Algebra Proofs:
Inequalities
ERHS Math Club
1. Squares are positive x2≥0, Factoring, Quadratics
a. Prove the Arithmetic Mean- Geometric Mean inequality for two variables:
ab
 ab for all a,b≥0.
2
b. Prove a 2  b 2  c 2  ab  bc  ca by multiplying by 2 and factoring as
squares.
c. Prove that a 3  b 3  c 3  3abc by factoring. Conclude the AM-GM inequality
for 3 variables.
d. Find all values of k such that x2+y2≥kxy for all real numbers x and y.
e. Find the largest value of k such that x4+4x2y2+y4≥k(x3y+xy3) for all real
numbers x and y.
f. Prove that x4+24x2y2+16y4≥8x3y+32xy3 for all real x,y.
g. Prove that if a>b>c>d and a+d=b+c then bc >ad. (Hint: let s=a+b. Factor bcad)
2
2
2
h. Prove Aczel’s Inequality: if a1  a2  ...an then
(a1b1  a2b2  ...  an bn ) 2  (a1  a2  ... a n )(b1  b2  ... b n ) by
considering the determinant of the quadratic
(a1 x  b1 ) 2  (a 2 x  b2 ) 2  ...  (a n x  bn ) 2
i. [Romanian Olympiad 1983] Let a1,…a5 and x,y be real numbers such that
2
2
2
2
2
(a1  a2  a3  a4  a5 ) 2  3(a1  a2  a3  a4  a5 ) . Prove that
2
2
2
2
2
2
(a1  a2  a3  a4  a5  x  y) 2  a1  a2  a3  a4  a5  x 2  y 2
j. [IMO 2008/2] Prove that if , and are three real numbers, all different from
2
2
2
2
2
, such that
, then
k. Problem 38 on Olympiad Inequalities
2. Power-Mean
a. Prove the Quadratic Mean- Arithmetic Mean Inequality for two
.
ab
a2  b2

for a, b≥0.
2
2
b. Prove the Harmonic Mean- Arithmetic Mean Inequality for two variables:
2
ab

for a,b>0.
1 1
2

a b
a b
c. Prove that   2 for a,b >0
b a
d. Prove that for a,b>0, 2 a  33 b  5 ab
variables:
e. Problem 1 on Olympiad Inequalities
f. Prove that a 3  b 3  c 3  a 2 b  b 2 c  c 2 a for a,b,c ≥0
g. Prove that for a,b,c>0, a 2 b 2  b 2 c 2  c 2 a 2  abc(a  b  c)
a m1  b m1 a n 1  b n1
h. Prove that for positive integers m≥n, a,b>0,

am  bm
an  bn
i. Prove that for a,b,c,d>0,
1
1
1
1
1 1 1 1 1



 (    )
abc abd acd bcd 3 a b c d
j. A die has 6 sides. Each number has a fixed probability of coming up (not
necessarily 1/6). Find the minimum possible probability of getting the same
number after rolling it twice.
k. Problem 15 on Olympiad Inequalities
l. Problem 20 on Olympiad Inequalities
3. Rearrangement
a. Four boxes contain $10, $20, $50, $100 bills, respectively. You may take 3
bills from one box, 4 bills from another, 5 bills from another, and 6 bills from
the remaining box. What is the maximum amount of money you can get?
b. Prove 1(b) by rearrangement.
c. Prove 2(f) by rearrangement.
a2 b2
d. Prove that for a,b>0,

 ab
b
a
e. (Problem 13 on Olympiad Inequalities) Prove that for x,y,z>0,
x2 y2 z2 x y z


  
y2 z2 x2 y z x
f. [IMO 1978] Let a1,…an be pairwise distinct positive integers. Show
a
a1 a2
1
1
 2  ...  n2  1   ... 
2
2
n
1
2
n
4. Cauchy-Schwarz
a. Prove that for a,b>0, (a  b)( a 3  b 3 )  (a 2  b 2 ) 2
(a  c)(b  d )  ab  cd
1 1 4 16
64

Prove that for a,b,c,d>0,   
a b c d abcd
Let x1,…xn, y1,…yn be positive reals such that a+b+c+d=1. Prove
a2
b2
c2
d2
1




ab bc cd d a 2
Let p be a polynomial with positive real coefficients. Prove that if
p(1/x)≥1/p(x) holds for x=1, it holds for all x>0.
Show that the sequence a0,a1,…an of positive reals is a geometric sequence if
and only if
2
2
2
2
2
2
(a0  a1  ...  an1 )(a1  a2  ...  an )  (a0 a1  a1a2  ...  an1an ) 2
Prove that for all reals a,b,c,
(a 2 b  b 2 c  c 2 a)(ab 2  bc 2  ca 2 )  (a 2  b 2  c 2 )(a 2 b 2  b 2 c 2  c 2 a 2 )
b. Prove that for a,b,c,d≥0,
c.
d.
e.
f.
g.
a
b
c
9



1  bc 1  ca 1  ab 10
i. [HMMT 2008] Let ABC be a triangle with AB = 5, BC = 4 and AC = 3. Let P
and Q be squares inside ABC with disjoint interiors such that they both have
one side lying on AB. Also, the two squares each have an edge lying on a
common line perpendicular to AB, and P has one vertex on AC and Q has one
vertex on BC. Determine the minimum value of the sum of the areas of the
two squares.
j. [IMO 2004/4] Let
be an integer. Let , , ..., be positive real
numbers such that
h. Prove that for a,b,c>0,a+b+c=1,
.
Show that ,
,
are side lengths of a triangle for all , , with
k. [IMO 1995/2] Let , , be positive real numbers such that
. Prove
that
l. [IMO 2004/5] Let be a positive integer and let
numbers.
Prove that
Show that the equality holds if and only if
sequence.
m. Problem 6 on Olympiad Inequalities
n. Problem 12 (uses Holder’s Inequality)
o. Problem 26
be real
is an arithmetic