Heinonen J., Kilpelftine T.'s A -superharmonic functions and supersolutions of degenerate PDF

By Heinonen J., Kilpelftine T.

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Answer: A . Let 5 be the set of all those 2007-place decimal integers 2a\a2a 3 ••• a 2006 which contain odd number of digit '9 ' in each sequencea l ' a 2 ' a 3 ' "', a 2006. 1( 102006 - 810(6 ) (C) 10 2006 + 8 2006 ( D ) 102006 - 82006 2 Solution 5 2 Define A as the number of the e lements In S. we have On the ot her hand , and (9 - 1) 2006 = - (2006) (- 1) *92006-*. L:: *_0 k So = ~ (10<006 - 820(6 ). Answe r: B. Part U Short-Answer Questions (Questions 7 to 12. each carries 9 marks. ) . Let f(x) = sin4 x - sin x cosx + cos4 x, the range of f(x) is _ __ Solution As f(x) = sin4 x - sinxcosx + cos4 x MathematicaIOI}'mpiad in China 6 we define t = sin 2x.

3 pieces. That means pieces in squares (i, j) ( 6 ~ i < 7, 6 ~ j :;:;;; 8 ) are untouched. By symmetry, we conclude that the picces in the shadowed areas of the fou r corners of the board as shown in Fig. 1 are untouched while removing 10 pieces satisfying the requirement. Further. it is easy to check tha t in rows 1, 2. 6. 7 and columns 1, 2, 7. 8, at least one piece should be removed from each row and each column , i . e. 8 pieces. Then at most two of pieces named However, CD, @, (j) , ® in Fig.

From (a) - ( d), we proved that M ~ [-2 ' -'4- J. Popularization Committee of Chinese Mathematical Society and Tianjin Mathematical Society were responsible for the assignment of competition problems in the first round and the extra round of the competition. Part I Multiple-Choice Questions (Questions 1 to 6, each carries 6 marks. ) . . Given a right sq uare pyramid P-ABCD with L APC = MathematicaIOI}'mpiad in China 14 60°, as shown in the fi gure. prove p that the cosine of the plane angle of the dihedral angle A-PB-C is ( ).

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A -superharmonic functions and supersolutions of degenerate elliptic equations by Heinonen J., Kilpelftine T.


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