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View Poll Results: Look at the two parts given below. Q: Math geniuses should solve the first part very easily, but not
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March 7th, 2008, 05:47 PM   #1
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Difference in the subject of Mathematics

Here are two sets of mathematical problems, and see which one you like more:
The first hard part of mathematics (Rule on this part: You may use a pencil and a paper, possibly using geometric protractors etc., but possibly not an calculator.)
A1. ABC is acute-angled. O is its circumcenter. X is the foot of the perpendicular from A to BC. Angle C ≥ angle B + 30deg. Prove that angle A + angle COX < 90deg.
A2. a, b, c are positive reals. Let a' = √(a^2 + 8bc), b' = √(b^2 + 8ca), c' = √(c^2 + 8ab). Prove that a/a' + b/b' + c/c' ≥ 1.
A3. Integers are placed in each of the 441 cells of a 21 x 21 array. Each row and each column has at most 6 different integers in it. Prove that some integer is in at least 3 rows and at least 3 columns.
B1. Let n_1, n_2, ... , n_m be integers where m is odd. Let x = (x_1, ... , x_m) denote a permutation of the integers 1, 2, ... , m. Let f(x) = x_1n_1 + x_2n_2 + ... + x_mn_m. Show that for some distinct permutations a, b the difference f(a) - f(b) is a multiple of m!.
B2. ABC is a triangle. X lies on BC and AX bisects angle A. Y lies on CA and BY bisects angle B. Angle A is 60deg. AB + BX = AY + YB. Find all possible values for angle B.
B3. K > L > M > N are positive integers such that KM + LN = (K + L - M + N)(-K + L + M + N). Prove that KL + MN is composite.
The second hard part of Mathematics (Rule on this part: Solve the following stuff in your head. Don't use paper and pencil, or calculators.)
A1. Simplify (a + 2b + 3c - 6c)^5 - d^3/dx^3[sin(x^2 + x + 1)].
A2. Find the 10th derivative of y = cos(sin(2x^2)).
A3. Simplify (x - y + z)^6 + d^9/dx^9[cos(sec(x^2))].
B1. What is the area of the circle with the radius of r = x^5 + x^4 + 3x^3 + x + 24, where x > 0. Express the area in terms of pi and x variable.
B2. Find the area under the curve of y = x^2 from x = k + k^2 + k^3 to x = k^5 + 6k^9 + k^15 + 8k^27, where k > 0. Express the area under the curve in terms of k.
B3. What's the 20th derivative of sin(cos(x))?
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March 7th, 2008, 05:49 PM   #2
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I think it's true, in my opinion.
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