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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
:
r
Pro 2. (Iranian National Olympiad (3rd Round) 2008). Find the
smallest real K such that for each x; y; z 2 R+:
xpy + ypz + zpx K
p
(x + y)(y + z)(z + x)
r
Pro 3. (Iranian National Olympiad (3rd Round) 2008). Let x; y; z 2 R+ and x + y + z = 3. Prove that:
x3
y3 + 8
+
y3
z3 + 8
+
z3
x3 + 8
1
9
+
2
27
(xy + xz + yz)
r
Pro 4. (Iran TST 2008.) Let a; b; c > 0 and ab+ac+bc = 1. Prove that:
pa3 + a + pb3 + b + pc3 + c 2pa + b + c
r
4
Inequalities from 2008 Mathematical Competition ? ? ? ? ?
Pro 5. (Macedonian Mathematical Olympiad 2008.) Positive num-
bers a, b, c are such that (a + b) (b + c) (c + a) = 8. Prove the inequality
a + b + c
3 27
r
a3 + b3 + c3
3
r
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.
r
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 maxf
3(x y)2
4
;
3(y z)2
4
;
3(y z)2
4 g:
r
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
r
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
r
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
Inequalities from 2008 Mathematical Competition ? ? ? ? ?
r
Pro 11. (Moldova 2008 IMO-BMO Second TST Problem 2) Let
a1; : : : ; an be positive reals so that a1 + a2 + : : : + an n
2 . Find the minimal
value of
A =
s
a2
1 +
1
a2
2
+
s
a2
2 +
1
a2
3
+ : : : +
s
a2
n +
1
a2
1
r
Pro 12. (RMO 2008, Grade 8, Problem 3) Let a; b 2 [0; 1]. Prove that
1
1 + a + b 1
a + b
2
+
ab
3
:
r
Pro 13. (Romanian TST 2 2008, Problem 1) Let n 3 be an odd
integer. Determine the maximum value of
p
jx1 x2j +
p
jx2 x3j + : : : +
p
jxn1 xnj +
p
jxn x1j;
where xi are positive real numbers from the interval [0; 1]
r
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
:
r
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.
r
Inequalities from 2008 Mathematical Competition ? ? ? ? ?
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
:
r
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
:
r
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.
r
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.
r
Pro 20. (Greek national mathematical olympiad 2008, P1) For the
positive integers a1; a2; :::; an prove that
Pn
i=1 a2
i Pn
i=1 ai
kn
t
Yn
i=1
ai
where k = max fa1; a2; :::; ang and t = min fa1; a2; :::; ang. When does the
equality hold?
r
non-negative real numbers. Prove that
1
(x y)2 +
1
(y z)2 +
1
(z x)2
4
xy + yz + zx
:
r
Pro 2. (Iranian National Olympiad (3rd Round) 2008). Find the
smallest real K such that for each x; y; z 2 R+:
xpy + ypz + zpx K
p
(x + y)(y + z)(z + x)
r
Pro 3. (Iranian National Olympiad (3rd Round) 2008). Let x; y; z 2 R+ and x + y + z = 3. Prove that:
x3
y3 + 8
+
y3
z3 + 8
+
z3
x3 + 8
1
9
+
2
27
(xy + xz + yz)
r
Pro 4. (Iran TST 2008.) Let a; b; c > 0 and ab+ac+bc = 1. Prove that:
pa3 + a + pb3 + b + pc3 + c 2pa + b + c
r
4
Inequalities from 2008 Mathematical Competition ? ? ? ? ?
Pro 5. (Macedonian Mathematical Olympiad 2008.) Positive num-
bers a, b, c are such that (a + b) (b + c) (c + a) = 8. Prove the inequality
a + b + c
3 27
r
a3 + b3 + c3
3
r
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.
r
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 maxf
3(x y)2
4
;
3(y z)2
4
;
3(y z)2
4 g:
r
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
r
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
r
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
Inequalities from 2008 Mathematical Competition ? ? ? ? ?
r
Pro 11. (Moldova 2008 IMO-BMO Second TST Problem 2) Let
a1; : : : ; an be positive reals so that a1 + a2 + : : : + an n
2 . Find the minimal
value of
A =
s
a2
1 +
1
a2
2
+
s
a2
2 +
1
a2
3
+ : : : +
s
a2
n +
1
a2
1
r
Pro 12. (RMO 2008, Grade 8, Problem 3) Let a; b 2 [0; 1]. Prove that
1
1 + a + b 1
a + b
2
+
ab
3
:
r
Pro 13. (Romanian TST 2 2008, Problem 1) Let n 3 be an odd
integer. Determine the maximum value of
p
jx1 x2j +
p
jx2 x3j + : : : +
p
jxn1 xnj +
p
jxn x1j;
where xi are positive real numbers from the interval [0; 1]
r
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
:
r
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.
r
Inequalities from 2008 Mathematical Competition ? ? ? ? ?
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
:
r
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
:
r
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.
r
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.
r
Pro 20. (Greek national mathematical olympiad 2008, P1) For the
positive integers a1; a2; :::; an prove that
Pn
i=1 a2
i Pn
i=1 ai
kn
t
Yn
i=1
ai
where k = max fa1; a2; :::; ang and t = min fa1; a2; :::; ang. When does the
equality hold?
r
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