Tài liệu Molympiad inequalities 2016

MOlympiad – Collection of Mathematical Competitions AoPS – The Art of Problem Solving Community Inequalities From 2016 Mathematical Competition Over The World www.molympiad.blogspot.com Abstract MOLYMPIAD collects all inequalities from 2016 mathematical competition over the world. For the solution to each problem, we refer to a topic on AoPS. This product is created for educational purpose. Please don’t use it for any commecial purpose unless you got the right of the author. Please visit www.molympiad.blogspot.com for more details and for more problem-solving mathematical contests. MOLYMPIAD Collection of Mathematical Competitions from Around the World Website: http://www.molympiad.blogspot.com E-mail: [email protected] 2 www.molympiad.blogspot.com Problem 1 (All Russian Olympiad 2016, grade 9, P8) Let a, b, c, d be positive a + b + c + d = 3. Prove that 1 1 1 1 1 + 2+ 2+ 2 ≤ 2 2 2 2 a2 b c d a b c d Proof https://artofproblemsolving.com/community/c6h1236063p6282109 Problem 2 (All Russian Olympiad 2016, grade 11, P7) Let a, b, c, d be positive real numbers such that a + b + c + d = 3. Prove that 1 1 1 1 1 + 3+ 3+ 3 ≤ 3 3 3 3 3 a b c d a b c d Proof https://artofproblemsolving.com/community/c6h1236063p6282109 Problem 3 (Azerbaijan BMO TST) Let a, b, c be non-negative real numbers. Prove that √ √
√  3 a2 + b2 + c2 ≥ (a + b + c) ab + bc + ca + 2 2 2 + (a − b) + (b − c) + (c − a) 2 ≥ (a + b + c) Proof https://artofproblemsolving.com/community/c6h1323474p7131589 Problem 4 (Azerbaijan Junior Mathematical Olympiad) Let x, y, z be real numbers. Prove that r r r √ 1 1 1 2 2 x + 2 + y + 2 + z2 + 2 ≥ 3 2 y z x Proof https://artofproblemsolving.com/community/c6h1195521p5852654 Problem 5 (Chinese Girls Mathematical Olympiad) Let n is a positive integers, a1 , a2 , · · · , an ∈ {0, 1, · · · , n}. For the integer j (1 ≤ j ≤ n), define bj is the number of elements in the set {i|i ∈ {1, · · · , n}, ai ≥ j}. For example, when n = 3, if a1 = 1, a2 = 2, a3 = 1, then b1 = 3, b2 = 1, b3 = 0. a) Prove that n n X X (i + ai )2 ≥ (i + bi )2 . i=1 i=1 b) Prove that n n X X (i + ai )k ≥ (i + bi )k , i=1 for the integer k ≥ 3. i=1 Inequalities From 2016 Mathematical Competition Over The World 3 Proof https://artofproblemsolving.com/community/c6h1288394p6805932 Problem 6 (China Mathematical Olympiad) Let a1 , a2 , · · · , a31 ; b1 , b2 , · · · , b31 be positive integers such that a1 < a2 < · · · < a31 ≤ 2015, b1 < b2 < · · · < b31 ≤ 2015 and a1 + a2 + · · · + a31 = b1 + b2 + · · · + b31 . Find the maximum value of S = |a1 − b1 | + |a2 − b2 | + · · · + |a31 − b31 |. Proof https://artofproblemsolving.com/community/c6h1174692p5660033 Problem 7 (China Second Round Olympiad) Let a1 , a2 , . . . , a2016 be reals such that 9ai > 11a2i+1 (i = 1, 2, . . . , 2015), find the maximum value of (a1 − a22 )(a2 − a23 ) . . . (a2015 − a22016 )(a2016 − a21 ) Proof https://artofproblemsolving.com/community/c6h1365571p7500146 Problem 8 (China Team Selection Test) Find the smallest  positive number λ, such that for any complex numbers z1 , z2 , z3 ∈ {z ∈ C |z| < 1}, if z1 + z2 + z3 = 0, then 2 2 |z1 z2 + z2 z3 + z3 z1 | + |z1 z2 z3 | < λ. Proof https://artofproblemsolving.com/community/c6h1212046p6011767 Problem 9 (China Team Selection Test) Let n > 1 be an integer, 0 < α < 2, a1 , a2 , a3 , · · · , an ; c1 , c2 , c3 , · · · , cn be positive real numbers, for y > 0,  21  f (y) =  X ai ≤y ci a2i  ! α1 + X ci aα i ai >y Prove that if x be positive real number such that x ≥ f (y), for one positive 1 real number y, then f (x) ≤ 8 α · x. Proof https://artofproblemsolving.com/community/c6h1217114p6068847 4 www.molympiad.blogspot.com Problem 10 (China Western Mathematical Olympiad) k P Let a1 , a2 , . . . , an be non-negative real numbers, Sk = ai with 1 ≤ k ≤ i=1 n. Prove that n X i=1  ai Si n X  a2j  ≤ j=i n X (ai Si ) 2 i=1 Proof https://artofproblemsolving.com/community/c6h1290752p6827152 Problem 11 (Croatia Team Selection Test) Let n ≥ 1 and x1 , . . . , xn ≥ 0. Prove that  (n + 1)2 xn  x2 2 (x1 + 2x2 + . . . + nxn ) ≤ + ... + (x1 + x2 + . . . + xn ) x1 + 2 n 4n https://artofproblemsolving.com/community/c6h1234376p6260511 Problem 12 (EGMO) Let n be an odd positive integer, and let x1 , x2 , · · · , xn be non-negative real numbers. Show that
min x2i + x2i+1 ≤ max (2xj xj+1 ) j=1,...,n i=1,...,n where xn+1 = x1 . Proof https://artofproblemsolving.com/community/c6h1226674p6171538 Problem 13 (EGMO TST Turkey) Prove that
x4 y + y 4 z + z 4 x + xyz x3 + y 3 + z 3 ≥ (x + y + z) (3xyz − 1) for all positive real numbers x, y, z. Proof https://artofproblemsolving.com/community/c6h1248639p6419840 Problem 14 (Germany National Olympiad – 4th Round) Let f (x1 , x2 , x3 , x4 , x5 , x6 , x7 ) =x1 x2 x4 + x2 x3 x5 + x3 x4 x6 + + x4 x5 x7 + x5 x6 x1 + x6 x7 x2 + x7 x1 x3 be defined for non-negative real numbers x1 , x2 , . . . , x7 with sum 1. Prove that f (x1 , x2 , . . . , x7 ) has a maximum value and find that value. Inequalities From 2016 Mathematical Competition Over The World 5 Proof https://artofproblemsolving.com/community/c6h1259588p6531595 Problem 15 (Korea National Olympiad Final Round) If x, y, z satisfies x2 + y 2 + z 2 = 1, find the maximum possible value of


x2 − yz y 2 − zx z 2 − xy Proof https://artofproblemsolving.com/community/c6h1214642p6037409 Problem 16 (Hong Kong Team Selection Test) Let a, b, c be positive real numbers satisfying abc = 1. Determine the smallest possible value of b3 + 8 c3 + 8 a3 + 8 + 3 + 3 + c) b (a + c) c (b + a) a3 (b Proof https://artofproblemsolving.com/community/c6h1155589p5481554 Problem 17 (IMC) Let n be a positive integer. Also let a1 , a2 , . . . , an , b1 , b2 , . . . , bn be real numbers such that ai + bi > 0 for i = 1, 2, . . . , n. Prove that n X n X a i bi − b2 i i=1 ai + bi ≤ i=1 ai · n X bi − i=1 n X n X !2 bi i=1 (ai + bi ) i=1 Proof https://artofproblemsolving.com/community/c7h1279182p6722142 Problem 18 (IMC) Let (x1 , x2 , . . .) be a sequence of positive real numbers satisfying ∞ X xn = 1. 2n −1 n=1 Prove that ∞ X k X xn ≤ 2. k2 n=1 k=1 Proof https://artofproblemsolving.com/community/c7h1279761p6727191 6 www.molympiad.blogspot.com Problem 19 (India International Mathematical Olympiad Training Camp) Let a, b, c, d be real numbers satisfying |a|, |b|, |c|, |d| > 1 and abc + abd + acd + bcd + a + b + c + d = 0. Prove that 1 1 1 1 + + + >0 a−1 b−1 c−1 d−1 Proof https://artofproblemsolving.com/community/c6h1092902p4873866 Problem 20 (India International Mathematical Olympiad Training Camp) Suppose that a sequence a1 , a2 , . . . of positive real numbers satisfies ak+1 ≥ kak a2k + (k − 1) for every positive integer k. Prove that a1 + a2 + . . . + an ≥ n for every n ≥ 2. Proof https://artofproblemsolving.com/community/c6h1268809p6621766 Problem 21 (India Regional Mathematical Olympiad) Let a, b, c be positive real numbers such that a + b + c = 3. Determine, with certainty, the largest possible value of the expression a3 a b c + 3 + 3 2 2 + b + c b + c + a c + a2 + b Problem 22 (India Regional Mathematical Olympiad) Let a, b, c be positive real numbers such that a b c + + = 1. 1+a 1+b 1+c Prove that abc ≤ 18 . Proof https://artofproblemsolving.com/community/c6h1317749p7087195 Problem 23 (India Regional Mathematical Olympiad) Let x, y, z be non-negative real numbers such that xyz = 1. Prove that (x3 + 2y)(y 3 + 2z)(z 3 + 2x) ≥ 27. Proof https://artofproblemsolving.com/community/c6h1317762p7087228 Inequalities From 2016 Mathematical Competition Over The World 7 Problem 24 (India Regional Mathematical Olympiad) Let a, b, c be three distinct positive real numbers such that abc = 1. Prove that b3 c3 a3 + + ≥3 (a − b)(a − c) (b − c)(b − a) (c − a)(c − b) Proof https://artofproblemsolving.com/community/c6h1320099p7110375 Problem 25 (India National Math Olympiad – 3rd Round) Let a, b, c ∈ R+ and abc = 1. Prove that b+c c+a 2 a+b + + ≥ (a + b + 1)2 (b + c + 1)2 (c + a + 1)2 a+b+c Problem 26 (International Zhautykov Olympiad) Let a1 , a2 , ..., a100 are permutation of 1, 2, ..., 100. S1 = a1 , S2 = a1 + a2 , ..., S100 = a1 + a2 + ... + a100 Find the maximum number of perfect squares from Si Proof https://artofproblemsolving.com/community/c6h1185316p5756328 Problem 27 (Iran TST) Let a, b, c, d be positive real numbers such that 1 1 1 1 + + + = 2. a+1 b+1 c+1 d+1 Prove that r r r r a2 + 1 b2 + 1 c2 + 1 d2 + 1 + + + 2 2 2 2√ √ √  √ ≥3 a+ b+ c+ d −8 Proof https://artofproblemsolving.com/community/c6h1272961p6661060 Problem 28 (Iran TST ) Suppose that a sequence a1 , a2 , . . . of positive real numbers satisfies kak ak+1 ≥ 2 ak + (k − 1) for every positive integer k. Prove that a1 + a2 + . . . + an ≥ n for every n ≥ 2. Proof https://artofproblemsolving.com/community/c6h1268809p6621766 8 www.molympiad.blogspot.com Problem 29 (Iran MO – 3rd Round) Let a, b, c ∈ R+ and abc = 1 prove that b+c c+a 2 a+b + + ≥ 2 2 2 (a + b + 1) (b + c + 1) (c + a + 1) a+b+c Proof https://artofproblemsolving.com/community/c6h1299715p6917170 Problem 30 (Israel Team Selection Test) Let a, b, c be positive numbers satisfying ab + bc + ca + 2abc = 1. Prove that 4a + b + c ≥ 2. Proof https://artofproblemsolving.com/community/c6h1313735p7047187 Problem 31 (Japan Mathematical Olympiad Preliminary) Let a, b, c, d be real numbers satisfying the system of equation (a + b)(c + d) = 2(a + c)(b + d) = 3(a + d)(b + c) = 4. Find the minimum value of a2 + b2 + c2 + d2 . Proof https://artofproblemsolving.com/community/c6h1195505p5852511 Problem 32 (Team Selection Test for JBMO - Turkey) Prove that (x4 + y)(y 4 + z)(z 4 + x) ≥ (x + y 2 )(y + z 2 )(z + x2 ) for all positive real numbers x, y, z satisfying xyz ≥ 1. Proof https://artofproblemsolving.com/community/c6h1246265p6393959 Problem 33 (Junior Balkan MO) Let a, b, c be positive real numbers. Prove that 8 8 8 + + + a2 + b2 + c2 (a + b)2 + 4abc (b + c)2 + 4abc (a + c)2 + 4abc 8 8 8 + + ≥ a+3 b+3 c+3 Proof https://artofproblemsolving.com/community/c6h1263180p6565536 Problem 34 (Junior Balkan Team Selection Test) Let a, b, c ∈ R+ , prove that √ p 2a 2b 2c +√ +√ ≤ 3(a + b + c) 3c + a 3a + b 3b + c Inequalities From 2016 Mathematical Competition Over The World 9 Proof https://artofproblemsolving.com/community/c6h1245779p6388664 Problem 35 (Junior Balkan Team Selection Tests - Romania) Let a, b, c > 0 and abc ≥ 1. Prove that 1 1 1 1 + + ≤ a3 + 2b3 + 6 b3 + 2c3 + 6 c3 + 2a3 + 6 3 Proof https://artofproblemsolving.com/community/c6h1257383p6509502 Problem 36 (Junior Balkan Team Selection Tests - Romania) Let a, b, c be real numbers such that a ≥ b ≥ 1 ≥ c ≥ 0 and a + b + c = 3. a) Prove that 2 ≤ ab + bc + ca ≤ 3 b) Prove that 25 24 + ≥ 14 a3 + b3 + c3 ab + bc + ca Proof https://artofproblemsolving.com/community/c6h1319224p7102074 Problem 37 (Junior Balkan Team Selection Tests - Romania) Let m, n are positive intergers and x, y, z positive real numbers such that 0 ≤ x, y, z ≤ 1. Let m + n = p. Prove that: 0 ≤ xp + y p + z p − xm y n − y m z n − z m xn ≤ 1 Proof https://artofproblemsolving.com/community/c6h1233243p6247077 Problem 38 (Korea Winter Program Practice Test) Let x, y, z ≥ 0 be real numbers such that (x + y − 1)2 + (y + z − 1)2 + (z + x − 1)2 = 27. Find the maximum and minimum of x4 + y 4 + z 4 Proof https://artofproblemsolving.com/community/c6h1189501p5795909 Problem 39 (Korea Winter Program Practice Test) Pn PnLet ai , bi (1 ≤ i ≤ n, n ≥ 2) be positive real numbers such that i=1 ai = i=1 bi . Prove that n X i=1 1 (ai+1 + bi+1 )2 Pn ≥ n−1 n(ai − bi )2 + 4(n − 1) j=1 aj bj 10 www.molympiad.blogspot.com Proof https://artofproblemsolving.com/community/c6h1189503p5795912 Problem 40 (Kosovo National Mathematical Olympiad) If a, b, c are sides of right triangle with c hypothenuse then show that for every positive integer n > 2 we have cn > an + bn . Proof https://artofproblemsolving.com/community/c6h1366039p7504772 Problem 41 (Kosovo National Mathematical Olympiad) If α is an acute angle and a, b ≥ 0 then show that:  a+ b sin α  b+ a  ≥ a2 + b2 + 3ab cos α Proof https://artofproblemsolving.com/community/c6h1366048p7504796 Problem 42 (Latvia National Olympiad) Assume that real numbers x, y and z satisfy x + y + z = 3. Prove that xy + xz + yz ≤ 3. Proof https://artofproblemsolving.com/community/c6h1276654p6698742 Problem 43 (Macedonian National Olympiad) Let n ≥ 3 and a1 , a2 , ..., an ∈ R+ , such that 1 1 1 + + ... + = 1. 1 + a41 1 + a42 1 + a4n n Prove that a1 a2 ...an ≥ (n − 1) 4 Proof https://artofproblemsolving.com/community/c6h1225111p6154409 Problem 44 (Mediterranean Mathematics Olympiad) Let a, b, c be positive real numbers with a + b + c = 3. Prove that r r r r b c a 34 1 + + ≤ a2 + 3 b2 + 3 c2 + 3 2 abc Proof https://artofproblemsolving.com/community/c6h1252005p6455069 Inequalities From 2016 Mathematical Competition Over The World 11 Problem 45 (Middle European Mathematical Olympiad) Let n ≥ 2 be an integer, and let x1 , x2 , . . . , xn be reals for which (a) xj > −1 for j = 1, 2, . . . , n and (b) x1 + x2 + . . . + xn = n. Prove that n n X X 1 xj ≥ 1 + xj 1 + x2j j=1 j=1 Proof https://artofproblemsolving.com/community/c6h1295276p6869616 Problem 46 (Pan-African Mathematical Olympiad) Let x, y, z be positive real numbers such that xyz = 1. Prove that (x + 1 1 1 1 + + ≤ 2 2 2 2 2 + y + 1 (y + 1) + z + 1 (z + 1) + x + 1 2 1)2 Proof https://artofproblemsolving.com/community/c6h1235293p6271434 Problem 47 ( Philippines Mathematical Olympiad) Let n be any positive integer. Prove that n X i=1 (i2 1 2 >2− √ 3/4 + i) n+1 Proof https://artofproblemsolving.com/community/c6h1371354p7556839 Problem 48 (Romanian Masters in Mathematic) Let x and y be positive real numbers such that: x + y 2016 ≥ 1. Prove that x2016 + y > 1 − 1 . 100 Proof https://artofproblemsolving.com/community/c6h1204702p5942450 Problem 49 (San Diego Math Olympiad) √ √ √ Let u, v, w be positive real numbers such that u vw + v wu + w uv ≥ 1. Find the smallest value of u + v + w. Proof https://artofproblemsolving.com/community/c6h247578p1358629 12 www.molympiad.blogspot.com Problem 50 (Selection round of Kiev team to UMO) Let a, b, c > 0 such that a + b + c = 3, prove that b2 c2 3 a2 + + ≥ 2 2 2 a+b b+c c+a 2 Proof https://artofproblemsolving.com/community/c6h1200061p5895510 Problem 51 (Selection round of Kiev team to UMO) Let be positive real numbers x, y, z. Prove that y2 z2 (x + y + z)3 x2 + + ≥ xy + z yz + x xz + y 3(x2 (y + 1) + y 2 (z + 1) + z 2 (x + 1) Proof https://artofproblemsolving.com/community/c6h1202171p5914965 Problem 52 (South East Mathematical Olympiad) Let n be positive integer, x1 , x2 , · · · , xn be positive real numbers such that x1 x2 · · · xn = 1 . Prove that n X i=1 xi q n + 1√ x21 + x22 + · · · x2i ≥ n 2 Proof https://artofproblemsolving.com/community/c6h1280874p6737599 Problem 53 (Spain Mathematical Olympiad) Let n ≥ 2 an integer. Find the least value of γ such that for any positive real numbers x1 , x2 , . . ., xn with x1 + x2 + ... + xn = 1 and y1 + y2 + ... + yn = 1 with 0 ≤ y1 , y2 , ..., yn ≤ 12 the following inequality holds x1 x2 ...xn ≤ γ (x1 y1 + x2 y2 + ... + xn yn ) Proof https://artofproblemsolving.com/community/c6h1222243p6118293 Problem 54 ( Taiwan TST– 1st Round) Let a, b, c be nonnegative real numbers such that (a + b)(b + c)(c + a) 6= 0. Find the minimum of   1 1 1 (a + b + c)2016 + + a2016 + b2016 b2016 + c2016 c2016 + a2016 Proof https://artofproblemsolving.com/community/c6h1269064p6624314 Inequalities From 2016 Mathematical Competition Over The World 13 Problem 55 ( Taiwan TST– 2nd Round) Let x, y > 0 such that x + y = 1. Prove that   y x y x + 3 ≤2 + 2 x2 + y 3 x + y2 x + y2 x +y Proof https://artofproblemsolving.com/community/c6h1274240p6673240 Problem 56 ( Taiwan TST– 2nd Round) Suppose that a sequence a1 , a2 , . . . of positive real numbers satisfies ak+1 ≥ kak a2k + (k − 1) for every positive integer k. Prove that a1 + a2 + . . . + an ≥ n for every n ≥ 2. Proof https://artofproblemsolving.com/community/c6h1268809p6621766 Problem 57 (Taiwan TST – 3rd Round) Let x, y, z > 0 such that x + y + z = 1. Find the smallest k such that y2 z2 z 2 x2 x2 y 2 + + ≤ k − 3xyz 1−z 1−x 1−y Proof https://artofproblemsolving.com/community/c6h1276974p6701808 Problem 58 (Turkey EGMO TST) For all x, y, z > 0. Prove that
x4 y + y 4 z + z 4 x + xyz x3 + y 3 + z 3 ≥ (x + y + z)(3xyz − 1) Proof https://artofproblemsolving.com/community/c6h1248639p6419840 Problem 59 (Turkey Team Selection Test) Let a, b, c ≥ 0 such that a2 + b2 + c2 ≤ 3. Prove that
(a + b + c)(a + b + c − abc) ≥ 2 a2 b + b2 c + c2 a Proof https://artofproblemsolving.com/community/c6h1222296p6119503 Problem 60 (Turkmenistan Regional Math Olympia) If a, b, c are triangle sides. Prove that r r r a b c + + ≥3 −a + b + c −b + c + a −c + a + b 14 www.molympiad.blogspot.com Proof https://artofproblemsolving.com/community/c6h1201857p5911968 Problem 61 (VJIMC) Let a, b, c be positive real numbers such that a + b + c = 1. Show that     1 1 1 1 1 1 + + + ≥ 1728 a bc b ca c ab Proof https://artofproblemsolving.com/community/c7h1225703p6160187