Download 101 Problems in Algebra From the Training of the USA IMO by Andreescu T., Feng Z. PDF

By Andreescu T., Feng Z.

ISBN-10: 187642012X

ISBN-13: 9781876420123

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Additional info for 101 Problems in Algebra From the Training of the USA IMO Team (Enrichment Series, Volume 18)

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0 d1 0 c2 0 a (3 Yl ... Y2 cc Y. 0 ... 0 0 (i) Compute the determinant of Problem 230 using the Laplace theorem. (j) Compute the determinant of Problem 171 using the Laplace theorem. PART I. X42 be third-order determinants formed b1 C1 dl b2 b, C2 d2 C3 d3 ) by deleting the first, second, third and fourth columns, respectively. Prove that a1 a2 a3 0 0 0 b1 c1 d1 0 0 b2 c2 d2 0 0 b3 c3 d3 0 0 0 al b1 Ci dl 0 a2 b2 c2 d2 0 a3 b3 C3 d3 AD—BC. (*m) Compute the fifteenth-order determinant A Al Al Al A Al Al Al A formed (as indicated) from the following blocks: a A= x x —x \—x x —x 0 0 0 0 2a a x 2a a a 2a 0 0 0 0 0 0 0 0 2 1 0 0 0 , A1 = 0 1 2 0 0 a 0 0 0 2 1 2a/ \ 0 0 0 1 2, —x\ 1 51 CH 2.

0 X1 Y2 XI Y3 • • • X1 yn 1 0 1 x 1 X1 Y2 X2 Y2 X2 Y3 • • • X2 Yn xi y3 X2 Y3 x3 Ya • • • x3 Yn 0 0 0 ... x Yn x2Yn X3Yn x *223. . 1 + al 1 1 1 1 + a, 1 1 1 1 ... 1 ... 1 + a, . . 1 • • • xnY n 1 1 1 . l+ an PART I. PROBLEMS 40 224. 1 1 1 1 1 . a2+1 1 an _,+ 1 1 an +1 *225. a1 x x x a2 x x x a3 . . *226. x1 a2 a3 a, x2 a3 a1 a2 x3 a1 a2 a3 x . . an *227. )2 al b3 a2 b3 x3 x„_, al b„ a2 bn as bn a1 a2 a3 an-1 *228. X1— in x2 xl xl 1 1 x x x . . x 1 1 . . x (21+1 1 a„ an b1 an b2 an b3 .

Sin (a, + „) sin 2a2 sin (a„+ *295. So sin (an+ cc:2) • • • S2 . . s„ S2 Sn _1 Sn Sn S,,÷1 S,,_i sin 2a„ 1 X Sn+1 • • • S2,, 2 S„+2 • • 52n-1 xn - 1 xn where sk =xlf + 4+ . . + n m 1 *296. a p m —1 p —n b —a —d —c n —p —1 m d —a —b c n —m —1 p b —a d —c c d 1 —m —n —p —a b —n —b —a d —c m1 b m —c —d —a 1 n —p c —b —a 1 —d p n —m sin y cos cp sin ? *297. cos cp cos 2y sin 2? 2 cos 2cp 2 sin 2

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101 Problems in Algebra From the Training of the USA IMO Team (Enrichment Series, Volume 18) by Andreescu T., Feng Z.


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