Math + Logic

[dw51688]

Please don't post anything irrelevant in here. In this event (first set of questions will be posted this coming Friday after opening ceremony) you will be tested in Logic and mathamatics. Everything will be revolving around mathamatics. The logic will most of the time require math. Multani asked this to write 4 to the second power, write 4^2

For square roots, just write out the words "square root of.

Your background

For logic, just bring your mind!

For math, you should already be familiar with algebra and geometry. From algebra, you should be comfortable with manipulating algebraic expressions and solving equations. From geometry, you should know about similar triangles, the Pythagorean theorem, and a few other things, but not a great deal. You need some trigonometry background and etc.


Some people have PM'ed me about trigonometry. I have prepared a little lesson.

How should you go about trigonometry?

Trigonometry is like other mathematics. Take your time. Write things down. Draw figures.
Work out the exercises. There aren't many, so do them all. There are hints if you need them. There are short answers given, too, so you can check to see that you did it right. But remember, the answers are not the goal of doing the exercises. The reason you're doing the exercises is to learn trigonometry. Knowing how to get the answer is your goal.

When will you ever deal with trig?

Historically, it was developed for astronomy and geography, but scientists have been using it for centuries for other purposes, too. Besides other fields of mathematics, trig is used in physics, engineering, and chemistry. Within mathematics, trig is used in primarily in calculus (which is perhaps its greatest application), linear algebra, and statistics. Since these fields are used throughout the natural and social sciences, trig is a very useful subject to know. And you need it in this competition .

What is trig?

Trig in computational geometry...

Trigonometry began as the computational component of geometry. For instance, one statement of plane geometry states that a triangle is determined by a side and two angles. In other words, given one side of a triangle and two angles in the triangle, then the other two sides and the remaining angle are determined. Trigonometry includes the methods for computing those other two sides. The remaining angle is easy to find since the sum of the three angles equals 180 degrees (usually written 180°). Not to hard right?

Angle measurements and tables...

Angle measurement and tables

If there is anything that distinguishes trigonometry from the rest of geometry, it is that trig depends on angle measurement and quantities determined by the measure of an angle. Of course, all of geometry depends on treating angles as quantities, but in the rest of geometry, angles aren't measured, they're just compared or added or subtracted.
Trigonometric functions such as sine, cosine, and tangent are used in computations in trigonometry. These functions relate measurements of angles to measurements of associated straight lines as described later in this short course.

Trig functions are not easy to compute like polynomials are. So much time goes into computing them in ancient times that tables were made for their values. Even with tables, using trig functions takes time because any use of a trig function involves at least one multiplication or division, and, when several digits are involved, even multiplication and division are slow. The thing a long time ago that made trig long and tedious was the computation. Not a problem anymore. THank you technology and calculators!

The concept of angle is one of the most important concepts in geometry. The concepts of equality, sums, and differences of angles are important and used throughout geometry, but the subject of trigonometry is based on the measurement of angles.

Radians and arc length

An alternate definition of radians is sometimes given as a ratio. Instead of taking the unit circle with center at the vertex of the angle, take any circle with center at the vertex of the angle. Then the radian measure of the angle is the ratio of the length of the subtended arc to the radius of the circle. For instance, if the length of the arc is 3 and the radius of the circle is 2, then the radian measure is 1.5.
The reason that this definition works is that the length of the subtended arc is proportional to the radius of the circle. In particular, the definition in terms of a ratio gives the same figure as that given above using the unit circle. This alternate definition is more useful, however, since you can use it to relate lengths of arcs to angles. The formula for this relation is


radian measure times radius = arc length
For instance, an arc of 0.3 radians in a circle of radius 4 has length 0.3 times 4, that is, 1.2.

Below is a table of common angles in both degree measurement and radian measurement. Note that the radian measurement is given in terms of . It could, of course, be given decimally, but radian measurement often appears with a factor of .

What is a chord?

As used in mathematics, the word chord refers to a straight line drawn between two points on a circle (or more generally, on any curve). The known first trigonometric table was a table of chords. In modern times, the sine is used instead (sines and chords are closely related), but, perhaps, chords are more intuitive.


The relation between sines and chords

A sine is half of a chord. More accurately, the sine of an angle is half the chord of twice the angle.

the sine of an angle in a right triangle equals the opposite side divided by the hypotenuse

Cosine:

The cosine of an angle is defined as the sine of the complementary angle. The complementary angle equals the the given angle subtracted from a right angle, 90°. For instance, if the angle is 30°, then its complement is 60°. Generally, for any angle t,

cos t = sin (90° – t).
Written in terms of radian measurement, this identity becomes


cos t = sin (pi2 – t).

the cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse:


cos = adjacent / hypoteneuse

Also, cos A = sin B = b/c.

The Pythagorean identity for sines and cosines

Everyone knows the pythagorean theorem. I assume...

Recall the Pythagorean theorem for right triangles. It says that

a^2 + b^2 = c^2
where c is the hypotenuse. This translates very easily into a Pythagorean identity for sines and cosines. Divide both sides by c2 and you get


a^2/c^2 + b^2/c^2 = 1.
But a^2/c^2 = (sin A)2, and b^2/c^2 = (cos A)^2. In order to reduce the number of parentheses that have to be written, it is a convention that the notation sin^2 A is an abbreviation for (sin A)^2, and similarly for powers of the other trig functions. Thus, we have proven that


sin2 A + cos^2 A = 1
when A is an acute angle. We haven't yet seen what sines and cosines of other angles should be, but when we do, we'll have for any angle t one of most important trigonometric identities, the Pythagorean identity for sines and cosines:


sin^2 t + cos^2 t = 1.

Sines and cosines for special common angles...

We can easily compute the sines and cosines for certain common angles. Consider first the 45° angle. It is found in an isosceles right triangle, that is, a 45°-45°-90° triangle. In any right triangle c^2 = a^2 + b^2, but in this one a = b, so c2 = 2a2. Hence c = a^2. Therefore, both the sine and cosine of 45° equal 1/ square root of 2 which may also be written: square root of 2 / 2.

Next consider 30° and 60° angles. In a 30°-60°-90° right triangle, the ratios of the sides are 1 : square root of3 : 2. It follows that sin 30° = cos 60° = 1/2, and sin 60° = cos 30° = quare root of 3 / 2.


Tangent in terms of sine and cosine...

Are you still with me or are you

tan A = sin A / cos A


We'll use three relations we already have. First, tan A = sin A / cos A. Second, sin A = a/c. Third, cos A = b/c. Dividing a/c by b/c and cancelling the c's that appear, we conclude that tan A = a/b. That means that the tangent is the opposite side divided by the adjacent side:

tan = opposite / adjacent

One reason tangents are so important is that they give the slopes of straight lines. Angle of elevation.


There is much more to trigonometry but these are the basics... Got it? Well trig really isn't that hard.

 

[Multani]

I swore, I posted on this. Oh well, anyway. I have a few concerns about the mathematics section.

1. We can't draw figures. This is a cripling blow to the geometry questions. As a matter of fact, most figures are rather hard to explain. I recommend going to either Zadok or Ed.

2. Do the questions have to be original. It's harder to make questions, than to answer them. I was thinking of making up some of the questions, and pulling a few from SAT books and other sources.

3. Dw: Should I post the questions, or do we combine our questions? And how many questions per event, are there?

 

[Hawaiian mage]

You can just use the code "code." Observe.

           |  
             ^
           /   \
         /       \
       /           \
      ---------------

Or something like that.


I'm just going to skip this, I fell asleep 4 times trying to read that Overveiw thing before giveing up.

[Edited by Hawaiian mage on 11-04-00 at 09:12 PM]

 

[Multani]

What overview?
I still don't get the code thing, and for the cos definitions, you got the definitions switched.

I might have pm'd you, then again, I don't remember.
Oh while you're at it, you might as well, include secants and other equations and "lessons".

 

[dw51688]

Hey Multani!

Um you can describe things like figure blah has a diameter of blah if blah semicircle... blah... blah

get my drift?

The questions don't have to be original. I am going to get some out of SAT books but alter the numbers slightly.

Sure let's put ours in the same thread. Try to make at least uh 3 questions. The less there are, the harder there should be.

 

[Multani]

With the figures, it's very hard to describe it accurately. It's just very inconvenient. Well, guess I'd better work on my explanation skills.

 

[dw51688]

Or you could open a geocities webpage and post the link so you can upload the photos onto the sight for visual?

 

[Multani]

Nah. Not worth it. Too lazy.
(sigh.)
It was so much easier in Kindergarten...

 

[dw51688]

Sample questions...

1. Point B is on the line segment AC, and point E is on line segment DF. If AB>DE and BC=EF, then

a)AC<DF
b)AC=DF
c)AC>DF
d)DF>AC
e)EF>DF

2. What is the value of (x+y)^2 if x^2+y^2=8 and xy=7?

3. If 10 parts of alcohol are mixed with 14 parts of xylol, what part of the mixture is alcohol?

4. The sum of two integers which are in a ratio of 5:4, is 54. How much larger than the smaller number is the bigger number?


THESE ARE JUST WARMUPS! TO GET YOUR BRAIN WORKING AGAIN. ON A SCALE OF 1-7 THIS IS LIKE A 1! ON EACH QUESTION POSTING, THERE WILL BE SOME 1'S SOME 2'S A MODERATE AMOUNT 3'S A MODERATE AMOUNT 4'S A LOT OF OF 5'S SOME 6'S AND SOME 7'S.

 

[dw51688]

I am also judging English.... so! I am going to make the first questions, combined into short essay question. PM your essays to me.

Choose your topic:

Riemann's Theorem-(10) What is it? Why does it work?

Parseval's Theorem-(10) What is it? Why does it work?

Pythagorean Theorem- What is it, why does it work?


The tens beside the first 2 theorems are the amount of bonus points possible your essay is correct. THEY ARE VERY DIFFICULT, so that is why you could possibly get bonus points. Each essay is based on 10 possible points.

2-Intro
3-Body
2-Conclusion
3-Correct

Total- 10

****Bonus for the first two if correct*****

They only need to be about 6 sentences at shortest.

 

[dw51688]

All essays are to be PMed and labeled Math ESSAY. Thanks,

 

[Multani]

I'm posting my questions on Dw's thread.
PM anwers to me.

1. The length of a rectangle is four more that twice the width. The perimeter is 44 meters. Find the length.

2. Explain and describe the significance of E=MC^2

3. Side AC is 3. Side AB is the hypotenuse, Angle C is 90. Angle B is 30. Find the area of right triangle ABC.

4. The measure of an inscribed angle is equal to one-half the measure of it's inscribed arc. Triangle ABC is inscribed in circle O, and line BD is tangent to circle at point B. If the measure of angle CBD is 70 degrees, what is the measure of angle BAC?

I apoligize for the lack of diagrams.
I will accept answers until Sunday.
Good luck!

 

[dw51688]

NEW QUESTIONS (no essays )

These are somewhat easy...


1. (10 points) Find the sum of all positive integers less than 1000 that are both perfect squares and perfect cubes.

2. (5 points) How many whole numbers between 200 and 999 (excluding 200 and 999) do not contain the digits 3,5, or 7?

3. (5 points) What is the greatest integer that will always divide the product of four consecutive integers?


These are very easy. Have fun!

 

[Multani]

I'm accepting answers until Monday.

1. What is the difference when -x-y is subtracted from -x^2+2y?

2. ST is tangent to circle O at T. RT is a diameter. RT and ST form a right angle. If RS=12, and ST=18, what is the area of the circle?

3.M and N are midpoints of the triangle legs PQ and PR of triangle PQR. There is a segment going from N to R. There is also a segment going from M to Q. S is the point where MQ and NR intersect, and forms triangle MNS. What is the ratio of the area of triangle MNS to that of triangle PQR?

........p......................
...... ....................
... n----------m..................
. ...............
q...................r...........

Each question is worth 5 except number 3 which is worth 10.




[Edited by Multani on 11-18-00 at 08:03 PM]

 

[Multani]

Here are this week's Math questions. Due to Dw's absence, this week's questions will count double. That means all the questions will amount to 40 points, instead of 20.

Fill in the blank.
1.)An angle formed outside a circle by two secants, a secant and a tangent, ot two tangents is equal in degrees to a.)____ the b.)_____________ of it's intercepted c.)____.
1d.) If arc AD=120 degrees and BD= 70 degrees, then angle ACD=___ degrees.
ABC is a secant and with tangent DC, forms a 70 degree angle, out side the circle. AD and BD are intercepted arcs of angle ACD.

2.)What is 10% of 1/3x if 2/3x is 10% of 60?
3.)A recipe for a cake calls for 2.5 cups of milk and 3 cups of flour. With this recipe, a cake was baked using 14 cups of flour. How many cups of milk was required?

Due to multiple holiday delays (Not to mention my incompetence, and general forgetness,) I'm allowing answers until Tuesday.

Good Luck!

 

[dw51688]

Very difficult (maybe?)...

1) Solve for x: log(3x^2+2x-4)=0

2)

Given
F(x,y) -> (-y,x)
G(x,y) -> (1/2x,1/2y)
H(x,y) -> (x+1/2,y+1/2)
I(x,y) -> (-x,y)

Two of these transformations are direct isometries, name them.

3) FInd the preimage of (4,-3) under a translation where (-6,-5) is the image of (-3,1)


1 is a logarithmical problem. 2 and 3 are transformational geometry.

1- 6 points
2- 8 points
3- 6 points

Have fun

 

[dw51688]

I am very sorry, I meant log base 10. I thought you would asume that, but it occured to me that it was rather confusing...

 

[Multani]

Here are my problems...

1.)Square root of 1/16 + 1/9

2.)xy/z - 1/A = 0
What's A in terms of x,y, and z equal to?

3.)234, 256, 273, 281, 218x
If it's assumed that x is greater than 1, which number cannot be the average of the numbers above?

a 218
b 255
c 271
d 281
e 2839

4.) Triangle ABC is a right triangle.
Angle C is 90 degrees. Angle A is 24 degrees. Side AB is 27 and is the hypotenuse. Find the lengths of the other two sides. Round to nearest thousandth.