POTD 2006-02

This is the Problem of the Day section for February 2006.

The problems are shown in reverse order.

Table of contents [hide]

Tuesday, Feb. 28, 2006

from Anil

Find all primes p, q and even n > 2 such that pn + pn − 1 + ... + p + 1 = q2 + q + 1.

Solution by HiLander

Sunday, Feb. 26, 2006

from landen If a, b, and c are the sides of a nondegenerate triangle then

{{a}\over{c+b}}+{{b}\over{c+a}}+{{c}\over{b+a}}\leq 2

Clever solution

Saturday, Feb. 25, 2006

from Karlsen Show that

(\sum_{j=1}^n a_j)^2 + (\sum_{j=1}^n (-1)^j a_j)^2 \le (n+2)\sum_{j=1}^n a_j^2

Can you improve upon (n+2)?

bonus problem: show for a,b,c > 0:

\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\ge \frac{3}{2}

solution from landen

Thursday, Feb. 23, 2006

from Karlsen

Prove that:

\sum_{k=1}^{\infty}(a_1...a_k)^{1/k} \le e\sum_{k=1}^{\infty}a_k

note from landen: This is Carleman's inequality. This paper (http://jipam-old.vu.edu.au/v4n3/135_02_www.pdf) has several proofs.

Wednesday, Feb. 22, 2006

from Karlsen

Show that if a,b,c > 0 then

a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}

Short solution by argz

Solution by landen

HiLander solved the same inequality for n variables instead of just three. General solution by HiLander (http://www.efnet-math.org/math_tech/abc.pdf). landen shortened HiLander's solution to make a quick read solution.

(Note from HiLander: landen's solution generalizes to the n variable case with very little effort, and has the benefit of being noticeably more clever than my own. And argz's solution generalizes much faster, and looks much prettier. )

Monday, Feb. 20, 2006

from Anil

The sum of a certain number of consecutive positive intergers is 1000. Find the integers. Solution by landen

Thursday, Feb. 16, 2006

(College Mathematics Journal Vol. 37 No. 1 Jan. 2006) Two elements x,y in a commutative ring R are said to be associates if there exists a unit u \in R such that x = uy. Show that there exists a commutative ring D (with identity) and elements x,y \in D such that x \mid y and y \mid x, but x and y are not associates.

Solution: dvi (http://jgarrett.org/math/problem820.dvi) pdf (http://jgarrett.org/math/problem820.pdf)

Tuesday, Feb 14, 2006

from Anil

[my valentine's day gift for you all ;)]

Show that a2 + b2 + c2 = (ab)(bc)(ca) has infinitely many integral solutions.

Below are some of the integral solutions computed by brute force. The problem has been solved by Arne Smeets and independently by Inept. There is a discussion of their Solution by landen

a b c

0 1 -1

14 21 7

100 125 75

250 265 155

266 497 259

330 385 275

Monday, Feb 13, 2006

from Anil

Find all integers m, n such that (5 + 3 sqrt(2))^m = (3 + 5 sqrt(2))^n. Solution by landen

Saturday, Feb 11, 2006

from Anil. Solved by landen. Rated pre-calc, very easy. Symmetric Polynomials (http://www.cs.berkeley.edu/~oholtz/H90/equations.pdf) are overkill on this problem but they are worth learning about and are cool. Many math contests have symmetric polynomial problems.

Solve for integers x, y, z:

x + y = 1 − z

x3 + y3 = 1 − z3

Friday, Feb 10, 2006

from Anil. For precalc students or maybe freshpersons.

if x5x3 + x = a then, prove that x^6 \geq 2a - 1

Wednesday, Feb 8, 2006

from Anil

Find all pairs (a, b) of positive integers that satisfy (a^b)^2 = b^a. landen thought this problem was hard and got a version of this hint from Polytope.

Thursday, Feb 2, 2006

Prove by induction that (2n)!/(n!)^2 < (4^n)/sqrt(3n+1) for all n>1.