Download POLYA SEMINAR WEEK 4: INEQUALITIES Ralph Furmaniak, Bob

POLYA SEMINAR WEEK 4: INEQUALITIES
Ralph Furmaniak, Bob Hough and K. Soundararajan
The Rules. There are too many problems to consider. Pick a few problems that
you find fun, and play around with them. The only rule is that you may not pick
a problem that you already know how to solve: where’s the fun in that?
General problem solving strategies. Try small cases; plug in smaller numbers.
Search for a pattern. Draw pictures. Choose effective notation. Work in groups.
Divide into cases. Look for symmetry. Work backwards. Argue by contradiction.
Parity? Pigeonhole? Induction? Generalize the problem, sometimes that makes it
easier. Be flexible: consider many possible approaches before committing to one.
Be stubborn: don’t give up if your approach doesn’t work in five minutes. Ask.
Eat pizza, have fun!
1. (Larson 7.2.10) Given that all the roots of x6 − 6x5 + ax4 + bx3 + cx2 + dx + 1 = 0
are positive, find a, b, c, and d.
2. (1996 A1) Find the least number A such that for any two squares of combined
area 1, a rectangle of area A exists such that the two squares can be packed in the
rectangle (without the interiors of the squares overlapping). You may assume that
the sides of the squares will be parallel to the sides of the rectangle.
3. (1996 B2) Show that for every positive integer n
2n − 1 2n−1
2
e
< 1 · 3 · 5 · · · (2n − 1) <
2n + 1 2n+1
2
e
.
4. If a1 , . . . , an and b1 , . . . , bn are complex numbers with |ai |, |bi | ≤ 1 then prove
that
n
X
|a1 · · · an − b1 · · · bn | ≤
|aj − bj |.
j=1
5. Let a1 , a2 , . . . , an be real numbers greater than 1. Prove that
n
X
i=1
1
n
≥
1 .
1 + ai
1 + (a1 · · · an ) n
(Hint: First prove this for n a power of 2).
6. If α, β and γ are the angles of a triangle prove that
sin(α/2) sin(β/2) sin(γ/2) ≤ 1/8.
7. Find all positive integers n such that 3n + 4n + . . . + (n + 2)n = (n + 3)n .
Typeset by AMS-TEX
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RALPH FURMANIAK, BOB HOUGH AND K. SOUNDARARAJAN
8. Let a1 , . . . , an be positive and put sn = a1 + a2 + . . . + an . Prove that
(1 + a1 )(1 + a2 ) · · · (1 + an ) < 1 +
sn
s2
sn
+ n + ... + n.
1!
2!
n!
9. (due to Pólya!) Let a1 , a2 , . . . be positive real numbers. Prove that
lim sup
1 + a
n→∞
n+1
an
n
≥ e.
Prove that the inequality need not hold if e is replaced by a larger number.
10. If ai > 0, and a1 a2 · · · an = `n then prove that
(1 + a1 )(1 + a2 ) · · · (1 + an ) ≥ (1 + `)n ,
with equality only if the ai are equal.
POLYA SEMINAR WEEK 4: INEQUALITIES
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Extra Problems.
11. (Jaehyun Park). You are given a sequence a1 , a2 , . . . , a2n of distinct positive
integers. You are also given a sequence b1 , . . . , bn where each bi is 1, 2, 3 or 4 (and
denotes the corresponding quadrant in the plane). You must find a sequence of n
vectors in the plane (u1 , v1 ), (u2 , v2 ), . . . , (un , vn ) where (i) the 2n values |uj | and
|vj | form a permutation of the 2n numbers a1 , . . . , a2n , and (ii) the sum of the
first i vectors lies in the bi -th quadrant. Determine under what conditions this is
possible. (Hint: Think first of the 1 D case).
12. (from Michael Haynes last week; Cseth Ramlagan) Let f (x) be a polynomial
with integer coefficients and suppose that for every natural number n the value f (n)
is a perfect k-th power. Prove that f (x) must be the k-th power of a polynomial g
with integer coefficients.
13. (1999 B3) For 0 < x, y < 1 let
S(x, y) =
X
xm y n .
1/2≤m/n≤2
Find the limit of (1 − xy 2 )(1 − x2 y)S(x, y) as (x, y) → (1, 1).
14. (from Ralph Furmaniak; Nathaniel Shar) Consider a game where you start
with a pair of natural numbers (x, y) and players alternate subtracting the smaller
number from the larger ones as many times as they want (at least once). So an
example sequence is (10, 3) → (4, 3) → (1, 3) → (1, 1) → (1, 0). The first player to
reach 0 or a negative number loses. For what (x, y) is there a winning strategy for
the first player.
Pk
15. Let x1 , . . . , xn be positive real numbers, and let ak = k1 j=1 xj denote the
sequence of averages. Prove that for some constant C
n
X
k=1
a2k ≤ C
n
X
x2k .
k=1
(This result is a special case of a more general inequality due to G. H. Hardy.)
16. (Jaehyun Park) Consider the plane and put two red dots at (0, 0) and (1, 0)
and two blue dots at (0, 1) and (1, 1). Inside the square formed by these vertices
are more red dots and blue dots with the property that no three dots (including
the vertices of the square) are collinear. Prove that it is always possible to make
two trees, one with all red dots and the other with all blue dots, such that no edges
intersect. The edges are to be straight-line segments of course.
17. (1992 A3) For a given positive integer m find all triples (n, x, y) of positive
integers, with (n, m) = 1, and such that (x2 + y 2 )m = (xy)n .