Dr. Steve's Challenge Questions for an 800 in math

Test Preparation SAT Preparation
81

<p>Yes… I got to this part(though I’m not sure you might follow my train of thought…):</p>

<p>f(0,2) + f(1,2) + f(0,1) + f(1,1) = 0
f(0,1) + f(1,1) + f(0,0) + f(1,0) = 0
f(1,2) + f(2,2) + f(1,1) + f(2,1) = 0
f(1,1) + f(2,1) + f(1,0) + f(2,0) = 0</p>

<p>Then, I add all four equations.</p>

<p>4f(1,1) + 2f(1,2) + 2f(0,1) + 2f(1,0) + 2f(2,1) + f(0,2) + f(2,2) + f(0,0) + f(1,1) = 0</p>

<p>f(0,2) + f(2,2) + f(0,0) + f(1,1) → This part is 0.</p>

<p>I’m left with 4f(1,1) + 2f(1,2) + 2f(0,1) + 2f(1,0) + 2f(2,1) = 0.</p>

<p>What do I do from here?</p>

82

<p>Silly Jeffery! (0,1), (1,0), (1,2) and (2,1) for a square, don’t they? Silly Jeffery!</p>

<p>Cool question, Drsteve. It reminded me of a question I encountered in my country’s national math Olympiad- only that one was kind of harder. Maybe I’ll post a question from the Olympiad practice tests I have in my drawer. By the way, isn’t this a little too much for SAT prep?</p>

83

<p>Oh oops. I did not see that. I was so close!!! :-)</p>

84

<p>I have found some questions for everyone’s amusement, especially Jeffery’s :p. They aren’t much harder than level five math questions.</p>

<p>1) Assume that the cuboid with dimensions x, y and z has a volume of 500 units. If (1/x)+(1/y)+(1/z)= 2, find the total surface area. </p>

<p>2) Without using a calculator, find 2^0.5 - (3-2^1.5)^0.5. </p>

<p>3) Assume that S(n)= 9+99+999+…+99999 (9 n times). Find S when n equals 15. (This one is definitely not SAT level, but I put it anyway. No calculator, by the way)</p>

<p>Enjoy.</p>

85

<p>I just have a quick question…</p>

<p>3) 1,111,111,111,111,110 - 15 = 1,111,111,111,111,095</p>

<p>Am I doing this incorrectly?</p>

86

<p>It’s actually right, sorry. Try to make a general (n) solution. Here’s a hint I wrote backwards: ecneuqes cirtemoeg.</p>

87

<p>@Cardgames </p>

<ol>
<li>Simplifying (1/x)+(1/y)+(1/z)= 2 yields (xy+xz+yz)/(xyz)=2. Because the volume is lwh=xyz=500, substituting yields xy+xz+yz=1000. The total surface area is 2(lw+lh+wh), so the surface area is 2(xy+xz+yz)=2000.</li>
</ol>

88

<p>@Cardgames</p>

<p>The questions that I’m posting here are not necessary for the average SAT student. BUT if you’re going for an 800 in math, then struggling with problems of this nature will expose you to the type of thinking that will help you get those really tough level 5 questions correct. </p>

<p>Nice questions. I think that something like question 3 could appear on the Level 2 Math Subject Test.</p>

<p>And by the way, nobody has yet solved the last question I asked about the cube inscribed in the cone. I think that there are enough hints in this thread that one of you should be able to solve it (just go back one page to find that question).</p>

89

<p>I agree with you, Drsteve. However, some of these topics are more math level 2 than average SAT. What we need is hard questions that require only very basic math-- the ones that need might you go “eureka”. The selfish set question is honestly more annoying than anything else. I think that the second question I posted is extremely difficult, not because you need to do induction or use combinations, but because discovering the trick behind the question requires cunning.</p>

90

<p>By the way, is the solution to the cone question 0.8284 blah blah? I just did a quick solution, and I was wondering if I got it right.</p>

91

<p>That is a decimal approximation to the answer. What is the exact answer? And how did you get it?</p>

92

<p>Umm, I’m not really good at explaining how I solve things. Moreover, I don’t know math terminology in English because I studied math in Arabic. Also, I didn’t really follow the hints you gave, so my answer is somewhat messy. The only part that the others got wrong is that the base of the cube never overlaps with the base of the cone, because of the shape of the cone. Just use this picture I drew: [View</a> image: Cone problem](<a href=“http://postimage.org/image/i4ntr7ftt/]View”>http://postimage.org/image/i4ntr7ftt/)</p>

<p>The answer is -2 +2^1.5.</p>

<p>You got any number theory questions? Nothing too complicated would be nice- I’m out of practice. </p>

<p>I’ll post questions soon.</p>

93

<p>Ok. Not a bad solution - a bit messier than necessary. You can just use the following ratio: 1/2=(1-x/sqrt(2))/x. (the big triangle is similar to triangle B in your picture)</p>

<p>Cross multiplying gives x=2-2x/sqrt(2), or xsqrt(2)=2sqrt(2)-2x.</p>

<p>So 2x+xsqrt(2)=2sqrt(2), and x=2sqrt(2)/(2+sqrt(2)).</p>

<p>This can be simplified to x=2sqrt(2)-2</p>

<p>I’ll try to think of a nice number theory question in the next couple of days. I don’t have time today.</p>

94

<p>Lol, I use to get that a lot when I was in school. All the math teachers would complain about my iffy solutions, but praise how they get the job done. This is why I love multiple choice questions!</p>

<p>Here are some questions to keep people busy till DrSteve comes up with a new question-courtesy of my hometown’s math Olympiad: (I handpicked the most suitable ones for the SAT. Consider them level six/seven questions)</p>

<p>1) Find all solutions for (x^2 -5x+5)^(x^2 -9x +20)=1. </p>

<p>2) What is the first digit for the following operation 2^14 *3^12? </p>

<p>3) Find x for the following equation: x=(1+(1+(1+…keep doing the same thing…)^0.5)^0.5)^0.5 Every pair of parentheses represents the square root of whatever is inside the parentheses. The thing goes on forever.</p>

<p>4) If x^2 + (1/x^2)=7, find x^5 + (1/x^5). (x>0)</p>

<p>5) You have three distinct numbers (a,b and c) By assembling these three numbers, you can form 6 discrete numbers comprised of two digits. If the sum of the six numbers is 484, find all possibilities for a, b and c.</p>

<p>If people actually like these questions, I’ll look through more of the practice tests I have.</p>

95

<p>5) You have three distinct numbers (a,b and c) By assembling these three numbers, you can form 6 discrete numbers comprised of two digits. If the sum of the six numbers is 484, find all possibilities for a, b and c.</p>

<p>589 and 679</p>

96

<p>Do really need to answer these questions, Xiggi :p? You should at least let the people who are actually preparing for the SAT solve the questions, because they are similar to level 5 SAT questions (I’m also a culprit, by the way). Perhaps I can direct you to more appropriate questions? Here’s something that I hope can stall you for an hour: </p>

<p>Write the general solution for the following:
sin x + sin 2x + sin3x= 0.</p>

<p>Now revise your trigonometric identities!</p>

97

<p>@cardgames</p>

<ol>
<li>1 is the value when the exponent is equal to 0, so x^2-9x+20=0 and x=5 and x=4</li>
</ol>

98

<p>You forgot something, cortana. Think about it a little more.</p>

99

<p>@cortana431
What about when the base is equal to 1?</p>

<p>x^2 - 5x + 5 = 1
x^2 - 5x + 4 = 0
(x-4)(x-1) = 0
x = 4 and x = 1</p>

100

<p>I want to see somebody solve this one. I remember having to think for quite a while to solve this sucker:</p>

<p>2) Without using a calculator, find 2^0.5 - (3-2^1.5)^0.5.</p>