Document Sample

```					www.batdangthuc.net                                                                  5

Inequalities From 2007 Mathematical
Competition Over The World

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

a+b b+c c+a
+   +    > 1.
a−b b−c c−a
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
√       √     √     √     √
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 + d2 ab ≤ 4.

Example 4 (Middle European Mathematical Olympiad 2007). Let a, b, c, d be real num-
1
bers which satisfy ≤ a, b, c, d ≤ 2 and abcd = 1. Find the maximum value of
2
1          1         1             1
a+           b+       c+             d+       .
b          c         d             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
4abc
a2 + b2 + c2 +        .
3
Example 6 (China Northern Mathematical Olympiad 2007). Let α, β be acute angles.
Find the maximum value of
√            2
1 − tan α tan β
.
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   bk   ck  3
+    +    ≥ ,
a+b b+c c+a    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 b2 c2
+     +    ≥ 3(a2 + b2 + c2).
b     c   a
Example 9 (Romania Junior Balkan Team Selection Tests 2007). Let a, b, c three pos-
itive reals such that
1     1     1
+     +      ≥ 1.
a+b+1 b+c+1 c+a+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 + y 3 + z 3        3
≥ xyz + |(x − y)(y − z)(z − x)|.
3                4
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             yk+2             z k+2       1
+ k+1            + k+1            ≥ .
xk+1 + yk + z k  y    + z k + xk  z    + xk + y k  7
Example 12 (Cezar Lupu & Tudorel Lupu, Romania TST 2007). For n ∈ N, n ≥
n             n
n
2, ai, bi ∈ R, 1 ≤ i ≤ n, such that            a2 =
i             b2 = 1,
i            i=1   aibi = 0. Prove that
i=1            i=1

n          2             n        2

ai        +             bi       ≤ n.
i=1                     i=1

Example 13 (Macedonia Team Selection Test 2007). Let a, b, c be positive real numbers.
Prove that
3             6
1+               ≥           .
ab + bc + ca    a+b+c
Example 14 (Italian National Olympiad 2007). a) For each n ≥ 2, find the maximum
constant cn such that
1      1           1
+       +...+        ≥ cn,
a1 + 1 a2 + 1       an + 1
for all positive reals a1 , a2, . . ., an such that a1 a2 · · · an = 1.
b) For each n ≥ 2, find the maximum constant dn such that
1       1              1
+        + ...+         ≥ dn
2a1 + 1 2a2 + 1        2an + 1
for all positive reals a1 , a2, . . ., an such that a1 a2 · · · an = 1.
www.batdangthuc.net                                                                           7

Example 15 (France Team Selection Test 2007). Let a, b, c, d be positive reals such taht
a + b + c + d = 1. Prove that
1
6(a3 + b3 + c3 + d3) ≥ a2 + b2 + c2 + d2 + .
8
Example 16 (Irish National Mathematical Olympiad 2007). Suppose a, b and c are
positive real numbers. Prove that

a+b+c               a2 + b2 + c2   1        ab bc ca
≤                          ≤            +   +         .
3                      3         3        c   a   b

For each of the inequalities, find conditions on a, b and c such that equality holds.

Example 17 (Vietnam Team Selection Test 2007). Given a triangle ABC. Find the
minimum of
cos2 A cos2 B
2      2   cos2 B cos2 C
2      2   cos2 C cos2 A
2      2
+               +               .
cos2 C2         cos2 A2         cos2 B
2

Example 18 (Greece National Olympiad 2007). Let a,b,c be sides of a triangle, show
that
(c + a − b)4   (a + b − c)4 (b + c − a)4
+             +             ≥ ab + bc + ca.
a(a + b − c) b(b + c − a) c(c + a − b)
Example 19 (Bulgaria Team Selection Tests 2007). Let n ≥ 2 is positive integer. Find
the best constant C(n) such that
n
√
xi ≥ C(n)            (2xi xj +    xi xj )
i=1               1≤j<i≤n

1
is true for all real numbers xi ∈ (0, 1), i = 1, ..., n for which (1 − xi)(1 − xj ) ≥ , 1 ≤
4
j < i ≤ n.

Example 20 (Poland Second Round 2007). Let a, b, c, d be positive real numbers satisfying
the following condition:
1 1 1 1
+ + + = 4.
a b      c d
Prove that:

3   a3 + b3     3   b3 + c3     3   c3 + d3      3   d3 + a3
+               +               +                ≤ 2(a + b + c + d) − 4.
2               2               2                2
Example 21 (Turkey Team Selection Tests 2007). Let a, b, c be positive reals such that
their sum is 1. Prove that
1               1               1              1
2 + 2c
+        2 + 2a
+        2 + 2b
≥              .
ab + 2c         bc + 2a         ac + 2b         ab + bc + ac
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Example 22 (Moldova National Mathematical Olympiad 2007). Real numbers
1
a1 , a2, . . . , an satisfy ai ≥ , for all i = 1, n. Prove the inequality
i
1                         1          2n
(a1 + 1) a2 +                      · · · · · an +       ≥            (1 + a1 + 2a2 + · · · + nan ).
2                         n       (n + 1)!
Example 23 (Moldova Team Selection Test 2007). Let a1, a2, . . . , an ∈ [0, 1]. Denote
S = a3 + a3 + . . . + a3 , prove that
1    2            n

a1                a2                        an         1
3 + 2n + 1 + S − a3 + . . . + 2n + 1 + S − a3 ≤ 3 .
2n + 1 + S − a1                 2                         n

Example 24 (Peru Team Selection Test 2007). Let a, b, c be positive real numbers, such
that
1 1 1
a+b+c≥ + + .
a b       c
Prove that
3          2
a+b+c ≥                +      .
a + b + c abc
Example 25 (Peru Team Selection Test 2007). Let a, b and c be sides of a triangle. Prove
that           √                √                   √
b+c−a            c+a−b               a+b−c
√    √    √   +√       √     √ +√          √     √ ≤ 3.
b+ c− a          c+ a− b             a+ b− c
Example 26 (Romania Team Selection Tests 2007). If a1, a2, . . . , an ≥ 0 satisfy a2 +
1
· · · + a2 = 1, find the maximum value of the product (1 − a1) · · · (1 − an ).
n

Example 27 (Romania Team Selection Tests 2007). Prove that for n, p integers, n ≥ 4
and p ≥ 4, the proposition P(n, p)
n              n                                                                     n
1                  p
≥         xi         for xi ∈ R,        xi > 0,    i = 1, . . ., n ,           xi = n,
i=1
xi p     i=1                                                                  i=1

is false.
Example 28 (Ukraine Mathematical Festival 2007). Let a, b, c be positive real numbers
and abc ≥ 1. Prove that
(a).
1           1             1         27
a+        b+            c+           ≥     .
a+1        b+1           c+1          8
(b).

27(a3 +a2 +a+1)(b3 +b2 +b+1)(c3 +c2 +c+1) ≥≥ 64(a2 +a+1)(b2 +b+1)(c2 +c+1).

Example 29 (Asian Pacific Mathematical Olympiad 2007). Let x, y and z be positive
√   √    √
real numbers such that x + y + z = 1. Prove that
x2 + yz                   y2 + zx             z 2 + xy
+                          +                   ≥ 1.
2x2(y + z)                2y2 (z + x)         2z 2(x + y)
www.batdangthuc.net                                                                    9

Example 30 (Brazilian Olympiad Revenge 2007). Let a, b, c ∈ R with abc = 1. Prove
that
1 1 1               1 1 1                          b  c a c c b
a2 +b2 +c2+     + + +2 a + b + c + + +                  ≥ 6+2      + + + + +                .
a2 b2 c2            a b  c                         a b c  a b c

Example 31 (India National Mathematical Olympiad 2007). If x, y, z are positive real
numbers, prove that

(x + y + z)2 (yz + zx + xy)2 ≤ 3(y2 + yz + z 2 )(z 2 + zx + x2)(x2 + xy + y2 ).

Example 32 (British National Mathematical Olympiad 2007). Show that for all positive
reals a, b, c,

(a2 + b2)2 ≥ (a + b + c)(a + b − c)(b + c − a)(c + a − b).

Example 33 (Korean National Mathematical Olympiad 2007). For all positive reals
a, b, and c, what is the value of positive constant k satisfies the following inequality?
a      b      c      1
+      +       ≥      .
c + kb a + kc b + ka   2007
Example 34 (Hungary-Isarel National Mathematical Olympiad 2007). Let a, b, c, d be
real numbers, such that

a2 ≤ 1, a2 + b2 ≤ 5, a2 + b2 + c2 ≤ 14, a2 + b2 + c2 + d2 ≤ 30.

Prove that a + b + c + d ≤ 10.
Chapter 1

Problems

Pro 1. (Vietnamese National Olympiad 2008) Let x, y, z be distinct
non-negative real numbers. Prove that
1           1           1            4
2
+         2
+        2
≥              .
(x − y)     (y − z)     (z − x)    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    z3  1  2
3+8
+ 3   + 3   ≥ + (xy + xz + yz)
y      z +8 x +8   9 27

Pro 4. (Iran TST 2008.) Let a, b, c > 0 and ab + ac + bc = 1. Prove that:
√         √         √           √
a3 + a + b 3 + b + c 3 + c ≥ 2 a + b + c
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         27   a3 + b 3 + c 3
≥
3                     3

Pro 6. (Mongolian TST 2008) Find the maximum number C such that
for any nonnegative x, y, z the inequality
x3 + y 3 + z 3 + C(xy 2 + yz 2 + zx2 ) ≥ (C + 1)(x2 y + y 2 z + z 2 x).
holds.

Pro 7. (Federation of Bosnia, 1. Grades 2008.) For arbitrary reals
x, y and z prove the following inequality:
3(x − y)2 3(y − z)2 3(y − z)2
x2 + y 2 + z 2 − xy − yz − zx ≥ max{                  ,         ,          }.
4         4         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 + b 5   b5 + c 5   c 5 + a5
+          +          ≥ 3(ab + bc + ca) − 2
ab(a + b) bc(b + c) ca(a + b)

Pro 9. (Federation of Bosnia, 1. Grades 2008.) If a, b and c are
positive reals prove inequality:
4a        4b        4c
(1 +       )(1 +     )(1 +     ) > 25
b+c       a+c       a+b

Pro 10. (Croatian Team Selection Test 2008) Let x, y, z be positive
numbers. Find the minimum value of:
x2 + y 2 + z 2
(a)
xy + yz
x2 + y 2 + 2z 2
(b)
xy + yz
Inequalities from 2008 Mathematical Competition

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
1            1                                  1
A = a2 + 2 + a2 + 2 + . . . + a2 +
1             2                   n
a2           a3                                 a2
1

Pro 12. (RMO 2008, Grade 8, Problem 3) Let a, b ∈ [0, 1]. Prove that

1       a + b ab
≤1−      + .
1+a+b       2   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                       1                 1
+                       +                       ≤       .
1+   a2 (b   + c)       1+   b2 (a   + c)       1+   c2 (b   + a)       abc

Pro 15. (Romanian Junior TST Day 4 Problem 4 2008) Determine
the maximum possible real value of the number k, such that

1   1   1
(a + b + c)           +   +    −k                              ≥k
a+b c+b a+c

for all real numbers a, b, c ≥ 0 with a + b + c = ab + bc + ca.
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                  1                   1              27
1   +              1    +              1   ≤      .
yz + x +   x
xz + y +   y
xy + z +   z
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 b − ca c − ab  3
+      +       ≤ .
a + bc b + ca c + ab  2

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 + y 2 )
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

a2 b2 cd + +ab2 c2 d + abc2 d2 + a2 bcd2 + a2 bc2 d + ab2 cd2 ≤ 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
kn        n
n    2
i=1 ai
t

n              ≥         ai
i=1 ai             i=1

where k = max {a1 , a2 , ..., an } and t = min {a1 , a2 , ..., an }. When does the
equality hold?
Inequalities from 2008 Mathematical Competition

Pro 21. (Greek national mathematical olympiad 2008, P2)
If x, y, z are positive real numbers with x, y, z < 2 and x2 + y 2 + z 2 = 3 prove
that
3     1 + y 2 1 + z 2 1 + x2
<          +       +         <3
2     x+2      y+2       z+2

Pro 22. (Moldova National Olympiad 2008) Positive real numbers
a, b, c satisfy inequality a + b + c ≤ 3 . Find the smallest possible value for:
2

1
S = abc +
abc

Pro 23. (British MO 2008) Find the minimum of x2 + y 2 + z 2 where
x, y, z ∈ R and satisfy x3 + y 3 + z 3 − 3xyz = 1

Pro 24. (Zhautykov Olympiad, Kazakhstan 2008, Question 6) Let
a, b, c be positive integers for which abc = 1. Prove that
1      3
≥ .
b(a + b)  2

Pro 25. (Ukraine National Olympiad 2008, P1) Let x, y and z are
non-negative numbers such that x2 + y 2 + z 2 = 3. Prove that:
x                 y                 z             √
+                +                  ≤    3
x2 + y + z       x + y2 + z         x + y + z2

Pro 26. (Ukraine National Olympiad 2008, P2) For positive a, b, c, d
prove that
√
4
(a + b)(b + c)(c + d)(d + a)(1 + abcd)4 ≥ 16abcd(1 + a)(1 + b)(1 + c)(1 + d)
Inequalities from 2008 Mathematical Competition

Pro 27. (Polish MO 2008, Pro 5) Show that for all nonnegative real
values an inequality occurs:
√        √     √
4( a3 b3 + b3 c3 + c3 a3 ) ≤ 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
n       m         2          m       n        2
n      i=1 (   j=1 aij )     +m     j=1 (   i=1 aij )
f=         n      m        2            n       m    2
(      i=1    j=1 aij )     + mn    i=1     i=j aij

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 bk = 1, 2bk ≥ bk−1 + bk+1 , k = 2, 3, · · · , n − 1;
k=1
(2):a2 ≤ 1 + k ai bi , k = 1, 2, 3, · · · , n, an ≡ M .
k             i=1
Weird Inequalities
Black Lecture, June 29

1. (IMO 2003/05) Let x1 ≤ x2 ≤ · · · ≤ xn be real numbers. Prove that
n                                               n
2       2 2
|xi − xj |            ≤     (n − 1)       (xi − xj )2
i,j=1
3         i,j=1

and determine when equality occurs.
2. (IMO 1999/2) Let n ≥ 2 be a ﬁxed integer. Find the smallest constant C such that for all non-negative
reals x1 , x2 , . . . , xn ,
n         4
xi xj (x2 + x2 ) ≤ C
i    j                            xi       .
1≤i<j≤n                                         i=1

3. (Russia 2004) Let n > 3 be an integer and let x1 , x2 , . . . , xn be positive reals with product 1. Prove
that
1                  1                         1
+                  +···+                     > 1.
1 + x 1 + x 1 x2   1 + x 2 + x 2 x3          1 + x n + x n x1
4. (Romania 2004) Let n ≥ 2 be an integer and let a1 , a2 , . . . , an be real numbers. Prove that for any
non-empty subset S ⊂ {1, 2, 3, . . . , n}, we have
2
ai       ≤                (ai + · · · + aj )2 .
i∈S                1≤i≤j≤n

5. (Romania 1996) Let x1 , . . . , xn+1 be positive real numbers such that x1 + · · · + xn = xn+1 . Prove that

n                                         n
xi (xn+1 − xi ) ≤                     xn+1 (xn+1 − xi ).
i=1                                       i=1

6. (IMO 2001 short list) Let x1 , x2 , . . . , xn be arbitrary real numbers. Prove the inequality
x1         x2                     xn            √
+              +··· +                     < n.
1 + x2
1   1 + x 2 + x2
1    2        1 + x2 + · · · + x2
1            n

n                     n   1
7. Let x1 , x2 , . . . , xn be positive real numbers with                   i=1    xi =           i=1 xi .      Prove that
n
1
≤ 1.
i=1
n − 1 + xi

8. (IMO 2004 short list) Let n be a positive integer and let (x1 , . . . , xn ) and (y1 , . . . , yn ) be two sequences of
2
positive real numbers. Suppose (z2 , . . . , z2n ) is a sequence of positive real numbers such that zi+j ≥ xi yj
for all 1 ≤ i, j ≤ n. Let M = max{z2 , . . . , z2n }. Prove that

M + z2 + · · · + z2n           2           x1 + · · · + x n           y1 + · · · + y n
≥                                                   .
2n                                        n                          n
7. Let n > 1 be a positive integer and a1 , a2 , . . . , an positive reals such that a1 a2 . . . an = 1.
Show that
1                1            a1 + · · · + an + n
+ ··· +               ≤
1 + a1          1 + an                     4
8. (Aaron Pixton) Let a, b, c be positive reals with product 1. Show that
a b c
5+     + + ≥ (1 + a)(1 + b)(1 + c)
b c a

9. (Valentin Vornicu13 ) Let a, b, c, x, y, z be arbitrary reals such that a ≥ b ≥ c and either
x ≥ y ≥ z or x ≤ y ≤ z. Let f : R → R+ be either monotonic or convex, and let k be
0
a positive integer. Prove that

f (x)(a − b)k (a − c)k + f (y)(b − c)k (b − a)k + f (z)(c − a)k (c − b)k ≥ 0

10. (IMO 01/2) Let a, b, c be positive reals. Prove that
a           b           c
√            +√          +√          ≥1
a2 + 8bc    b2 + 8ca    c2 + 8ab

11. (Vasile Cirtoaje) Show that for positive reals a, b, c,
a3                         b3                       c3                1
2 + b2 )(2a2 + c2 )
+     2 + c2 )(2b2 + a2 )
+     2 + a2 )(2c2 + b2 )
≤
(2a                      (2b                       (2c                       a+b+c

12. (USAMO 04/5) Let a, b, c be positive reals. Prove that

a5 − a2 + 3      b5 − b2 + 3   c5 − c2 + 3 ≥ (a + b + c)3

13. (Titu Andreescu) Show that for all nonzero reals a, b, c,
a2 b2 c2   a c b
2
+ 2+ 2 ≥ + +
b   c  a   c b a

14. (Darij Grinberg) Show that for positive reals a, b, c,
b2 + c2 − a2 c2 + a2 − b2 a2 + b2 − c2   3
+            +             ≥
a(b + c)     b(c + a)     c(a + b)     2

15. (IMO 96 Shortlist) Let a, b, c be positive reals with abc = 1. Show that
ab          bc            ca
+ 5          + 5          ≤1
a5   +b5 + ab  b +c 5 + bc  c + a5 + ca
13
This improvement is more widely known than the other one in this packet, and is published in his book,
Olimpiada de Matematica... de la provocare la experienta, GIL Publishing House, Zalau, Romania. (In
English, “The Math Olympiad... from challenge to experience.”)

41
16. Let a, b, c be positive reals such that a + b + c = 1. Prove that
√           √          √              √      √   √
ab + c + bc + a + ca + b ≥ 1 + ab + bc + ca

17. (IMO 00/2) Positive reals a, b, c have product 1. Prove that

1                  1                   1
a−1+                 b−1+                     c−1+            ≤1
b                  c                   a

18. (APMO 2005/2) Let a, b, c be positive reals with abc = 8. Prove that

a2                                b2                               c2                  4
+                                   +                              ≥
(a3   +   1) (b3   + 1)           (b3   +   1) (c3   + 1)          (c3   +   1) (a3   + 1)       3

19. Show that for all positive reals a, b, c,

a3          b3          c3
+ 2         + 2          ≥a+b+c
b2 − bc + c2 c − ca + a2 a − ab + b2

20. (USAMO 97/5) Prove that for all positive reals a, b, c,
1              1            1          1
+ 3           + 3          ≤
a3   +b 3 + abc  b +c 3 + abc  c +a3 + abc   abc

21. (Moldova 1999) Show that for all positive reals a, b, c,

ab       bc       ca       a   b   c
+        +         ≥    +   +
c(c + a) a(a + b) b(b + c)   c+a a+b b+c

1
22. (Tuymaada 2000) Prove that for all reals 0 < x1 , . . . , xn ≤ 2 ,
n       n
n                                        1
−1                   ≤            −1
x1 + · · · + xn                          i=1
xi

23. (Mathlinks Lore) Show that for all positive reals a, b, c, d with abcd = 1, and k ≥ 2,
1          1          1          1
k
+        k
+        k
+          ≥ 22−k
(1 + a)    (1 + b)    (1 + c)    (1 + d)k

24. (Tiks) Show that for all reals a, b, c > 0,

a2               b2               c2          1
+                +                 ≤
2a + b)(2a + c) (2b + c)(2b + a) (2c + a)(2c + b)   3

42
25. (Hyun Soo Kim) Let a, b, c be positive reals with product not less than one. Prove that
1                              1                           1
+                            +                          ≤1
a+   b2005   +   c2005        b+   c2005   +   a2005       c+   a2005   + b2005

26. (IMO 05/3) Prove that for all positive a, b, c with product at least 1,

a5 − a2       b5 − b2     c5 − c2
+ 5         + 5          ≥0
a5 + b2 + c2 b + c2 + a2 c + a2 + b2

27. (Mildorf) Let a, b, c, k be positive reals. Determine a simple, necessary and suﬃcient
condition for the following inequality to hold:

(a + b + c)k ak bk + bk ck + ck ak ≤ (ab + bc + ca)k (ak + bk + ck )

28. Let a, b, c be reals with a + b + c = 1 and a, b, c ≥ − 3 . Prove that
4

a     b     c    9
+ 2   + 2   ≤
a2   +1 b +1 c +1    10

29. (Mildorf) Show that for all positive reals a, b, c,
√
3
√
3
√
3                      4a2   4b2   4c2
4a3 + 4b3 +           4b3 + 4c3 +         4c3 + 4a3 ≤            +     +
a+b b+c c+a

30. Let a, b, c, x, y, z be real numbers such that

(a + b + c)(x + y + z) = 3,                    (a2 + b2 + c2 )(x2 + y 2 + z 2 ) = 4

Prove that
ax + by + cz ≥ 0

31. (Po-Ru Loh) Let a, b, c be reals with a, b, c > 1 such that
1     1     1
+ 2   + 2   =1
a2   −1 b −1 c −1
Prove that
1   1   1
+   +    ≤1
a+1 b+1 c+1
32. (Weighao Wu) Prove that
(sin x)sin x < (cos x)cos x
for all real numbers 0 < x < π .
4

43
33. (Michael Rozenberg) Show that for all positive reals a, b, c,
a2   b2   c2  3 a3 + b3 + c3
+    +    ≥ · 2
b+c c+a a+b    2 a + b2 + c2

34. (Hungktn) Prove that for all positive reals a, b, c,
a2 + b2 + c2         8abc
+                     ≥2
ab + bc + ca (a + b)(b + c)(c + a)

35. (Mock IMO 05/2) Let a, b, c be positive reals. Show that
√
a          b          c       3 2
1< √          +√         +√         ≤
a 2 + b2    b2 + c2    c2 + a2    2

36. (Gabriel Dospinescu) Let n ≥ 2 be a positive integer. Show that for all positive reals
a1 , a2 , . . . , an with a1 a2 . . . an = 1,

a2 + 1
1                    a2 + 1
n
+ ··· +               ≤ a1 + · · · + an
2                     2
n−1
37. Let n ≥ 2 be a positive integer, and let k ≥               n
be a real number. Show that for all
positive reals a1 , a2 , . . . , an ,
k                               k                                  k
(n − 1)a1                    (n − 1)a2                             (n − 1)an
+                                + ··· +                           ≥n
a2 + · · · + an            a3 + · · · + an + a1                   a1 + · · · + an−1

38. Show that for reals x, y, z which are not all positive,
16 2
x −x+1              y2 − y + 1       z 2 − z + 1 ≥ (xyz)2 − xyz + 1
9

39. (Mildorf) Let a, b, c be arbitrary reals such that a ≥ b ≥ c, and let x, y, z be nonnegative
reals with x + z ≥ y. Prove that

x2 (a − b)(a − c) + y 2 (b − c)(b − a) + z 2 (c − a)(c − b) ≥ 0

and determine where equality holds.
40. (IMO 06/3) Determine the least real number M such that for all reals a, b, c,
2
a3 b + b3 c + c3 a − a3 c − b3 a − c3 b ≤ M · a2 + b2 + c2

41. (Kiran Kedlaya) Show that for all nonnegative a1 , a2 , . . . , an ,
√               √
a1 + a1 a2 + · · · + n a1 · · · an          a1 + a2        a1 + · · · + an
≤ n a1 ·           ···
n                                 2                 n

44
42. (Vasile Cirtoaje) Prove that for all positive reals a, b, c such that a + b + c = 3,
a      b      c     3
+      +       ≥
ab + 1 bc + 1 ca + 1   2

43. (Gabriel Dospinescu) Prove that ∀a, b, c, x, y, z ∈ R+ | xy + yz + zx = 3,
a(y + z) b(z + x) c(x + y)
+        +         ≥3
b+c      c+a      a+b

44. (Mildorf) Let a, b, c be non-negative reals. Show that for all real k,
max(ak , bk )(a − b)2                                            min(ak , bk )(a − b)2
≥             ak (a − b)(a − c) ≥
cyc
2                   cyc                          cyc
2

(where a, b, c = 0 if k ≤ 0) and determine where equality holds for k > 0, k = 0, and
k < 0 respectively.
45. (Vasile Cirtoaje) Let a, b, c, k be positive reals. Prove that
ab + (k − 3)bc + ca bc + (k − 3)ca + ab ca + (k − 3)ab + bc   3(k − 1)
2 + kbc
+          2 + kca
+          2 + kab
≥
(b − c)             (c − a)             (a − b)               k

46. (Darij Grinberg and Vascile Cirtoaje) Show that for positive reals a, b, c, d,
1       1      1      1       2
+ 2    + 2    + 2     ≥√
a2   + ab b + bc c + cd d + da    abcd

47. (Vasile Cirtoaje; inspired by the next problem) Show the for all positive reals a, b, c,
3a2 + ab 3b2 + bc 3c2 + ca
+          +          ≥3
(a + b)2   (b + c)2   (c + a)2

48. (Vasile Cirtoaje; inspired by the next problem) Show that for all positive reals a, b, c,
3a2 − 2ab − b2 3b2 − 2bc − c2 3c2 − 2ca − a2
+              +               ≥0
a2 + b2        b2 + c2        c2 + a2

49. (Mildorf) Show that for all positive reals a, b, c,
3a2 − 2ab − b2  3b2 − 2bc − c2  3c2 − 2ca − a2
+ 2             + 2             ≥0
3a2 + 2ab + 3b2 3b + 2bc + 3c2 3c + 2ca + 3a2

50. (Vasile Cirtoaje) Show that for real numbers a, b, c,
2
2 2                           4          2 2                 3
4            a b − abc           a         a −         ab    ≥3           a b − abc         a
cyc                 cyc       cyc         cyc                cyc               cyc

45
Chapter 1

Warm-up problem set

1.1     Applications
1. Let a, b, c, d be real numbers such that a2 + b2 + c2 + d2 = 4. Prove that

a3 + b3 + c3 + d3 ≤ 8.

2. If a, b, c are non-negative numbers, then
3
b+c
a3 + b3 + c3 − 3abc ≥ 2          −a          .
2
3. Let a, b, c be positive numbers such that abc = 1. Prove that

a+b+c         5    a2 + b2 + c2
≥                         .
3                     3
4. Let a, b, c be non-negative numbers such that a3 + b3 + c3 = 3. Prove that

a4 b4 + b4 c4 + c4 a4 ≤ 3.

ırtoaje, GM-A, 1, 2003)
(Vasile Cˆ

5. If a, b, c are non-negative numbers, then

a2 + b2 + c2 + 2abc + 1 ≥ 2(ab + bc + ca).

(Darij Grinberg, MS, 2004)

6. If a, b, c are distinct real numbers, then
a2       b2       c2
+        +         ≥ 2.
(b − c)2 (c − a)2 (a − b)2

5
6                                                        1. Warm-up problem set

7. If a, b, c are non-negative numbers, then
√                 √                 √
(a2 − bc) b + c + (b2 − ca) c + a + (c2 − ab) a + b ≥ 0.

8. If a, b, c, d are non-negative real numbers, then

a−b         b−c        c−d       d−a
+          +          +           ≥ 0.
a + 2b + c b + 2c + d c + 2d + a d + 2a + b

9. Let a, b, c be non-negative numbers such that

a2 + b2 + c2 = a + b + c.

Prove that
a2 b2 + b2 c2 + c2 a2 ≤ ab + bc + ca.

ırtoaje, MS, 2006)
(Vasile Cˆ

10. Let a, b, c be non-negative numbers, no two of them are zero. Then,

a2          b2          c2
+ 2         + 2          ≥ 1.
a2 + ab + b2 b + bc + c2 c + ca + a2

11. If a, b, c are non-negative numbers, then

a3                  b3                  c3
+                  +                   ≥ 1.
a3 + (b + c)3      b3 + (c + a)3       c3 + (a + b)3

12. Let a, b, c be positive numbers and let

E(a, b, c) = a(a − b)(a − c) + b(b − c)(b − a) + c(c − a)(c − b).

Prove that:
a) (a + b + c)E(a, b, c) ≥ ab(a − b)2 + bc(b − c)2 + ca(c − a)2 ;
1 1 1
b) 2    + +      E(a, b, c) ≥ (a − b)2 + (b − c)2 + (c − a)2 .
a b c
ırtoaje, MS, 2005)
(Vasile Cˆ

13. Let a, b, c and x, y, z be real numbers such that a + x ≥ b + y ≥ c + z ≥ 0
and a + b + c = x + y + z. Prove that

ay + bx ≥ ac + xz.
1.1. Applications                                                                        7

1
14. Let a, b, c ∈      , 3 . Prove that
3
a   b   c   7
+   +    ≥ .
a+b b+c c+a  5
15. Let a, b, c and x, y, z be non-negative numbers such that

a + b + c = x + y + z.

Prove that

ax(a + x) + by(b + y) + cz(c + z) ≥ 3(abc + xyz).

ırtoaje, MS, 2005)
(Vasile Cˆ

16. If a, b, c are non-negative numbers, then

4(a + b + c)3 ≥ 27(ab2 + bc2 + ca2 + abc).

17. Let a, b, c be non-negative numbers such that a + b + c = 3. Prove that
1               1               1
+               +               ≥ 1.
2ab2    +1       2bc2   +1       2ca2   +1
18. If a, b, c, d are positive numbers, then
1       1      1      1        4
+ 2    + 2    + 2     ≥         .
a2    + ab b + bc c + cd d + da   ac + bd

1 √
19. If a, b, c ∈ √ , 2 , then
2
3       3      3      2   2   2
+      +       ≥    +   +    .
a + 2b b + 2c c + 2a   a+b b+c c+a
20. Let a, b, c be non-negative numbers such that ab + bc + ca = 3. Prove
that
1        1       1
2+2
+ 2     + 2      ≤ 1.
a        b +2 c +2
21. Let a, b, c be non-negative real numbers such that ab+bc+ca = 3. Prove
that
1          1       1      3
2+1
+ 2      + 2      ≥ .
a         b +1 c +1         2
ırtoaje, MS, 2005)
(Vasile Cˆ
8                                                            1. Warm-up problem set

22. Let a, b, c be non-negative numbers such that a2 + b2 + c2 = 3. Prove
that
a       b       c
+     +        ≤ 1.
b+2 c+2 a+2
ırtoaje, MS, 2005)
(Vasile Cˆ

23. Let a, b, c be positive numbers such that abc = 1. Prove that
a−1 b−1 c−1
a)        +       +        ≥ 0;
b         c       a
a−1 b−1 c−1
b)        +       +        ≥ 0.
b+c c+a a+b

24. Let a, b, c, d be non-negative numbers such that a2 −ab+b2 = c2 −cd+d2 .
Prove that
(a + b)(c + d) ≥ 2(ab + cd).

25. Let a1 , a2 , . . . , an be positive numbers such that a1 a2 . . . an = 1. Prove
that
1                   1                   1
+              + ··· +                    ≥ 1.
1 + (n − 1)a1 1 + (n − 1)a2                 1 + (n − 1)an
ırtoaje, GM-B, 10, 1991)
(Vasile Cˆ

26. Let a, b, c, d be non-negative real numbers such that a2 +b2 +c2 +d2 = 1.
Prove that
(1 − a)(1 − b)(1 − c)(1 − d) ≥ abcd.

ırtoaje, GM-B, 9-10, 2001)
(Vasile Cˆ

27. If a, b, c are positive real numbers, then

2a            2b        2c
+             +         ≤ 3.
a+b           b+c       c+a

ırtoaje, GM-B, 7-8, 1992)
(Vasile Cˆ

28. If a, b, c, d are positive real numbers, then

2             2              2              2
a              b             c              d
+             +              +              ≥ 1.
a+b            b+c           c+d            d+a

ırtoaje, GM-B, 6, 1995)
(Vasile Cˆ
1.1. Applications                                                                   9

1 1 1
29. Let a, b, c be positive numbers such that a + b + c = + + . If
a b c
a ≤ b ≤ c, then
ab2 c3 ≥ 1.

ırtoaje, GM-B, 11, 1998)
(Vasile Cˆ

30. Let a, b, c be non-negative numbers, no two of them are zero. Then

a2       b2       c2      a   b   c
2 + c2
+ 2    2
+ 2    2
≥    +   +    .
b        c +a     a +b      b+c c+a a+b
ırtoaje, GM-B, 10, 2002)
(Vasile Cˆ

31. If a, b, c are non-negative numbers, then

2(a2 + 1)(b2 + 1)(c2 + 1) ≥ (a + 1)(b + 1)(c + 1)(abc + 1).

ırtoaje, GM-A, 2, 2001)
(Vasile Cˆ

32. If a, b, c are non-negative numbers, then

3(1 − a + a2 )(1 − b + b2 )(1 − c + c2 ) ≥ 1 + abc + a2 b2 c2 .

ırtoaje, Mircea Lascu, RMT, 1-2, 1989)
(Vasile Cˆ

33. If a, b, c, d are non-negative numbers, then
2
1 + abcd
(1 − a + a2 )(1 − b + b2 )(1 − c + c2 )(1 − d + d2 ) ≥                    .
2

ırtoaje, GM-B, 1, 1992)
(Vasile Cˆ

34. If a, b, c are non-negative numbers, then

(a2 + ab + b2 )(b2 + bc + c2 )(c2 + ca + a2 ) ≥ (ab + bc + ca)3 .

ırtoaje, Mircea Lascu, ONI, 1995)
(Vasile Cˆ

35. Let a, b, c, d be positive numbers such that abcd = 1. Prove that

1                 1                1               1
+                +                +                 ≤ 1.
1 + ab + bc + ca 1 + bc + cd + db 1 + cd + da + ac 1 + da + ab + bd
10                                                         1. Warm-up problem set

36. If a, b, c and x, y, z are real numbers, then

4(a2 + x2 )(b2 + y 2 )(c2 + z 2 ) ≥ 3(bcx + cay + abz)2 .

ırtoaje, MS, 2004)
(Vasile Cˆ

37. If a ≥ b ≥ c ≥ d ≥ e, then

(a + b + c + d + e)2 ≥ 8(ac + bd + ce).

For e ≥ 0, determine when equality occurs.
ırtoaje, MS, 2005)
(Vasile Cˆ

38. If a, b, c, d are real numbers, then

6(a2 + b2 + c2 + d2 ) + (a + b + c + d)2 ≥ 12(ab + bc + cd).

ırtoaje, MS, 2005)
(Vasile Cˆ

39. If a, b, c are positive numbers, then

1 1 1                                          1    1  1
(a + b + c)    + +   ≥1+            1+     (a2 + b2 + c2 )    2
+ 2+ 2 .
a b  c                                         a   b  c

ırtoaje, GM-B, 11, 2002)
(Vasile Cˆ

40. If a, b, c, d are positive numbers, then

1    1   1                   1 1 1
5+   2(a2 + b2 + c2 )    2
+ 2 + 2 − 2 ≥ (a + b + c)  + +   .
a   b   c                    a b  c
ırtoaje, GM-B, 5, 2004)
(Vasile Cˆ

41. If a, b, c, d are positive numbers, then

a−b b−c   c−d d−a
+    +    +    ≥ 0.
b+c c+d d+a a+b
42. If a, b, c > −1, then

1 + a2      1 + b2      1 + c2
2
+         2
+         ≥ 2.
1+b+c       1+c+a       1+a+b
¸
(Laurentiu Panaitopol, Junior BMO, 2003)
1.1. Applications                                                               11

43. Let a, b, c and x, y, z be positive real numbers such that

(a + b + c)(x + y + z) = (a2 + b2 + c2 )(x2 + y 2 + z 2 ) = 4.

Prove that
1
abcxyz <      .
36
ırtoaje, Mircea Lascu, ONI, 1996)
(Vasile Cˆ

44. Let a, b, c be positive numbers such that a2 + b2 + c2 = 3. Prove that

a2 + b2 b2 + c2 c2 + a2
+       +        ≥ 3.
a+b     b+c     c+a
(Cezar Lupu, MS, 2005)

45. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
1        1        1           3
+        +         ≥              .
a2 + bc b2 + ca c2 + ab     ab + bc + ca
ırtoaje, MS, 2005)
(Vasile Cˆ

46. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
1             1           1               3
+ 2          + 2           ≥              .
b2   − bc + c2  c − ca + a2  a − ab + b 2   ab + bc + ca
47. Let a, b, c be positive numbers such that a + b + c = 3. Prove that
12
abc +                ≥ 5.
ab + bc + ca

48. Let a, b, c be non-negative numbers such that a2 + b2 + c2 = 3. Prove
that
12 + 9abc ≥ 7(ab + bc + ca).

ırtoaje, MS, 2005)
(Vasile Cˆ

49. Let a, b, c be non-negative numbers such that ab + bc + ca = 3. Prove
that
a3 + b3 + c3 + 7abc ≥ 10.

ırtoaje, MS, 2005)
(Vasile Cˆ
12                                                        1. Warm-up problem set

50. If a, b, c are positive numbers such that abc = 1, then

(a + b)(b + c)(c + a) + 7 ≥ 5(a + b + c).

ırtoaje, MS, 2005)
(Vasile Cˆ

51. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
a3                      b3                    c3               1
2 +b2 )(2a2 +c2 )
+    2 +c2 )(2b2 +a2 )
+    2 +a2 )(2c2 +b2 )
≤       .
(2a                    (2b                    (2c                    a+b+c
ırtoaje, MS, 2005)
(Vasile Cˆ

52. Let a, b, c be non-negative numbers such that a + b + c ≥ 3. Prove that
1     1          1
+       +            ≤ 1.
a2   +b+c a+b2+c   a + b + c2
53. Let a, b, c be non-negative numbers such that ab + bc + ca = 3. If r ≥ 1,
then
1            1            1           3
2 + b2
+     2 + c2
+     2 + a2
≤        .
r+a           r+b          r+c           r+2
(Pham Van Thuan, MS, 2005)

54. Let a, b, c be positive numbers such that abc = 1. Prove that
1        1        1               5
+        +        +                      ≥ 1.
(1 + a)3 (1 + b)3 (1 + c)3 (1 + a)(1 + b)(1 + c)
(Pham Kim Hung, MS, 2006)

55. Let a, b, c be positive numbers such that abc = 1. Prove that
2   1       3
+ ≥              .
a+b+c 3  ab + bc + ca
56. If a, b, c are real numbers, then

2(1 + abc) +     2(1 + a2 )(1 + b2 )(1 + c2 ) ≥ (1 + a)(1 + b)(1 + c).

(Wolfgang Berndt, MS, 2006)

57. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
a(b + c) b(c + a) c(a + b)
+ 2       + 2       ≥ 2.
a2 + bc    b + ca    c + ab
(Pham Kim Hung, MS, 2006)
1.1. Applications                                                                     13

58. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
a(b + c)    b(c + a)    c(a + b)
2 + bc
+    2 + ca
+           ≥ 2.
a            b           c2 + ab
ırtoaje, MS, 2006)
(Vasile Cˆ

59. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
1      1       1       a         b         c
+       +       ≥ 2      + 2       + 2       .
b+c c+a a+b             a + bc b + ca c + ab
60. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
1        1      1        2a         2b        2c
+      +        ≥ 2       + 2       + 2        .
b+c c+a a+b               3a + bc 3b + ca 3c + ab
ırtoaje, MS, 2005)
(Vasile Cˆ

61. Let a, b, c be positive numbers such that a2 + b2 + c2 = 3. Prove that

3
5(a + b + c) +        ≥ 18.
abc
ırtoaje, MS, 2005)
(Vasile Cˆ

62. Let a, b, c be non-negative numbers such that a + b + c = 3. Prove that

1      1      1     3
+      +       ≤ .
6 − ab 6 − bc 6 − ca  5
63. Let n ≥ 4 and let a1 , a2 , . . . , an be real numbers such that

a1 + a2 + · · · + an ≥ n and a2 + a2 + · · · + a2 ≥ n2 .
1    2            n

Prove that
max{a1 , a2 , . . . , an } ≥ 2.
(Titu Andreescu, USAMO, 1999)

64. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
a       b       c    13 2(ab + bc + ca)
+       +      ≥     −                   .
b+c c+a a+b            6    3(a2 + b2 + c2 )
ırtoaje, MS, 2006)
(Vasile Cˆ
14                                                     1. Warm-up problem set

65. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
a2 (b + c) b2 (c + a) c2 (a + b)
+ 2        + 2         ≥ a + b + c.
b2 + c2    c + a2     a + b2
(Darij Grinberg, MS, 2004)

66. Let a, b, c be non-negative numbers such that

(a + b)(b + c)(c + a) = 2.

Prove that
(a2 + bc)(b2 + ca)(c2 + ab) ≤ 1.
ırtoaje, MS, 2005)
(Vasile Cˆ
Chapter 8

Final problem set

8.1    Applications
19. Let a, b, c be positive numbers such that abc = 1. Prove that

a+b        b+c       c+a
+          +         ≥ 3.
b+1        c+1       a+1

ırtoaje, MC, 2005)
(Vasile Cˆ

20. Let a, b, c be positive numbers such that abc = 1. Prove that

a         b          c   3
+         +          ≥ .
b+3       c+3        a+3  2

ırtoaje, MS, 2005)
(Vasile Cˆ

21. Let a, b, c be non-negative numbers such that a + b + c = 3. Prove that

5 − 3bc 5 − 3ca 5 − 3ab
+       +        ≥ ab + bc + ca.
1+a     1+b     1+c

ırtoaje, MS, 2005)
(Vasile Cˆ

22. Let a, b, c, d be non-negative numbers such that a2 + b2 + c2 + d2 = 4.
Prove that
(abc)3 + (bcd)3 + (cda)3 + (dab)3 ≤ 4.

ırtoaje, MS, 2004)
(Vasile Cˆ

371
372                                                                 8. Final problem set

23. Let a, b, c be non-negative numbers, no two of which are zero. Then,

a               b             c
+               +             ≤ 1.
4a + 5b         4b + 5c       4c + 5a
ırtoaje, GM-A, 1, 2004)
(Vasile Cˆ

24. Let a1 , a2 , . . . , an be positive numbers. Prove that

(a1 + a2 + · · · + an )2       (n − 1)n−1
(a)
(a2   + 1)(a22 + 1) . . . (a2 + 1) ≤    nn−2
;
1                         n
1
a1 + a2 + · · · + an         (2n − 1)n− 2
(b)              2 + 1)(a2 + 1) . . . (a2 + 1) ≤
(a1                                 2n nn−1
.
2              n

ırtoaje, GM-B, 6, 1994)
(Vasile Cˆ

25. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be real numbers. Prove that
2
a1 b1 +· · ·+an bn + (a2 + · · · + a2 )(b2 + · · · + b2 ) ≥ (a1 +· · ·+an )(b1 +· · ·+bn ).
1            n    1            n
n
ırtoaje, Kvant, 11, 1989)
(Vasile Cˆ

26. Let k and n be positive integers with k < n, and let a1 , a2 , . . . , an be
real numbers such that a1 ≤ a2 ≤ · · · ≤ an . Prove that

(a1 + a2 + · · · + an )2 ≥ n(a1 ak+1 + a2 ak+2 + · · · + an ak )

in the following cases:
(a) for n = 2k;
(b) for n = 4k.
ırtoaje, CM, 5, 2005)
(Vasile Cˆ

27. Let a, b, c, d be positive numbers such that abcd = 1. Prove that
1              1              1                 1
2 + a3
+       2 + b3
+       2 + c3
+                 ≥ 1.
1+a+a          1+b+b          1+c+c          1 + d + d2 + d3
ırtoaje, GM-B, 11, 1999)
(Vasile Cˆ

28. If a, b, c are non-negative numbers, then

9(a4 + 1)(b4 + 1)(c4 + 1) ≥ 8(a2 b2 c2 + abc + 1)2 .

ırtoaje, GM-B, 3, 2004)
(Vasile Cˆ
8.1. Applications                                                               373

29. If a, b, c, d are non-negative numbers, then
(1 + a3 )(1 + b3 )(1 + c3 )(1 + d3 )   1 + abcd
2 )(1 + b2 )(1 + c2 )(1 + d2 )
≥          .
(1 + a                                    2
ırtoaje, GM-B, 10, 2002)
(Vasile Cˆ

30. Let a, b, c be non-negative numbers, no two of which are zero. Then,
1            1             1             9
+ 2          + 2          ≥              .
a2   + ab + b2  b + bc + c2  c + ca + a2   (a + b + c)2
ırtoaje, GM-B, 9, 2000)
(Vasile Cˆ

31. Let a, b, c be positive numbers, and let
1             1            1
x=a+         − 1, y = b + − 1, z = c + − 1.
b             c            a
Prove that
xy + yz + zx ≥ 3.
ırtoaje, GM-B, 1, 1991)
(Vasile Cˆ

32. Let a, b, c be positive numbers, no two of which are zero. If n is a positive
integer, then
2an − bn − cn 2bn − cn − an 2cn − an − bn
+ 2           + 2           ≥ 0.
b2 − bc + c2   c − ca + a2   a − ab + b2
ırtoaje, GM-B, 1, 2004)
(Vasile Cˆ

33. Let 0 ≤ a < b and let a1 , a2 , . . . , an ∈ [a, b]. Prove that
√                             √    √   2
a1 + a2 + · · · + an − n n a1 a2 . . . an ≤ (n − 1) b − a           .

ırtoaje and Gabriel Dospinescu, MS, 2005)
(Vasile Cˆ

34. Let a, b, c and x, y, z be positive numbers such that x + y + z = a + b + c.
Prove that
ax2 + by 2 + cz 2 + xyz ≥ 4abc.
ırtoaje, GM-A, 4, 1987)
(Vasile Cˆ

35. Let a, b, c and x, y, z be positive numbers such that x + y + z = a + b + c.
Prove that
x(3x + a) y(3y + a) z(3z + a)
+           +           ≥ 12.
bc            ca          ab
374                                                                   8. Final problem set

36. Let a, b, c be positive numbers such that a2 + b2 + c2 = 3. Prove that

a b c    9
+ + ≥       .
b c a  a+b+c
37. Let a1 , a2 , . . . , an be positive numbers such that a1 a2 . . . an = 1. Prove
that
1        1            1             4n
+         + ··· +     +                       ≥ n + 2.
a1 a2                 an n + a1 + a2 + · · · + an
ırtoaje, MS, 2005)
(Vasile Cˆ

38. Let a1 , a2 , . . . , an be positive numbers such that a1 a2 . . . an = 1. Prove
that

1   1          1
a1 + a2 + · · · + an − n + 1 ≥         n−1
+   + ··· +    − n + 1.
a1 a2         an

ırtoaje, MS, 2006)
(Vasile Cˆ

39. Let r > 1 and let a, b, c be non-negative numbers such that ab+bc+ca = 3.
Prove that
ar (b + c) + br (c + a) + cr (a + b) ≥ 6.

40. Let a, b, c be positive real numbers such that abc ≥ 1. Prove that
a     b   c
(a)     a b b c c a ≥ 1;
a     b
(b)     a b b c cc ≥ 1.

ırtoaje, CM, 4, 2005)
(Vasile Cˆ

41. Let a, b, c, d be non-negative numbers. Prove that

4(a3 + b3 + c3 + d3 ) + 15(abc + bcd + cda + dab) ≥ (a + b + c + d)3 .

42. Let a, b, c be positive numbers such that
1 1 1
(a + b − c)           + −   = 4.
a b  c
Prove that
1    1  1
(a4 + b4 + c4 )        4
+ 4+ 4        ≥ 2304.
a   b  c
ırtoaje, MC, 2005)
(Vasile Cˆ
8.1. Applications                                                          375

43. Let a, b, c be positive numbers. Prove that
1        1       1           2
+ 2     + 2      >              .
a2   + 2bc b + 2ca c + 2ab   ab + bc + ca
ırtoaje, MS, 2005)
(Vasile Cˆ

44. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
a(b + c)    b(c + a)   c(a + b)     ab + bc + ca
+           +          ≥1+ 2            .
a2 + 2bc b2 + 2ca c2 + 2ab           a + b2 + c2
ırtoaje, MS, 2006)
(Vasile Cˆ

45. Let a, b, c be non-negative numbers, no two of which are zero. Then

(b + c)2 (c + a)2 (a + b)2
+ 2      + 2       ≥ 6.
a2 + bc   b + ca   c + ab
(Peter Scholze and Darij Grinberg, MS, 2005)

46. Let a, b, c be non-negative numbers, no two of which are zero. Then

b+c       c+a     a+b        6
2 + bc
+ 2     + 2      ≥       .
2a        2b + ca 2c + ab   a+b+c
ırtoaje, MS, 2006)
(Vasile Cˆ

47. If a, b, c are non-negative numbers, then

a a2 + 3bc + b b2 + 3ca + c c2 + 3ab ≥ 2(ab + bc + ca).

ırtoaje, MS, 2005)
(Vasile Cˆ

48. Let a, b, c be non-negative numbers, no two of which are zero. Then

a2 − bc   b2 − ca   c2 − ab
√         +√        +√         ≥ 0.
a2 + bc   b2 + ca    c2 + ab
ırtoaje, MS, 2005)
(Vasile Cˆ

49. If a, b, c are non-negative numbers, then

(a2 − bc) a2 + 4bc + (b2 − ca) b2 + 4ca + (c2 − ab) c2 + 4ab ≥ 0.

ırtoaje, MS, 2005)
(Vasile Cˆ
376                                                              8. Final problem set

50. If a, b, c are positive numbers, then

a2 − bc               b2 − ca                 c2 − ab
+                      +                       ≥ 0.
8a2 + (b + c)2         8b2 + (c + a)2          8c2 + (a + b)2

ırtoaje, MS, 2006)
(Vasile Cˆ

51. If a, b, c are non-negative numbers, then
3
a2 + bc +       b2 + ca +     c2 + ab ≤     (a + b + c).
2
(Pham Kim Hung, MS, 2005)

52. Let a, b, c be non-negative numbers such that a2 + b2 + c2 = 3. Then,

21 + 18abc ≥ 13(ab + bc + ca).

ırtoaje, MS, 2005)
(Vasile Cˆ

53. Let a, b, c be non-negative numbers such that a2 + b2 + c2 = 3. Then
1       1       1
+       +        ≤ 1.
5 − 2ab 5 − 2bc 5 − 2ca
ırtoaje, MS, 2005)
(Vasile Cˆ

54. Let a, b, c be non-negative numbers such that a2 + b2 + c2 = 3. Then,

(2 − ab)(2 − bc)(2 − ca) ≥ 1.

ırtoaje, MS, 2005)
(Vasile Cˆ

55. Let a, b, c be non-negative numbers such that a + b + c = 2. Prove that
bc     ca     ab
+      +       ≤ 1.
a2 + 1 b2 + 1 c2 + 1
(Pham Kim Hung, MS, 2005)

56. Let a, b, c be non-negative numbers, no two of which are zero. Then,

a3 + 3abc b3 + 3abc c3 + 3abc
+          +          ≥ a + b + c.
(b + c)2   (c + a)2   (a + b)2

ırtoaje, MS, 2005)
(Vasile Cˆ
8.1. Applications                                                            377

57. Let a, b, c be positive numbers such that a4 + b4 + c4 = 3. Then,

a2 b2 c2
a)        +   +   ≥ 3;
b   c   a
a2   b2   c2  3
b)        +    +    ≥ .
b+c c+a a+b    2
(Alexey Gladkich, MS, 2005)

58. If a, b, c are positive numbers, then

a3 − b3 b3 − c3 c3 − a3   (a − b)2 + (b − c)2 + (c − a)2
+       +        ≤                                .
a+b     b+c     c+a                    8
(Marian Tetiva and Darij Grinberg, MS, 2005)

59. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that

a2               b2               c2         1
+                +                 ≤ .
(2a + b)(2a + c) (2b + c)(2b + a) (2c + a)(2c + b)  3

(Tigran Sloyan, MS, 2005)

60. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
1                  1                  1              1
+                  +                  ≥ 2          .
5(a2   +b2 ) − ab   5(b2 + c2 ) − bc   5(c2 + a2 ) − ca  a + b2 + c2

ırtoaje, MS, 2006)
(Vasile Cˆ

61. Let a, b, c be non-negative real numbers such that a2 +b2 +c2 = 1. Prove
that
bc        ca      ab      3
2+1
+ 2      + 2      ≤ .
a         b +1 c +1         4
(Pham Kim Hung, MS, 2005)

62. Let a, b, c be non-negative numbers such that a2 + b2 + c2 = 1. Prove
that
1              1             1         9
2 − 2bc
+     2 − 2ca
+     2 − 2ab
≤ .
3+a            3+b           3+c            8
ırtoaje and Wolfgang Berndt, MS, 2006)
(Vasile Cˆ
378                                                            8. Final problem set

63. If a, b, c are positive numbers, then
4a2 − b2 − c2 4b2 − c2 − a2 4c2 − a2 − b2
+             +              ≤ 3.
a(b + c)      b(c + a)      c(a + b)
ırtoaje, MS, 2006)
(Vasile Cˆ

64. If a, b, c are positive numbers such that abc = 1, then
3                 1 1 1
a2 + b2 + c2 + 6 ≥       a+b+c+         + +   .
2                 a b  c
ırtoaje, MS, 2006)
(Vasile Cˆ

65. Let a1 , a2 , . . . , an be positive numbers such that a1 + a2 + · · · + an = n.
Prove that
1     1           1
a1 a2 . . . an     +     + ··· +    − n + 3 ≤ 3.
a1 a2            an
ırtoaje, MS, 2004)
(Vasile Cˆ

66. Let a, b, c be the side lengths of a triangle. If a2 + b2 + c2 = 3, then

ab + bc + ca ≥ 1 + 2abc.

ırtoaje, MS, 2005)
(Vasile Cˆ

67. Let a, b, c be the side lengths of a triangle. If a2 + b2 + c2 = 3, then

a + b + c ≥ 2 + abc.

ırtoaje, MS, 2005)
(Vasile Cˆ

68. If a, b, c are the side lengths of a non-isosceles triangle, then
a+b b+c c+a
a)          +   +    > 5;
a−b b−c c−a
a2 + b2 b2 + c2 c2 + a2
b)              +       +        > 3.
a2 − b2 b2 − c2 c2 − a2
ırtoaje, GM-B, 3, 2003)
(Vasile Cˆ

69. Let a, b, c be the lengths of the sides of a triangle. Prove that
b          c          a
a2      − 1 + b2   − 1 + c2   − 1 ≥ 0.
c          a          b
ırtoaje, Moldova TST, 2006)
(Vasile Cˆ
8.1. Applications                                                                379

70. Let a, b, c be the lengths of the sides of an triangle. Prove that

1 1 1      a   b   c
(a + b + c)       + +   ≥6    +   +    .
a b  c    b+c c+a a+b

(Vietnam TST, 2006)

1 √
71. If a1 , a2 , a3 , a4 , a5 , a6 ∈ √ , 3 , then
3
a1 − a2 a2 − a3         a6 − a1
+        + ··· +         ≥ 0.
a2 + a3 a3 + a4         a1 + a2

ırtoaje, AJ, 7-8, 2002)
(Vasile Cˆ

72. Let a, b, c be positive numbers such that a2 + b2 + c2 ≥ 3. Prove that

a5 − a2        b5 − b2        c5 − c2
+ 2            + 2          ≥ 0.
a5   +b  2 + c2  a +b   5 + c2  a + b2 + c5
ırtoaje, MS, 2005)
(Vasile Cˆ

73. Let a, b, c be positive numbers such that x + y + z ≥ 3. Then,

1     1           1
+        +            ≤ 1.
x3   +y+z x+y 3+z   x + y + z3

ırtoaje, MS, 2005)
(Vasile Cˆ

74. Let x1 , x2 , . . . , xn be positive numbers such that x1 x2 . . . xn ≥ 1.
If α > 1, then
xα
1
≥ 1.
xα + x2 + · · · + xn
1

ırtoaje, CM, 2, 2006)
(Vasile Cˆ

75. Let x1 , x2 , . . . , xn be positive numbers such that x1 x2 . . . xn ≥ 1.
−2
If n ≥ 3 and              ≤ α < 1, then
n−2
xα
1
≤ 1.
xα + x2 + · · · + xn
1

ırtoaje, CM, 2, 2006)
(Vasile Cˆ
380                                                             8. Final problem set

76. Let x1 , x2 , . . . , xn be positive numbers such that x1 x2 . . . xn ≥ 1.
If α > 1, then
x1
≤ 1.
xα + x2 + · · · + xn
1
ırtoaje, CM, 2, 2006)
(Vasile Cˆ

77. Let x1 , x2 , . . . , xn be positive numbers such that x1 x2 . . . xn ≥ 1.
2
If −1 −         ≤ α < 1, then
n−2
x1
≥ 1.
xα + x2 + · · · + xn
1

ırtoaje, CM, 2, 2006)
(Vasile Cˆ

78. Let n ≥ 3 be an integer and let p be a real number such that 1 < p < n−1.
pn − p − 1
If 0 < x1 , x2 , . . . , xn ≤              such that x1 x2 . . . xn = 1, then
p(n − p − 1)
1       1               1       n
+        + ··· +         ≥     .
1 + px1 1 + px2         1 + pxn   1+p
ırtoaje, GM-A, 1, 2005)
(Vasile Cˆ

79. Let a, b, c be positive numbers such that abc = 1. Prove that
1          1          1                 2
2
+        2
+        2
+                       ≥ 1.
(1 + a)    (1 + b)    (1 + c)    (1 + a)(1 + b)(1 + c)
(Pham Van Thuan, MS, 2006)

80. Let a, b, c be positive numbers such that abc = 1. Prove that

a2 + b2 + c2 + 9(ab + bc + ca) ≥ 10(a + b + c).

81. Let a, b, c be non-negative numbers such that ab + bc + ca = 3. Prove
that
a(b2 + c2 ) b(c2 + a2 ) c(a2 + b2 )
+ 2         + 2          ≥ 3.
a2 + bc     b + ca      c + ab
(Pham Huu Duc, MS, 2006)

82. If a, b, c are positive numbers, then
a2 b2 c2     6(a2 + b2 + c2 )
a+b+c+           +   +   ≥                  .
b   c   a      a+b+c
(Pham Huu Duc, MS, 2006)
8.1. Applications                                                               381

83. If a, b, c are positive numbers, then

a2   b2   c2   3(a3 + b3 + c3 )
+    +    ≥                  .
b+c c+a a+b     2(a2 + b2 + c2 )

(Pham Huu Duc, MS, 2006)

84. If a, b, c are given non-negative numbers, ﬁnd the minimum value E(a, b, c)
of the expression
ax      by       cz
E=         +       +
y+z z+x x+y
for any positive numbers x, y, z.
ırtoaje, MS, 2006)
(Vasile Cˆ

85. Let a, b, c be positive real numbers such that a + b + c = 3. Prove that

1   1   1
+   +   ≥ a2 + b2 + c2 .
a2 b2 c2
ırtoaje, Romania TST, 2006)
(Vasile Cˆ

86. Let a, b, c be non-negative real numbers such that a + b + c = 3. Prove
that
(a2 − ab + b2 )(b2 − bc + c2 )(c2 − ca + a2 ) ≤ 12.

(Pham Kim Hung, MS, 2006)

87. Let a, b, c be non-negative real numbers such that a + b + c = 1. Prove
that
a + b2 + b + c2 + c + a2 ≥ 2.

(Phan Thanh Nam)

88. If a, b, c are non-negative real numbers, then

a3 + b3 + c3 + 3abc ≥        bc 2(b2 + c2 ).

89. If a, b, c are non-negative real numbers, then

15
(1 + a2 )(1 + b2 )(1 + c2 ) ≥      (1 + a + b + c)2 .
16
ırtoaje, MS, 2006)
(Vasile Cˆ
382                                                               8. Final problem set

90. Let a, b, c, d be positive real numbers such that abcd = 1. Prove that

(1 + a2 )(1 + b2 )(1 + c2 )(1 + d2 ) ≥ (a + b + c + d)2 .

(Pham Kim Hung, MS, 2006)

91. If x1 , x2 , . . . , xn are non-negative numbers, then

√                      x2 + x2 + · · · + x2
1    2            n
x1 + x2 + · · · + xn ≥ (n − 1) n x1 x2 . . . xn +                          .
n
ırtoaje, MS, 2006)
(Vasile Cˆ

92. If k is a real number and x1 , x2 , . . . , xn are positive numbers, then

(n−1) xn+k +xn+k + · · · +xn+k +x1 x2 . . . xn xk +xk + · · · +xk ≥
1     2             n                    1   2           n

≥ (x1 +x2 + · · · +xn ) xn+k−1 +xn+k−1 + · · · +xn+k−1 .
1       2               n

(Gjergji Zaimi and Keler Marku, MS, 2006

93. Let a, b, c be non-negative numbers, no two of which are zero. Prove
that
a4        b4       c4     a+b+c
3 + b3
+ 3    3
+ 3      ≥          .
a        b +c     c + a3        2

8.2      Solutions
1. Let a, b, c be positive numbers such that abc = 1. Prove that

a+b         b+c         c+a
+           +           ≥ 3.
b+1         c+1         a+1
Solution. By AM-GM Inequality, it follows that

a+b         b+c         c+a     (a + b)(b + c)(c + a)
+           +           ≥36                       .
b+1         c+1         a+1     (b + 1)(c + 1)(a + 1)
Thus, we still have to show that

(a + b)(b + c)(c + a) ≥ (a + 1)(b + 1)(c + 1).

Let A = a + b + c and B = ab + bc + ca. The AM-GM Inequality yields
A ≥ 3 and B ≥ 3. Since

(a + b)(b + c)(c + a) = (a + b + c)(ab + bc + ca) − abc = AB − 1
Inequalities
Reid Barton
June 13, 2005
1. (IMO ’95) Let a, b, c be positive real numbers with abc = 1. Prove that
1          1          1
+          +           ≥ 1.
a3 (b + c) b3 (c + a) c3 (a + b)

2. (MOP ’00?) Show that if k is a positive integer and x1 , x2 , . . . , xn are positive real numbers which
sum to 1, then
n
1 − xk
i
≥ (nk − 1)n .
i=1
xk
i

(Hint: the case k = 1 is equivalent to USAMO 98/3.)
3. (IMO ’01) Let a, b, c be positive real numbers. Prove that
a            b            c
√              +√           +√           ≥ 1.
a2   + 8bc    b 2 + 8ca    c 2 + 8ab

4. (USAMO ’04) Let a, b, c be positive reals. Prove that

(a5 − a2 + 3)(b5 − b2 + 3)(c5 − c2 + 3) ≥ (a + b + c)3 .

5. (IMO ’96 shortlist) Let a, b, c be positive reals with abc = 1. Prove that
ab           bc           ca
+            +             ≤ 1.
a5 + b5 + ab b5 + c5 + bc c5 + a5 + ca

6. (APMO ’05) Let a, b, c be real numbers with abc = 8. Prove that

a2                            b2                    c2          4
+                            +                     ≥ .
(a2 + 1)(b2 + 1)             (b2 + 1)(c2 + 1)     (c 2 + 1)(a2 + 1)  3

7. (Poland ’96?) Let a, b, c be real numbers with a + b + c = 1 and a, b, c ≥ −3/4. Prove that
a     b     c   9
+ 2   + 2   ≤    .
a2   +1 b +1 c +1    10

8. (Japan, ’97) Let a, b, c be positive real numbers. Prove that

(b + c − a)2     (c + a − b)2     (a + b − c)2  3
2 + a2
+         2 + b2
+               ≥ .
(b + c)          (c + a)          (a + b)2 + c2  5

9. (MOP ’02) Let a, b, c be positive real numbers. Prove that
2/3             2/3             2/3
2a                   2b              2c
+               +               ≥ 3.
b+c                  c+a             a+b

10. (USAMO ’97) Prove that for all positive real numbers a, b, c,
1              1            1           1
+ 3           + 3           ≤     .
a3   +b 3 + abc  b +c 3 + abc  c +a 3 + abc   abc

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