Inequalities from 2008 Mathematical Contests

Inequalities from 2008 Mathematical Contests Inequality Project
Inequalities from
2008 Mathematical Contests
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Acknowledgments
We thank a lot to Mathlinks Forum and their members for the reference to
problems and many nice solutions from them!
Hanoi, 10 October 2008
Inequalities from 2008 Mathematical Contests Inequality Project
Contributors Of The Book
• Chief editor:Manh Dung Nguyen, High School
for Gifted Students, HUS, Vietnam
• Editor:
• Editor:
Inequalities from 2008 Mathematical Contests Inequality Project
Chapter 1: Problems
Inequalities from 2008 Mathematical Contests Inequality Project
Pro 1. (Vietnamese National Olympiad 2008) Let x, y, z be distinct
non-negative real numbers. Prove that
1
(x− y)2 +
1
(y − z)2 +
1
(z − x)2 ≥
4
xy + yz + zx
.
∇
Pro 2. (Iranian National Olympiad (3rd Round) 2008). Find the
smallest real K such that for each x, y, z ∈ R+:
x
√
y + y
√
z + z
√
x ≤ K
√
(x+ y)(y + z)(z + x)
∇
Pro 3. (Iranian National Olympiad (3rd Round) 2008). Let x, y, z ∈
R+ and x+ y + z = 3. Prove that:
x3
y3 + 8
+
y3
z3 + 8
+
z3
x3 + 8
≥ 1
9
+
2
27
(xy + xz + yz)
∇
Pro 4. (Iran TST 2008.) Let a, b, c > 0 and ab+ac+ bc = 1. Prove that:
√
a3 + a+
√
b3 + b+
√
c3 + c ≥ 2√a+ b+ c
∇
Pro 5. Macedonian Mathematical Olympiad 2008. Positive numbers
a, b, c are such that (a+ b) (b+ c) (c+ a) = 8. Prove the inequality
a+ b+ c
3
≥ 27
√
a3 + b3 + c3
3
∇
Pro 6. (Mongolian TST 2008) Find the maximum number C such that
for any nonnegative x, y, z the inequality
x3 + y3 + z3 + C(xy2 + yz2 + zx2) ≥ (C + 1)(x2y + y2z + z2x).
holds.
∇
Inequalities from 2008 Mathematical Contests Inequality Project
Pro 7. (Federation of Bosnia, 1. Grades 2008.) For arbitrary reals
x, y and z prove the following inequality:
x2 + y2 + z2 − xy − yz − zx ≥ max{3(x− y)
2
4
,
3(y − z)2
4
,
3(y − z)2
4
}.
∇
Pro 8. (Federation of Bosnia, 1. Grades 2008.) If a, b and c are
positive reals such that a2 + b2 + c2 = 1 prove the inequality:
a5 + b5
ab(a+ b)
+
b5 + c5
bc(b+ c)
+
c5 + a5
ca(a+ b)
≥ 3(ab+ bc+ ca)− 2
∇
Pro 9. (Federation of Bosnia, 1. Grades 2008.) If a, b and c are
positive reals prove inequality:
(1 +
4a
b+ c
)(1 +
4b
a+ c
)(1 +
4c
a+ b
) > 25
∇
Pro 10. (Croatian Team Selection Test 2008) Let x, y, z be positive
numbers. Find the minimum value of:
(a)
x2 + y2 + z2
xy + yz
(b)
x2 + y2 + 2z2
xy + yz
∇
Pro 11. (Moldova 2008 IMO-BMO Second TST Problem 2) Let
a1, . . . , an be positive reals so that a1 + a2 + . . .+ an ≤ n2 . Find the minimal
value of
A =
√
a21 +
1
a22
+
√
a22 +
1
a23
+ . . .+
√
a2n +
1
a21
∇
Inequalities from 2008 Mathematical Contests Inequality Project
Pro 12. (RMO 2008, Grade 8, Problem 3) Let a, b ∈ [0, 1]. Prove that
1
1 + a+ b
≤ 1− a+ b
2
+
ab
3
.
∇
Pro 13. (Romanian TST 2 2008, Problem 1) Let n ≥ 3 be an odd
integer. Determine the maximum value of√
|x1 − x2|+
√
|x2 − x3|+ . . .+
√
|xn−1 − xn|+
√
|xn − x1|,
where xi are positive real numbers from the interval [0, 1]
∇
Pro 14. (Romania Junior TST Day 3 Problem 2 2008) Let a, b, c
be positive reals with ab+ bc+ ca = 3. Prove that:
1
1 + a2(b+ c)
+
1
1 + b2(a+ c)
+
1
1 + c2(b+ a)
≤ 1
abc
.
∇
Pro 15. (Romanian Junior TST Day 4 Problem 4 2008) Determine
the maximum possible real value of the number k, such that
(a+ b+ c)
(
1
a+ b
+
1
c+ b
+
1
a+ c
− k
)
≥ k
for all real numbers a, b, c ≥ 0 with a+ b+ c = ab+ bc+ ca.
∇
Pro 16. (Serbian National Olympiad 2008) Let a, b, c be positive real
numbers such that x+ y + z = 1. Prove inequality:
1
yz + x+ 1
x
+
1
xz + y + 1
y
+
1
xy + z + 1
z
≤ 27
31
.
∇
Pro 17. (Canadian Mathematical Olympiad 2008) Let a, b, c be
positive real numbers for which a+ b+ c = 1. Prove that
a− bc
a+ bc
+
b− ca
b+ ca
+
c− ab
c+ ab
≤ 3
2
.
Inequalities from 2008 Mathematical Contests Inequality Project
∇
Pro 18. (German DEMO 2008) Find the smallest constant C such that
for all real x, y
1 + (x+ y)2 ≤ C · (1 + x2) · (1 + y2)
holds.
∇
Pro 19. (Irish Mathematical Olympiad 2008) For positive real num-
bers a, b, c and d such that a2 + b2 + c2 + d2 = 1 prove that
a2b2cd++ab2c2d+ abc2d2 + a2bcd2 + a2bc2d+ ab2cd2 ≤ 3/32,
and determine the cases of equality.
∇
Pro 20. (Greek national mathematical olympiad 2008, P1) For the
positive integers a1, a2, ..., an prove that(∑n
i=1 a
2
i∑n
i=1 ai
) kn
t
≥
n∏
i=1
ai
where k = max {a1, a2, ..., an} and t = min {a1, a2, ..., an}. When does the
equality hold?
∇
Pro 21. (Greek national mathematical olympiad 2008, P2)
If x, y, z are positive real numbers with x, y, z < 2 and x2+ y2+ z2 = 3 prove
that
3
2
<
1 + y2
x+ 2
+
1 + z2
y + 2
+
1 + x2
z + 2
< 3
∇
Pro 22. (Moldova National Olympiad 2008) Positive real numbers
a, b, c satisfy inequality a + b + c ≤ 3
2
. Find the smallest possible value for:
S = abc+ 1
abc
∇
Inequalities from 2008 Mathematical Contests Inequality Project
Pro 23. (British MO 2008) Find the minimum of x2 + y2 + z2 where
x, y, z ∈ R and satisfy x3 + y3 + z3 − 3xyz = 1
∇
Pro 24. (Zhautykov Olympiad, Kazakhstan 2008, Question 6) Let
a, b, c be positive integers for which abc = 1. Prove that∑ 1
b(a+ b)
≥ 3
2
.
∇
Pro 25. (Ukraine National Olympiad 2008, P1) Let x, y and z are
non-negative numbers such that x2 + y2 + z2 = 3. Prove that:
x√
x2 + y + z
+
y√
x+ y2 + z
+
z√
x+ y + z2
≤
√
3
∇
Pro 26. (Ukraine National Olympiad 2008, P2) For positive a, b, c, d
prove that
(a+ b)(b+ c)(c+ d)(d+ a)(1 +
4
√
abcd)4 ≥ 16abcd(1 + a)(1 + b)(1 + c)(1 + d)
∇
Pro 27. (Polish MO 2008, Pro 5) Show that for all nonnegative real
values an inequality occurs:
4(
√
a3b3 +
√
b3c3 +
√
c3a3) ≤ 4c3 + (a+ b)3.
∇
Pro 28. (Chinese TST 2008 P5) For two given positive integers m,n >
1, let aij(i = 1, 2, · · · , n, j = 1, 2, · · · ,m) be nonnegative real numbers, not
all zero, find the maximum and the minimum values of f , where
f =
n
∑n
i=1(
∑m
j=1 aij)
2 +m
∑m
j=1(
∑n
i=1 aij)
2
(
∑n
i=1
∑m
j=1 aij)
2 +mn
∑n
i=1
∑m
i=j a
2
ij
∇
Pro 29. (Chinese TST 2008 P6) Find the maximal constant M , such
that for arbitrary integer n ≥ 3, there exist two sequences of positive real
number a1, a2, · · · , an, and b1, b2, · · · , bn, satisfying
(1):
∑n
k=1 bk = 1, 2bk ≥ bk−1 + bk+1, k = 2, 3, · · · , n− 1;
(2):a2k ≤ 1 +
∑k
i=1 aibi, k = 1, 2, 3, · · · , n, an ≡M .