Ebook chuyên đề bất đẳng thức của diễn đàn bất đẳng thức việt nam năm 2007

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Vietnam Inequality Forum - VIF -

www.batdangthuc.net

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Ebook Written by:

VIF Community

User Group: All

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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 contact www.batdangthuc.net for more details.

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Dien Dan Bat Dang Thuc Viet Nam

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Editors

Dien Dan Bat Dang Thuc Viet Nam

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Bài Viet Nay (cung voi file PDF di kem) duoc tao ra vi muc dich giao duc. Khong duoc su dung ban EBOOK nay duoi bat ky muc dich thuong mai nao, tru khi duoc su dong y cua tac gia. Moi chi tiet xin lien he: www.batdangthuc.net.

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Dien Dan Bat Dang Thuc Viet Nam

www.batdangthuc.net

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Contributors Of The Book

⋆Editor. Pham Kim Hung (hungkhtn)

Admin, VIF Forum, Student, Stanford University

⋆Editor. Nguyen Manh Dung (NguyenDungTN)

Super Mod, VIF Forum, Student, Hanoi National University

⋆Editor. Vu Thanh Van (VanDHKH)

Moderator, VIF Forum, Student, Hue National School

⋆Editor. Duong Duc Lam (dduclam)

Super Moderator, VIF Forum, Student, Civil Engineering University

⋆Editor. Le Thuc Trinh (pi3.14)

Moderator, VIF Forum, Student, High School

⋆Editor. Nguyen Thuc Vu Hoang (zaizai)

Super Moderator, VIF Forum, Student, High School

⋆Editors. And Other VIF members who help us a lot to complete this verion

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Inequalities From 2007 Mathematical Competition Over The World

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Example 1 (Iran National Mathematical Olympiad 2007). Assume that a, b, c are three different positive real numbers. Prove that a + b a −b + b + c

b −c + c + a

c −a

> 1.

Example 2 (Iran National Mathematical Olympiad 2007). Find the largest real T such that for each non-negative real numbers a, b, c, d, e such that a + b = c + d + e, then p

a2 + b2 + c2 + d2 + e2 ≥T(√a + √

b + √c + √

d + √e)2.

Example 3 (Middle European Mathematical Olympiad 2007). Let a, b, c, d be positive real numbers with a + b + c + d = 4. Prove that

a2bc + b2cd + c2da + d2ab ≤4.

Example 4 (Middle European Mathematical Olympiad 2007). Let a, b, c, d be real num-

bers which satisfy 1

2 ≤a, b, c, d ≤2 and abcd = 1. Find the maximum value of

a + 1

b

b + 1

c

c + 1

d

d + 1

a

.

Example 5 (China Northern Mathematical Olympiad 2007). Let a, b, c be side lengths of a triangle and a + b + c = 3. Find the minimum of

a2 + b2 + c2 + 4abc

3 .

Example 6 (China Northern Mathematical Olympiad 2007). Let α, β be acute angles. Find the maximum value of 1 −√tan α tan β 2

cot α + cot β .

Example 7 (China Northern Mathematical Olympiad 2007). Let a, b, c be positive real numbers such that abc = 1. Prove that

ak

a + b + bk

b + c + ck

c + a ≥3

2,

for any positive integer k ≥2.

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Example 8 (Croatia Team Selection Test 2007). Let a, b, c > 0 such that a + b + c = 1. Prove that a2

b + b2

c + c2

a ≥3(a2 + b2 + c2).

Example 9 (Romania Junior Balkan Team Selection Tests 2007). Let a, b, c three pos- itive reals such that

1 a + b + 1 + 1 b + c + 1 + 1 c + a + 1 ≥1.

Show that a + b + c ≥ab + bc + ca.

Example 10 (Romania Junior Balkan Team Selection Tests 2007). Let x, y, z ≥0 be real numbers. Prove that

x3 + y3 + z3

3 ≥xyz + 3

4|(x −y)(y −z)(z −x)|.

Example 11 (Yugoslavia National Olympiad 2007). Let k be a given natural number. Prove that for any positive numbers x, y, z with the sum 1 the following inequality holds

xk+2

xk+1 + yk + zk + yk+2

yk+1 + zk + xk + zk+2

zk+1 + xk + yk ≥1

7.

Example 12 (Cezar Lupu & Tudorel Lupu, Romania TST 2007). For n ∈N, n ≥

2, ai, bi ∈R, 1 ≤i ≤n, such that nP

i=1 a2 i = nP

i=1 b2 i = 1, Pn i=1 aibi = 0. Prove that

n X

i=1 ai

!2

+

n X

i=1 bi

!2

≤n.

Example 13 (Macedonia Team Selection Test 2007). Let a, b, c be positive real numbers. Prove that 1 + 3 ab + bc + ca ≥ 6 a + b + c.

Example 14 (Italian National Olympiad 2007). a) For each n ≥2, find the maximum constant cn such that

1 a1 + 1 + 1 a2 + 1 + . . . + 1 an + …