or tired, but this doesn't exactly tell us anything at all. This is more of a logic problem than a math problem.

The way I like to think of this is the further the numbers go, the smaller the number between them gets, but it never is exactly 1. The best way to write the difference would be this:

0.00...1

So now we are left with the equation: 0.999... + .00...1 = 1

But that doesn't make sense if .999... = 1. You can't simply say 0.00...1 has absolutely no value, whatsoever, no matter how infinitely small it is.

In other words, you can't make numbers disappear, no matter how unimaginably small they really are.

With this in mind, if they are exactly the same, why don't they both make linear equations? A single number can't make more than one line on a graph. There isn't really much math to it. Just logic.

]]>I'm struggling a bit with your working. The octagon consists of 8 isosceles triangles. If 'a' is the height of one of these then I agree with your answer. So next I worked out the area of 8 of these triangles 8 times 2.5 times a. Where did 40 come from ?

Please re-post your working with some notes at each line to say what you are calculating.

Bob

]]>But I think I can modify the plan to cover this. Draw a plane that intersects the three lines in A, B and C. Then choose D on that line as any point other than C. Then as before.

Bob

]]>A collinear means that there are three or more points on one line so if you think about it all you have to do is put two points on one line and one in the plane not on the line. The reason for this is because a line needs at least two points to be considered a line.

]]>︶ Convex (= convex down = concave up) function

︵ Concave (= convex up = concave down) function

According to the definitions above, we have:

1. All non-vertical lines (y = ax + b) are both concave up and concave down;

2. A vertical line (x = c) is neither concave up nor concave down.

]]>Now consider the following table:

Thus we see that outside of {0,±1} the equation has no solution as the LHS and RHS have opposite signs. This proves that there are no solutions other than *x* = 0, ±1.

]]>Can you help me solve this equation?

Here are the links to the contest questions and answers:

https://cemc.uwaterloo.ca/contests/past_contests/2018/2018Gauss8Contest.pdf (Questions, q. 24-25 near the end)

https://cemc.uwaterloo.ca/contests/past_contests/2018/2018GaussSolution.pdf (Answers, q. 24-25 near the end, the ones in the middle are for Gr. 7 contest)

This is my first time here, so tell me if I'm doing anything wrong!

Thank you.

]]>5, 1, 3, 9, 31, ?

15, 16, 20, ? , 303, 3428

]]>The way I do questions like these is to call the unknown number x, and then build an equation using the information given.

So for Q1, let the number be x. Then the true answer is 7x/8 and the wrong answer is 8x/7. These differ by 15/14. So make an equation out of this and solve for x.

If you post your attempt we can move on to the others.

Bob

]]>where we have used trigonometric substitution and observing that:

and hence

evaluating these integrations and make the substitution:

we would get the result:

This work must equal the elastic potential energy by the conservation of energy. So calculating the potential energies and stored in the AC and BC parts of the string as a result of the work done by the force F, respectively, we would get:

Therefore we have:

where k is a constant. Notice that we have treated the modulus of elasticity as a "constant". However, the particle could have any weight!! which is a contradiction. This leads to the conclusion that the modulus of elasticity depends in fact on the force applied to the string and since the force F is variable the modulus of elasticity is also variable. This should be taken into account in the previous calculation which makes it much more sophisticated. Therefore, the second argument which based on the Hook's law is much more simple than the conservation of energy mothed. I hope that will answer your question.]]>