Algebraic Inequalities in Mathematical

Olympiads: Problems and Solutions

Mohammad Mahdi Taheri

July 20, 2015

Abstract

This is a collection of recent algebraic inequalities proposed in mathOlympiads from around the world.

mohammadmahdit@gmail.comwww.hamsaze.com

1 Problems

1. (Azerbaijan JBMO TST 2015) With the conditions a, b, c R+ and a +b+ c = 1, prove that

7 + 2b

1 + a+

7 + 2c

1 + b+

7 + 2a

1 + c 69

4

2. (Azerbaijan JBMO TST 2015) a, b, c R+ and a2 + b2 + c2 = 48. Provethat

a2

2b3 + 16 + b2

2c3 + 16 + c2

2a3 + 16 242

3. (Azerbaijan JBMO TST 2015) a, b, c R+ prove that

[(3a2+1)2+2(1+3

b)2][(3b2+1)2+2(1+

3

c)2][(3c2+1)2+2(1+

3

a)2] 483

4. (AKMO 2015) Let a, b, c be positive real numbers such that abc = 1. Provethe following inequality:

a3 + b3 + c3 +ab

a2 + b2+

bc

b2 + c2+

ca

c2 + a2 9

2

5. (Balkan MO 2015) If a, b and c are positive real numbers, prove that

a3b6 + b3c6 + c3a6 + 3a3b3c3 abc (a3b3 + b3c3 + c3a3)+ a2b2c2 (a3 + b3 + c3).6. (Bosnia Herzegovina TST 2015) Determine minimum value of the follow-

ing expression:a+ 1

a(a+ 2)+

b+ 1

b(b+ 2)+

c+ 1

c(c+ 2)

for positive real numbers such that a+ b+ c 3

1

7. (China 2015) Let z1, z2, ..., zn be complex numbers satisfying |zi 1| rfor some r (0, 1). Show that

ni=1

zi

ni=1

1

zi

n2(1 r2).8. (China TST 2015) Let a1, a2, a3, , an be positive real numbers. For the

integers n 2, prove thatn

j=1

(jk=1 ak

) 1jn

j=1 aj

1n

+(ni=1 ai)

1nn

j=1

(jk=1 ak

) 1j

n+ 1n

9. (China TST 2015) Let x1, x2, , xn (n 2) be a non-decreasing monotonoussequence of positive numbers such that x1,

x22 , , xnn is a non-increasing

monotonous sequence .Prove thatni=1 xi

n (ni=1 xi)

1n

n+ 12 nn!

10. (Junior Balkan 2015) Let a, b, c be positive real numbers such that a+ b+c = 3. Find the minimum value of the expression

A =2 a3a

+2 b3b

+2 c3c

.

11. (Romania JBMO TST 2015) Let x,y,z > 0 . Show that :

x3

z3 + x2y+

y3

x3 + y2z+

z3

y3 + z2x 3

2

12. (Romania JBMO TST 2015) Let a, b, c > 0 such that a bc2 , b ca2and c ab2 . Find the maximum value that the expression :

E = abc(a bc2)(b ca2)(c ab2)

can acheive.

13. (Romania JBMO TST 2015) Prove that if a, b, c > 0 and a + b + c = 1,then

bc+ a+ 1

a2 + 1+ca+ b+ 1

b2 + 1+ab+ c+ 1

c2 + 1 39

10

14. (Kazakhstan 2015 ) Prove that

1

22+

1

32+ + 1

(n+ 1)2< n

(1 1

n

2

).

2

15. (Moldova TST 2015) Let c (

0,pi

2

),

a =( 1sin(c)

) 1cos2(c) , b =

( 1cos(c)

) 1sin2(c)

. Prove that at least one of a, b is bigger than 11

2015.

16. (Moldova TST 2015) Let a, b, c be positive real numbers such that abc = 1.Prove the following inequality:

a3 + b3 + c3 +ab

a2 + b2+

bc

b2 + c2+

ca

c2 + a2 9

2

17. (All-Russian MO 2014) Does there exist positive a R, such that| cosx|+ | cos ax| > sinx+ sin ax

for all x R?18. (Balkan 2014) Let x, y and z be positive real numbers such that xy+yz+

xz = 3xyz. Prove that

x2y + y2z + z2x 2(x+ y + z) 3and determine when equality holds.

19. (Baltic Way 2014) Positive real numbers a, b, c satisfy 1a +1b +

1c = 3. Prove

the inequality

1a3 + b

+1b3 + c

+1c3 + a

32.

20. (Benelux 2014) Find the smallest possible value of the expressiona+ b+ c

d

+

b+ c+ d

a

+

c+ d+ a

b

+

d+ a+ b

c

in which a, b, c, and d vary over the set of positive integers.

(Here bxc denotes the biggest integer which is smaller than or equal to x.)21. (Britain 2014) Prove that for n 2 the following inequality holds:

1

n+ 1

(1 +

1

3+ . . .+

1

2n 1)>

1

n

(1

2+ . . .+

1

2n

).

22. (Bosnia Herzegovina TST 2014) Let a,b and c be distinct real numbers.a) Determine value of

1 + ab

a b 1 + bc

b c +1 + bc

b c 1 + ca

c a +1 + ca

c a 1 + ab

a bb) Determine value of

1 aba b

1 bcb c +

1 bcb c

1 cac a +

1 cac a

1 aba b

3

c) Prove the following ineqaulity

1 + a2b2

(a b)2 +1 + b2c2

(b c)2 +1 + c2a2

(c a)2 3

2

When does quality holds?

23. (Canada 2014) Let a1, a2, . . . , an be positive real numbers whose productis 1. Show that the sum

a11+a1

+ a2(1+a1)(1+a2) +a3

(1+a1)(1+a2)(1+a3)+ + an(1+a1)(1+a2)(1+an)

is greater than or equal to 2n12n .

24. (CentroAmerican 2014) Let a, b, c and d be real numbers such that notwo of them are equal,

a

b+b

c+c

d+d

a= 4

and ac = bd. Find the maximum possible value of

a

c+b

d+c

a+d

b.

25. (China Girls Math Olympiad 2014) Let x1, x2, . . . , xn be real numbers,where n 2 is a given integer, and let bx1c, bx2c, . . . , bxnc be a permuta-tion of 1, 2, . . . , n. Find the maximum and minimum of

n1i=1

bxi+1 xic

(here bxc is the largest integer not greater than x).26. (China Northern MO 2014) Define a positive number sequence sequence{an} by

a1 = 1, (n2 + 1)a2n1 = (n 1)2a2n.

Prove that

1

a21+

1

a22+ + 1

a2n 1 +

1 1

a2n.

27. (China Northern MO 2014) Let x, y, z, w be real numbers such that x +2y + 3z + 4w = 1. Find the minimum of

x2 + y2 + z2 + w2 + (x+ y + z + w)2

28. (China TST 2014) For any real numbers sequence {xn} ,suppose that {yn}is a sequence such that: y1 = x1, yn+1 = xn+1 (

ni=1

x2i )12 (n 1) . Find

the smallest positive number such that for any real numbers sequence{xn} and all positive integers m ,we have

1

m

mi=1

x2i mi=1

miy2i .

4

29. (China TST 2014) Let n be a given integer which is greater than 1. Findthe greatest constant (n) such that for any non-zero complex z1, z2, , zn,we have

nk=1

|zk|2 (n) min1kn

{|zk+1 zk|2},

where zn+1 = z1.

30. (China Western MO 2014) Let x, y be positive real numbers .Find theminimum of

x+ y +|x 1|y

+|y 1|x

.

31. (District Olympiad 2014) Prove that for any real numbers a and b thefollowing inequality holds:(

a2 + 1) (b2 + 1

)+ 50 2 (2a+ 1) (3b+ 1)

32. (ELMO Shortlist 2014) Given positive reals a, b, c, p, q satisfying abc = 1and p q, prove that

p(a2 + b2 + c2

)+ q

(1

a+

1

b+

1

c

) (p+ q)(a+ b+ c).

33. (ELMO Shortlist 2014) Let a, b, c, d, e, f be positive real numbers. Giventhat def + de+ ef + fd = 4, show that

((a+ b)de+ (b+ c)ef + (c+ a)fd)2 12(abde+ bcef + cafd).

34. (ELMO Shortlist 2014) Let a, b, c be positive reals such that a + b + c =ab+ bc+ ca. Prove that

(a+ b)abbc(b+ c)bcca(c+ a)caab acababcbc.

35. (ELMO Shortlist 2014) Let a, b, c be positive reals with a2014 + b2014 +c2014 + abc = 4. Prove that

a2013 + b2013 cc2013

+b2013 + c2013 a

a2013+c2013 + a2013 b

b2013 a2012+b2012+c2012.

36. (ELMO Shortlist 2014) Let a, b, c be positive reals. Prove thata2(bc+ a2)

b2 + c2+

b2(ca+ b2)

c2 + a2+

c2(ab+ c2)

a2 + b2 a+ b+ c.

37. (Korea 2014) Suppose x, y, z are positive numbers such that x+y+z = 1.Prove that

(1 + xy + yz + zx)(1 + 3x3 + 3y3 + 3z3)

9(x+ y)(y + z)(z + x)(x

1 + x4

3 + 9x2+

y

1 + y4

3 + 9y2+

z

1 + z4

3 + 9z2

)2.

5

38. (France TST 2014) Let n be a positive integer and x1, x2, . . . , xn be posi-tive reals. Show that there are numbers a1, a2, . . . , an {1, 1} such thatthe following holds:

a1x21 + a2x

22 + + anx2n (a1x1 + a2x2 + + anxn)2

39. (Harvard-MIT Mathematics Tournament 2014) Find the largest real num-ber c such that

101i=1

x2i cM2

whenever x1, . . . , x101 are real numbers such that x1 + + x101 = 0 andM is the median of x1, . . . , x101.

40. (India Regional MO 2014) Let a, b, c be positive real numbers such that

1

1 + a+

1

1 + b+

1

1 + c 1.

Prove that (1 + a2)(1 + b2)(1 + c2) 125. When does equality hold?41. (India Regional MO 2014) Let x1, x2, x3 . . . x2014 be positive real numbers

such that2014

j=1 xj = 1. Determine with proof the smallest constant Ksuch that

K

2014j=1

x2j1 xj 1

42. (IMO Training Camp 2014) Let a, b be positive real numbers.Prove that

(1 + a)8 + (1 + b)8 128ab(a+ b)2

43. (Iran 2014) Let x, y, z be three non-negative real numbers such that

x2 + y2 + z2 = 2(xy + yz + zx).

Prove thatx+ y + z

3 3

2xyz.

44. (Iran 2014) For any a, b, c > 0 satisfying a+ b+ c+ab+ac+ bc = 3, provethat

2 a+ b+ c+ abc 3

45. (Iran TST 2014) n is a natural number. for every positive real numbersx1, x2, ..., xn+1 such that x1x2...xn+1 = 1 prove that:

x1n+ ...+ xn+1

n n n

x1 + ...+ n

nxn+1

46. (Iran TST 2014) if x, y, z > 0 are postive real numbers such that x2+y2+z2 = x2y2 + y2z2 + z2x2 prove that

((x y)(y z)(z x))2 2((x2 y2)2 + (y2 z2)2 + (z2 x2)2)

6

47. (Japan 2014) Suppose there exist 2m integers i1, i2, . . . , im and j1, j2, . . . , jm,of values in {1, 2, . . . , 1000}. These integers are not necessarily distinct.For any non-negative real numbers a1, a2, . . . , a1000 satisfying a1+a2+ +a1000 = 1, find the maximum positive integer m for which the followinginequality holds

ai1aj1 + ai2aj2 + + aimajm 1

2.014.

48. (Japan MO Finals 2014) Find the maximum value of real number k suchthat

a

1 + 9bc+ k(b c)2 +b

1 + 9ca+ k(c a)2 +c

1 + 9ab+ k(a b)2 1

2

holds for all non-negative real numbers a, b, c satisfying a+ b+ c = 1.

49. (Turkey JBMO TST 2014) Determine the smallest value of

(a+ 5)2 + (b 2)2 + (c 9)2

for all real numbers a, b, c satisfying a2 + b2 + c2 ab bc ca = 350. (JBMO 2014) For positive real numbers a, b, c with abc = 1 prove that(

a+1

b

)2+

(b+

1

c

)2+

(c+

1

a

)2 3(a+ b+ c+ 1)

51. (Korea 2014) Let x, y, z be the real numbers that satisfies the following.

(x y)2 + (y z)2 + (z x)2 = 8, x3 + y3 + z3 = 1Find the minimum value of

x4 + y4 + z4

52. (Macedonia 2014) Let a, b, c be real numbers such that a+ b+ c = 4 anda, b, c > 1. Prove that:

1

a 1 +1

b 1 +1

c 1 8

a+ b+

8

b+ c+

8

c+ a

53. (Mediterranean MO 2014) Let a1, . . . , an and b1 . . . , bn be 2n real numbers.Prove that there exists an integer k with 1 k n such that

ni=1

|ai ak| ni=1

|bi ak|.

54. (Mexico 2014) Let a, b, c be positive reals such that a+ b+ c = 3. Prove:

a2

a+ 3bc

+b2

b+ 3ca

+c2

c+ 3ab 3

2

And determine when equality holds.

7

55. (Middle European MO 2014) Determine the lowest possible value of theexpression

1

a+ x+

1

a+ y+

1

b+ x+

1

b+ y

where a, b, x, and y are positive real numbers satisfying the inequalities

1

a+ x 1

2

1

a+ y 1

2

1

b+ x 1

2

1

b+ y 1.

56. (Moldova TST 2014) Let a, b R+ such that a+b = 1. Find the minimumvalue of the following expression:

E(a, b) = 3

1 + 2a2 + 2

40 + 9b2.

57. (Moldova TST 2014) Consider n 2 positive numbers 0 < x1 x2 ... xn, such that x1 +x2 + ...+xn = 1. Prove that if xn 2

3, then there

exists a positive integer 1 k n such that1

3 x1 + x2 + ...+ xk < 2

3

58. (Moldova TST 2014) Let a, b, c be positive real numbers such that abc = 1.Determine the minimum value of

E(a, b, c) = a3 + 5

a3(b+ c)

59. (Romania TST 2014) Let a be a real number in the open interval (0, 1).Let n 2 be a positive integer and let fn : R R be defined by fn(x) =x+ x

2

n . Show that

a(1 a)n2 + 2a2n+ a3(1 a)2n2 + a(2 a)n+ a2 < (fn fn)(a) tdholds for all (not necessarily distinct) x, y, z X, all real numbers a andall positive real numbers d.

78. (China Northern MO 2013) If a1, a2, , a2013 [2, 2] anda1 + a2 + + a2013 = 0

, find the maximum of

a31 + a32 + + a32013

.

79. (China TST 2013) Let n and k be two integers which are greater than 1.Let a1, a2, . . . , an, c1, c2, . . . , cm be non-negative real numbers such thati) a1 a2 . . . an and a1 + a2 + . . . + an = 1; ii) For any integerm {1, 2, . . . , n}, we have that c1 + c2 + . . . + cm mk. Find themaximum of c1a

k1 + c2a

k2 + . . .+ cna

kn.

80. (China TST 2013) Let n > 1 be an integer and let a0, a1, . . . , an be non-

negative real numbers. Definite Sk =k

i=0

(ki

)ai for k = 0, 1, . . . , n. Prove

that

1

n

n1k=0

S2k 1

n2

(nk=0

Sk

)2 4

45(Sn S0)2.

81. (China TST 2013) Let k 2 be an integer and let a1, a2, , an, b1, b2, , bnbe non-negative real numbers. Prove that(

n

n 1)n1(

1

n

ni=1

a2i

)+

(1

n

ni=1

bi

)2

ni=1

(a2i + b2i )

1n .

11

82. (China Western MO 2013) Let the integer n 2, and the real numbersx1, x2, , xn [0, 1].Prove that

1k

91. (ELMO Shortlist 2013) Let a, b, c be positive reals, and let

2013

3

a2013 + b2013 + c2013= P

Prove thatcyc

((2P + 12a+b )(2P +

1a+2b )

(2P + 1a+b+c )2

)cyc

((P + 14a+b+c )(P +

13b+3c )

(P + 13a+2b+c )(P +1

3a+b+2c )

).

92. (Federal Competition for Advanced students 2013) For a positive integern, let a1, a2, . . . , an be nonnegative real numbers such that for all realnumbers x1 > x2 > . . . > xn > 0 with x1 + x2 + . . . + xn < 1, theinequality

nk=1

akx3k < 1

holds. Show that

na1 + (n 1)a2 + . . .+ (n j + 1)aj + . . .+ an n2(n+ 1)2

4.

93. (Korea 2013) For a positive integer n 2, define set T = {(i, j)|1 i 2. Prove thatz3 + 1 > 1.

96. (India Regional MO 2013) Given real numbers a, b, c, d, e > 1. Prove that

a2

c 1 +b2

d 1 +c2

e 1 +d2

a 1 +e2

b 1 20

97. (Iran TST 2013) Let a, b, c be sides of a triangle such that a b c.prove that:

a(a+ b

ab) +

b(a+ cac) +

c(b+ c

bc) a+ b+ c

13

98. (Macedonia JBMO TST 2013) a, b, c > 0 and abc = 1. Prove that

1

2(a +

b +c ) +

1

1 + a+

1

1 + b+

1

1 + c 3

.

99. (Turkey JBMO TST 2013) For all positive real numbers a, b, c satisfyinga+ b+ c = 1, prove that

a4 + 5b4

a(a+ 2b)+b4 + 5c4

b(b+ 2c)+c4 + 5a4

c(c+ 2a) 1 ab bc ca

100. (Turkey JBMO TST 2013) Let a, b, c, d be real numbers greater than 1and x, y be real numbers such that

ax + by = (a2 + b2)x and cx + dy = 2y(cd)y/2

Prove that x < y.

101. (JBMO 2013) Show that(a+ 2b+

2

a+ 1

)(b+ 2a+

2

b+ 1

) 16

for all positive real numbers a and b such that ab 1.102. (Kazakhstan 2013) Find maximum value of

|a2 bc+ 1|+ |b2 ac+ 1|+ |c2 ba+ 1|when a, b, c are reals in [2; 2].

103. (Kazakhstan 2013) Consider the following sequence a1 = 1; an =a[n2]

2 +a[n3]

3 + . . .+a[n

n]

n Prove that n Na2n < 2an

104. (Korea 2013) Let a, b, c > 0 such that ab+ bc+ ca = 3. Prove thatcyc

(a+ b)3

(2(a+ b)(a2 + b2))13

12

105. (Kosovo 2013) For all real numbers a prove that

3(a4 + a2 + 1) (a2 + a+ 1)2

106. (Kosovo 2013) Which number is bigger 2012

2012! or 2013

2013!?

107. (Macedonia 2013) Let a, b, c be positive real numbers such that a4 + b4 +c4 = 3. Prove that

9

a2 + b4 + c6+

9

a4 + b6 + c2+

9

a6 + b2 + c4 a6 + b6 + c6 + 6

14

108. (Mediterranean MO 2013) Let x, y, z be positive reals for which:(xy)2 = 6xyz

Prove that: xx+ yz

3

.

109. (Middle European MO 2013) Let a, b, c be positive real numbers such that

a+ b+ c =1

a2+

1

b2+

1

c2.

Prove that

2(a+ b+ c) 3

7a2b+ 1 +3

7b2c+ 1 +3

7c2a+ 1.

Find all triples (a, b, c) for which equality holds.

110. (Middle European MO 2013) Let x, y, z, w be nonzero real numbers suchthat x+ y 6= 0, z + w 6= 0, and xy + zw 0. Prove that(

x+ y

z + w+z + w

x+ y

)1+

1

2(xz

+z

x

)1+

(y

w+w

y

)1111. (Moldova TST 2013) For any positive real numbers x, y, z, prove that

x

y+y

z+z

x z(x+ y)y(y + z)

+x(z + y)

z(x+ z)+y(x+ z)

x(x+ y)

112. (Moldova TST 2013) Prove that for any positive real numbers ai, bi, ciwith i = 1, 2, 3,

(a31+b31+c

31+1)(a

32+b

32+c

32+1)(a

33+b

33+c

33+1)

3

4(a1+b1+c1)(a2+b2+c2)(a3+b3+c3)

113. (Moldova TST 2013) Consider real numbers x, y, z such that x, y, z > 0.Prove that

(xy + yz + xz)

(1

x2 + y2+

1

x2 + z2+

1

y2 + z2

)>

5

2.

114. (Olympic Revenge 2013) Let a, b, c, d to be non negative real numberssatisfying ab+ ac+ ad+ bc+ bd+ cd = 6. Prove that

1

a2 + 1+

1

b2 + 1+

1

c2 + 1+

1

d2 + 1 2

115. (Philippines 2013) Let r and s be positive real numbers such that

(r + s rs)(r + s+ rs) = rs. Find the minimum value of r + s rs and r + s+ rs

15

116. (Poland 2013) Let k,m and n be three different positive integers. Provethat (

k 1k

)(m 1

m

)(n 1

n

) kmn (k +m+ n).

117. (Rioplatense 2013) Let a, b, c, d be real positive numbers such that

a2 + b2 + c2 + d2 = 1

Prove that(1 a)(1 b)(1 c)(1 d) abcd

118. (Romania 2013) To be considered the following complex and distincta, b, c, d. Prove that the following affirmations are equivalent:

i) For every z C this inequality takes place :

|z a|+ |z b| |z c|+ |z d|

ii) There is t (0, 1) so that c = ta+ (1 t) b si d = (1 t) a+ tb119. (Romania 2013)

a)Prove that1

2+

1

3+ ...+

1

2m< m

for any m N.b)Let p1, p2, ..., pn be the prime numbers less than 2

100. Prove that

1

p1+

1

p2+ ...+

1

pn< 10

120. (Romania TST 2013) Let n be a positive integer and let x1, . . ., xn bepositive real numbers. Show that:

min

(x1,

1

x1+ x2, , 1

xn1+ xn,

1

xn

) 2 cos pi

n+ 2 max

(x1,

1

x1+ x2, , 1

xn1+ xn,

1

xn

).

121. (Serbia 2013) Find the largest constant K R with the following prop-erty: if a1, a2, a3, a4 > 0 are numbers satisfying

a2i + a2j + a

2k 2(aiaj + ajak + akai)

for every 1 i < j < k 4, then

a21 + a22 + a

23 + a

24 K(a1a2 + a1a3 + a1a4 + a2a3 + a2a4 + a3a4).

122. (Southeast MO 2013) Let a, b be real numbers such that the equationx3 ax2 + bx a = 0 has three positive real roots . Find the minimum of

2a3 3ab+ 3ab+ 1

16

123. (Southeast MO 2013) n 3 is a integer. , , (0, 1). For everyak, bk, ck 0(k = 1, 2, . . . , n) with

nk=1

(k + )ak ,nk=1

(k + )bk ,nk=1

(k + )ck

we always havenk=1

(k + )akbkck

Find the minimum of

124. (Todays Calculation of Integrals 2013) Let m, n be positive integer suchthat 2 m < n.(1) Prove the inequality as follows.

n+ 1mm(n+ 1)

0.

Given any spooky sequence a1, a2, a3, . . . , prove that

20133a1 + 20123a2 + 2011

3a3 + + 23a2012 + a2013 < 12345.

132. (Uzbekistan 2013) Let real numbers a, b such that a b 0. Prove thata2 + b2 +

3a3 + b3 +

4a4 + b4 3a+ b.

133. (Uzbekistan 2013) Let x and y are real numbers such that x2y2+2yx2+1 =0. If

S =2

x2+ 1 +

1

x+ y(y + 2 +

1

x)

find

(a)maxS

(b) minS.

134. (Albania TST 2012) Find the greatest value of the expression

1

x2 4x+ 9 +1

y2 4y + 9 +1

z2 4z + 9where x, y, z are nonnegative real numbers such that x+ y + z = 1.

135. (All-Russian MO 2012) The positive real numbers a1, . . . , an and k aresuch that

a1 + + an = 3ka21 + + a2n = 3k2

anda31 + + a3n > 3k3 + k

Prove that the difference between some two of a1, . . . , an is greater than1.

18

136. (All-Russian MO 2012) Any two of the real numbers a1, a2, a3, a4, a5 differby no less than 1. There exists some real number k satisfying

a1 + a2 + a3 + a4 + a5 = 2k

a21 + a22 + a

23 + a

24 + a

25 = 2k

2

Prove that k2 253 .137. (APMO 2012) Let n be an integer greater than or equal to 2. Prove that

if the real numbers a1, a2, , an satisfy a21 + a22 + + a2n = n, then1i 0.

Show that

a+ b+ c abcab+ bc+ ac

4

143. (China Girls Math Olympiad 2012) Let a1, a2, . . . , an be non-negative realnumbers. Prove that

1

1 + a1+

a1(1 + a1)(1 + a2)

+a1a2

(1 + a1)(1 + a2)(1 + a3)+ + a1a2 an1

(1 + a1)(1 + a2) (1 + an) 1.

144. (China 2012) Let f(x) = (x + a)(x + b) where a, b > 0. For any realsx1, x2, . . . , xn 0 satisfying x1 + x2 + . . .+ xn = 1, find the maximum of

F =

1i

145. (China 2012) Suppose that x, y, z [0, 1]. Find the maximal value of theexpression

|x y|+|y z|+

|z x|.

146. (china TST 2012) Complex numbers xi, yi satisfy |xi| = |yi| = 1 for i =1, 2, . . . , n. Let x = 1n

ni=1

xi, y =1n

ni=1

yi and zi = xyi + yxi xiyi. Provethat

ni=1

|zi| n.

147. (China TST 2012) Given two integers m,n which are greater than 1. r, sare two given positive real numbers such that r < s. For all aij 0 whichare not all zeroes,find the maximal value of the expression

f =(n

j=1(m

i=1 asij)

rs )

1r

(m

i=1)n

j=1 arij)

sr )

1s

.

148. (China TST 2012) Given an integer k 2. Prove that there exist kpairwise distinct positive integers a1, a2, . . . , ak such that for any non-negative integers b1, b2, . . . , bk, c1, c2, . . . , ck satisfying a1 bi 2ai, i =1, 2, . . . , k and

ki=1 b

cii 0 : abc = 1 prove that

a3 + b3 + c3 + 6 (a+ b+ c)2

180. (Puerto rico TST 2012) Let x, y and z be consecutive integers such that

1

x+

1

y+

1

z>

1

45.

Find the maximum value of

x+ y + z

181. (Regional competition for advanced students 2012) Prove that the inequal-ity

a+ a3 a4 a6 < 1holds for all real numbers a.

182. (Romania 2012) Prove that if n 2 is a natural number and x1, x2, . . . , xnare positive real numbers, then:

4

(x31 x32x1 + x2

+x32 x33x2 + x3

+ . . .+x3n1 x3nxn1 + xn

+x3n x31xn + x1

)

(x1 x2)2 + (x2 x3)2 + . . .+ (xn1 xn)2 + (xn x1)2

183. (Romania 2012) Let a , b and c be three complex numbers such thata+ b+ c = 0 and |a| = |b| = |c| = 1 . Prove that:

3 |z a|+ |z b|+ |z c| 4,for any z C , |z| 1 .

184. (Romania 2012) Let a, b R with 0 < a < b . Prove that:a)

2ab x+ y + z

3+

ab3xyz

a+ b

for x, y, z [a, b] .b)

{x+ y + z3

+ab

3xyz|x, y, z [a, b]} = [2

ab, a+ b] .

185. (Romania TST 2012) Let k be a positive integer. Find the maximumvalue of

a3k1b+ b3k1c+ c3k1a+ k2akbkck,

where a, b, c are non-negative reals such that a+ b+ c = 3k.

24

186. (Romania TST 2012) Let f, g : Z [0,) be two functions such thatf(n) = g(n) = 0 with the exception of finitely many integers n. Defineh : Z [0,) by

h(n) = max{f(n k)g(k) : k Z}.Let p and q be two positive reals such that 1/p+ 1/q = 1. Prove that

nZ

h(n) (nZ

f(n)p

)1/p(nZ

g(n)q

)1/q.

187. (South East MO 2012) Let a, b, c, d be real numbers satisfying inequality

a cosx+ b cos 2x+ c cos 3x+ d cos 4x 1holds for any real number x. Find the maximal value of

a+ b c+ dand determine the values of a, b, c, d when that maximum is attained.

188. (South East MO 2012) Find the least natural number n, such that thefollowing inequality holds:

n 20112012

n 2012

2011