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B, because you are four standard deviations above the mean score (as opposed to two with A).

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You're correct. All you need to do now is factor it to 4(k^2 + 2k + 8). Since k^2 + 2k + 8 is an integer, the expression is divisible by 4.

Another way to prove it is to use modular arithmetic, or arithmetic dealing with remainders. If m is an odd integer, then m ≡ 1 mod 2 (i.e. m divided by 2 leaves a remainder 1. The ≡ sign means "equivalent"). Then m^2 ≡ 1 (mod) 4, 2m ≡ 2 (mod 4), and 5 ≡ 1 (mod 4). We can add modulos or residues the same way we add numbers; in this case, m^2 + 2m + 5 ≡ 1 + 2 + 1 ≡ 4 ≡ 0 (mod 4). Note that if a number is m mod n, we can subtract any multiple of n and still obtain an equivalent result. Since the expression is 0 mod 4, it is divisible by 4 (since it leaves remainder 0).

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Counterexample: Let a = 2, b = 7, c = 3, d = 13. Here, the conditions are met, but a+c divides b+d (5 divides 20).

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Find the areas of all the faces, and add them up.

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Since angle 1 and angle 2 are equal, name each angle "x." The sum of angle 1 and angle 2 is 2x, which is equal to 180 degrees (since BD is a straight line). Hence 2x = 180, x = 90, so the angles are right angles. This implies that AC and BD are perpendicular.

Use this to write out your proof.

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We have

(if x is not equal to 0). Hence the three possible values of x (roots of unity) are:


, and
.

Hence, replacing each of these values into the expression x^2 + x + 1 and simplifying yields 3, 0, and 0 respectively, so the minimum value is 0.

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They're used in virtually every field of science, as well as economics, business, engineering, systems design, engineering, and so on.

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Proofs are very broad, and are found in an extremely wide range of mathematics. The questions found in the prestigious USAMO and IMO competitions contain lots of good proofs, if you want to check into them.

Here are some websites containing useful information about proofs:


http://www.artofproblemsolving.com/Resources/articles.php?page=howtowrite&

http://www.artofproblemsolving.com/Resources/articles.php?page=gallery

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How many dice are there?

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Replace "x" with "5"

f(5) = 5+6 = 11

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The nationality of your teacher shouldn't affect your ability to learn; you'll have to learn to cope with it. The high school I go to has teachers that are mostly Turkish or Russian.

The equation "y = mx + b" is simply a *general* form of a line. For example, 3x - y = 8 becomes y = 3x - 8 (after some algebraic manipulation), in which m = 3 (i.e. the slope is 3) and the y-intercept is -8 (i.e. the graph crosses the y-axis at (0,-8)).

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That question is asking, for a normal curve with and , what is the probability that the z-score is between 1.22 and 2.15. Use your calculator and the normal cdf (cumulative distribution function) to evaluate it. On a TI calculator, it might look something like

normcdf(1.22, 2.15, 0, 1)

The syntax might be slightly different on other calculators.

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It really depends on the what you are supposed to prove.

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Just prove what the question is asking for (note that "prove" means to justify or demonstrate the truth of a statement -- it does not mean to check only one case and claim it works, except for some existence proofs).

Then once you're familiar with various proof techniques such as induction, proof by contradiction, direct proof, etc., you can move onto more challenging proofs. The annual USAMO competition, for example, is a prestigious contest that has six proof questions to be completed in nine hours, spread over two days (yes, 1 1/2 hrs per question). The Art of Problem Solving website contains all of the past USAMO questions, if you're interested (note that some of the problems can be extremely difficult).

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They always say "practice makes perfect."

If I were you, I would look at some examples of proofs and ask myself, "Why must this always be true?" Then I would try some (or many) on my own. Note that proofs are not found exclusively in geometry courses; they come up in every branch of mathematics. Therefore you could try proving theorems from other areas of mathematics as well.

Above all, don't get scared! I see many students (including myself, sometimes) attempt to prove a question or theorem that they have never seen before, and shudder in fear, having absolutely no idea what to do (as a note, this happened to me when I had to prove that, in triangle ABC, tan B tan C = 3 if and only if the Euler line is parallel to BC).

In this case, you want to really sink your teeth into the problem: read word by word, draw pictures or diagrams, use whatever techniques are necessary. This enabled me to prove the Euler line problem in about half an hour.

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What is the significance of 386.9 miles? And you didn't provide any data to work with.

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Let L be the length of the shorter piece. The longer piece is 2 in longer so we can say the length of that piece is L+2. Hence, L + (L+2) = 72 --> 2L + 2 = 72 --> 2L = 70 --> L = 35. Therefore L+2 = 37, so the lengths of the two pieces are 35 and 37 inches.

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"All the math" is quite a bit of math. Please ask a more specific question.

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Not necessarily.

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Each number after the third number equals the sum of the previous three numbers. The first three are completely arbitrary. Use these facts to find the 7th, 8th and 9th terms.

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Five ninths divided into five parts yields one ninth per part, so (5/9)/5 = 1/9.

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where c is the amount spent.

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Presuming

18 - 2m = 0

We can subtract both sides by 18 (it will almost look like "moving" 18 to the right) to get

-2m = -18

Divide both sides by -2.

m = 9.

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You just asked the same question twice.

It's useful since we can write a difference of squares as a product of two factors.

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The easiest way to find the centroid is to take the average of the x-coordinates and the average of the y-coordinates (this corresponds to a "center of mass" encountered in physics). To find the orthocenter and circumcenter, use the methods that the other tutor suggested.

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No, because I don't know what the three squares are. Since you noted the topic as "Pythagorean theorem" you are probably referring to the squares of the lengths of the sides of a right triangle, in which the sum of the areas of the smaller two squares equals the area of the larger square.

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Area = length*width = (t-3)(t) = t^2 - 3t

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50-p

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Here are some for you to try on your own:

1 (easy). In triangle ABC, the altitude from A onto BC bisects BC. Show that triangle ABC is isosceles.

2 (medium). Prove that the incenter and circumcenter of a triangle are the same point if and only if the triangle is equilateral.

3 (hard). Prove that for any triangle, the circumcenter, orthocenter, and centroid all lie on a line.

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Just use a simple substitution and that is all. By letting u = x^5 + 1 and, du/dx = 5x^4, du = 5x^4 dx,



This is equal to

where C is a constant.