Shortlisted Problems | Đại học Sư Phạm Hà Nội
Shortlisted Problems | Đại học Sư Phạm Hà Nội với những kiến thức và thông tin bổ ích giúp sinh viên tham khảo, ôn luyện và phục vụ nhu cầu học tập của mình cụ thể là có định hướng, ôn tập, nắm vững kiến thức môn học và làm bài tốt trong những bài kiểm tra, bài tiểu luận, bài tập kết thúc học phần, từ đó học tập tốt và có kết quả cao cũng như có thể vận dụng tốt những kiến thức mình đã học vào thực tiễn cuộc sống
Môn: Khoa học tự nhiên và Công nghệ
Trường: Đại học Sư Phạm Hà Nội
Thông tin:
Tác giả:
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Preview text:
Shortlisted Problems (with solutions)
58th International Mathematical Olympiad
Rio de Janeiro, 12–23 July 2017 150 a1, a2, . . . , an, k M 1 1 1 ` ` ¨ ¨ ¨ ` a “ k a1a2 . . . an “ M. 1 a2 an M ą 1
P pxq “ Mpx ` 1qk ´ px ` a1qpx ` a2q ¨ ¨ ¨ px ` anq q • a ´ b a b • qab a b • a2 ` b2 ´ c2 ´ d2 a, b, c, d q S A S S f A T “ fpSq S f f ˝ g ˝ f ‰ g ˝ f ˝ g g A g ‰ f fpT q “ T a1, a2, . . . an “ ´ maxpai ` ajq n ą 2017 i`j“n M |an| ď M n n ě 3 n px1, x2, . . . , xnq y1, y2, . . . , yn n´1
ÿ yiyi`1 “ y1y2 ` y2y3 ` y3y4 ` ¨ ¨ ¨ ` yn´1yn ě ´1. i“1 K “ Kpnq ÿ xixj ě K 1ďiăjďn n px1, x2, . . . , xnq f : R Ñ R
fpfpxqfpyqq ` fpx ` yq “ fpxyq x, y P R a0, a1, a2, . . . b0, b1, b2, . . . a0 “ 0, a1 “ 1 # bn´1 “ 1 a n “ 1, 2, . . . a nbn ` an´1, anbn ´ an´1, b n`1 “ n´1 ą 1 a2017 a2018 2017 f : R Ñ R x, y P R ` ˘` ˘ fpxq ` y “ fpyq ` x fpxq ` y fpyq ` x ą 0 fpxq ` y ď fpyq ` x x ą y R R n 3n n a b c X Y X Y 3n2{2 9 2j j 2j 2j`1 2n n n N ě 2 NpN ` 1q NpN ´ 1q 2N H0 R0 n n ě 1 Rn´1 Rn Rn´1Rn “ 1 R1n RnR1n ď 1 Hn´1 Hn Hn´1Hn “ 1 109 100 n ą 1 n ˆ n ˆ n n3 n ˆ n ˆ 1 n2 3n n X Y fXpkq k X
X ˚ Y “ X Y tfXpyq: y P Y u. A a ą 0 B b ą 0 A ˚ B “ B ˚ A
A ˚ pA ˚ ¨ ¨ ¨ ˚ pA ˚ pA ˚ Aqq . . .“ qB ˚ pB ˚ ¨ ¨ ¨ ˚ pB ˚ pB ˚ Bqq . . ..q looooooooAooooooo b oooomooooooooolo o o o oooooo B oon ooooo a oooomooooooooooooooooooon n c p2n ` 1q ˆ p2n ` 1q c c N N ABCDE AB “ BC “ CD =EAB “ =BCD =EDC “ =CBA E BC AC BD R S Ω t Ω R R1 R S I RS Ω Γ ISR1 t A Γ t R AI Ω J JR1 Γ O ABC OA ABC B C P Q H P QH ABC ABC ω A D E F ω BC CA AB AEF BC P Q M AD MP Q ω ABCC1B1A1 AB “ BC AA1 BB1 CC1 AC1 A1C D ω ABC ω A1BC1 E ‰ B BB1 DE ω n ě 3 n A B A B A B A ABCD I Ia Ib Ic Id DAB ABC BCD CDA AIbId CIbId X BIaIc DIaIc Y =XIY “ 900 a0, a1, a2, . . . ? #? an a n ě 0 a n, an ` 3, n`1 “ a0 ą 1 a an “ a n p ě 2 i t0, 1, . . . , p´1u ai t0, 1, 2, 3, 4, 5, 6, 7, 8, 9u i P t0, 1, . . . , p ´ 1u p´1
M “ a0 ` 10 ¨ a1 ` ¨ ¨ ¨ ` 10p´1 ¨ ap´1 “ ÿ aj ¨ 10j. j“0 M p n ě 2 a1, a2, . . . , an n 1 ď i ď n
ai, ai ` ai`1, . . . , ai ` ai`1 ` ¨ ¨ ¨ ` ai`n´1 n ai “ ai´n i ą n m t m 10t ´ 1 10k ´ 1 c P t1, 2, 3, . . . , 2017u c ¨ m c ¨ m 1 ď k ă t Spmq m Spmq m “ 1, 2, . . . Spmq pp, qq p ą q pp ` qqp`qpp ´ qqp´q ´ 1 pp ` qqp´qpp ´ qqp`q ´ 1 n n n pa1, a2, . . . , anq 1 1 1 a ` ` ¨ ¨ ¨ ` 1 ` a2 ` ¨ ¨ ¨ ` an a1 a2 an px, yq x y S fpx, yq fpx, yq “ 1 px, yq S n fpx, yq “ a n n´1 n´2 n´1 0x ` a1x
y ` a2x y2 ` ¨ ¨ ¨ ` an´1xy ` anyn. p Zą0 f : Zą0 ˆ Zą0 Ñ t0, 1u • fp1, 1q “ 0 • fpa, bq ` fpb, aq “ 1 pa, bq 1 • fpa ` b, bq “ fpa, bq pa, bq p´1 ÿ a n“1 fpn2, pq ě 2p ´ 2. a1, a2, . . . , an, k M 1 1 1 ` ` ¨ ¨ ¨ ` a “ k a1a2 . . . an “ M. 1 a2 an M ą 1
P pxq “ Mpx ` 1qk ´ px ` a1qpx ` a2q ¨ ¨ ¨ px ` anq x ą 0 aipx ` 1q1{ai ď x ` ai, ai “ 1 ai “ 1 ai ą 1 x ` 1 ai ´ 1 1 ai´1 px ` 1q ` hk 1 k ` k 1 k`kk ¨ k ¨ i¨kk ` kk 1 kkkj ě ai a a i
px ` 1q ¨ 1ai´1 ùñ aipx ` 1q1{ai ď x ` ai. x ` 1 ą 1 ai ą 1 i “ 1, 2, . . . , n n n n ź aipx ` 1q1{ai ď ź řn i“1 1{ai ´
ź px ` aiq ď 0 ðñ P pxq ď 0 i“1 px ` aiq ðñ Mpx ` 1q i“1 i“1 ai “ 1 i P t1, 2, . . . , nu M “ 1 P pxq ă 0 x P R` P ai ě 1 ai ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙ x ai “ ÿ a x j a a x “ 1 ` x. 1 ` i ě i ` i ¨ ai ai j a 0 1 j“0 i ai x ˆ ˙ ě 1 ` a x ai i ¨ “ 1 ` x, 1 ` ai ai ˆ ˙ x x ` ai “ ai 1 ` ě a 1 ` x 1{ai, a ip q i 1{ai ď 1 1 p1 ` xq1{ai ď 1 ` x ` a ¨ x “ i ùñ a 1{a ip1 ` xq i ď x ` ai. a a i i P pxq P pxq ă 0 x ą 0 aj ě 1 j aj ą 1 j a1a2 . . . an “ M ą 1 k “ 1 ` 1 ` ¨ ¨ ¨ ` 1 a ă n xn P pxq ´1 ă 0 1 a2 an xr P pxq k ă r ď n “ degpP q 0 ď r ď k xr P pxq ˆ ˙ ˆ ˙ k ÿ k ÿ ai a ¨ ¨ ¨ ai , 1 i2 n´r M ¨ ´ 1ďi1ăi2㨨¨ăin´rďn a 1ďi1ăi2㨨¨ăin´rďn i a ¨ ¨ ¨ ai “ a ´ r 1 i2 n´r 1a2 ¨ ¨ ¨ an ¨ r 1 ˆ ˙ . k ÿ ď 1ďj aj a ¨ ¨ ¨ a 1ăj2㨨¨ăjr ďn 1 j2 jr r r r “ 0 1 P pxq P p0q “ 0 r “ 1 ř i“1 ai r ą 1 r ă k k “ n 1 ˆ ˙ ˆ ˙ , k k ´ r k k ´ r ÿ “ ¨ ď ¨ 1ďj aj a ¨ ¨ ¨ a 1ăj2㨨¨ăjrďn 1 j2 jr r ` 1 r ` 1 r r ` 1 k ´ r 1 1 ÿ ÿ , ¨ 1ďj aj a ¨ ¨ ¨ a a 1ăj2㨨¨ăjrďn ď 1ďj1㨨¨ăjrăjr`1ďn 1 j2 jr jr`1 r ` 1 aj a ¨ ¨ ¨ a 1 j2 jr 1 1 1 1 ˆ ` `¨¨¨` ˙ . 1a ÿ ÿ 1 a2 ´ r 1ďj aj a ¨¨¨a a 1ăj2㨨¨ăjrďn ďpr ` 1q 1ďj1㨨¨ăjrăjr`1ďn 1 j2 jr jr`1 an aj a ¨¨¨a 1 j2 jr 1 1 r ` 1 aj aj a ¨¨¨a i 1 j2 jr ajr`1 1 aj a ¨¨¨a r ` 1 1 j2 jr ajr`1 1 1 1 ˆ ` ` ¨ ¨ ¨ ` ˙ . 1a ÿ 1 a2 1ďj aj a ¨ ¨ ¨ a 1ăj2㨨¨ăjrďn 1 j2 jr an 1 1 1 r ` ` ¨ ¨ ¨ ` ´ a2 a ¨ ¨ ¨ a a 2 j a ¨ ¨ ¨ a j 2 1 j2 jr 1 j2 jr aj a ¨ ¨ ¨ a j aj a ¨ ¨ ¨ a 1 j2 r 1 j2 jr 1 1 1 “ ˆ ` ` ¨ ¨ ¨ ` ˙ 1 aj a 1 j2 ´ r , aj a ¨ ¨ ¨ a a 1 j2 jr jr ai ě 1 ùñ 1 ď 1 i “ 1, 2, . . . , n ai ai i “ 1, 2, . . . , n ai ě 1 M 1 P pxq ă 0 x ą ´1 x ‰ 0