
 JaneFairfax
 Legendary Member
Jane’s exercises II
I thought I’d start a new exercise thread since nobody seems to be reading the first thread any more.
1. Let a_{1}, a_{2}, … a_{k} be k consecutive terms of an arithmetic progression with common difference d (where the a_{i} are integers). Show that if gcd(k,d) = 1, then k divides a_{r} for some r = 1, …, k.
2. Let a, b, c be three consecutive integers. Prove that at least one of 10a−1, 10b−1 and 10c−1, is not prime.
Last edited by JaneFairfax (20071231 05:53:00)
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
Re: Jane’s exercises II
Man, what a pain...
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
Yeah, I made both problems up myself. On reflection, I now realize that #1 is a bit on the tough side. Sorry.
Last edited by JaneFairfax (20071106 23:25:32)
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
 Kurre
 Power Member
Re: Jane’s exercises II
edit: forgot to hide it
Last edited by Kurre (20071121 03:13:50)
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
3.
Let f and g be realvalued functions which are twice differentiable on an interval I. Show that if f″(x) ≠ g″(x) for all x ∈ I, the graphs y = f(x) and y = g(x) cannot intersect each other more than twice in the interval I.
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
Re: Jane’s exercises II
igloo myrtilles fourmis
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
3.
a, b, c, d are real numbers such that 0 ≤ a ≤ b ≤ c ≤ d and a + d ≤ b + c. Show that ad ≤ bc.
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
Re: Jane’s exercises II
a, b, c, d are real numbers such that 0 ≤ a ≤ b ≤ c ≤ d and a + d ≤ b + c. Show that ad ≤ bc.
let a+d≤b+c a+a+z≤a+x+a+y z≤x+y =a(zxy)xy≤0 ad ≤ bc
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
Nice work. (The last part of the proof needs a little bit of touching up, that’a all.)
4.
Let a, b, x, y be rational numbers. Consider the following statement.
(i) Give a counterexample to show that the above statement, as it stands, is not true. (ii) What condition must be imposed on the above statement to make it true?
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
Re: Jane’s exercises II
I had another method for 3 halfformed in my mind yesterday. After a good night's sleep I've fortified it, so I thought I'd post even though I'm late.
Let p and q be the mean and halfrange of a and d, and define r and s similarly for b and c. That is:
p = (a+d)/2 q = (da)/2 r = (b+c)/2 s = (cb)/2
Then it is a simple matter to show that:
pq = a p+q = d rs = b r+s = c
Therefore ad = (pq)(p+q) = p²q² and bc = (rs)(r+s) = r²s².
Further, using the fact that 0≤a≤b≤c≤d shows that cb≤db≤da and so s≤q. Also, since a+d≤b+c, then p≤r.
Hence p²q² ≤ r²q² ≤ r²s² and thusly ad ≤ bc, WWWWW.
Why did the vector cross the road? It wanted to be normal.
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
Nice solution, Mathsy.
Actually #3 wasn’t exactly mine; it was something I copied from the Web, which I managed to solve myself. Anyway, the following exercise question is mine.
#5
A cubic polynomial has roots . Another cubic polynomial has roots . If the leading term in is , find (in terms of p, q and r) the coefficient of the term in in .
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
 Kurre
 Power Member
Re: Jane’s exercises II
5# There is probably noone looking here, but anyway:P: lets call the requested coefficient t. according to vietes identity:
we have that p=a+b+c, q=ab+bc+ac, r=abc, so that gives: if we take (a+b+c)^3 we get: therefor thus
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
You could also make use of the following very useful identity:
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
#6
Prove that for all positive integers n,
. .
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
#7
Solve the following differential equation:
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
 Kurre
 Power Member
Re: Jane’s exercises II
Using the inequality between arithmetic and geometric metdium yields:
equality holds if and only if ab=2bc=4ca letting ab=1/c, bc=1/a, ca=1/b, yields 1/c=2/a=4/b > b=2a and c=a/2 abc=1 >a*2a*a/2=1 > a=1, which gives that b=2, c=1/2
 Kurre
 Power Member
Re: Jane’s exercises II
#7
integration gives same as with solutions: nice problem btw
Last edited by Kurre (20080516 23:59:06)
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
#8
Suppose the roots of the cubic equation are all real and nonnegative. Prove that What condition is necessary and sufficient for equality to occur?
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
Last edited by JaneFairfax (20080526 02:52:21)
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
 Kurre
 Power Member
Re: Jane’s exercises II
Its also possible to use the rearrangement inequality directly on the first inequality. However, I spotted an error on my solution to the second one:
 JaneFairfax
 Legendary Member
Re: Jane’s exercises II
#9
Let R and S be rings with multiplicative identities 1_{R} and 1_{S} respectively and suppose f:R → S is a function which preserves multiplication (i.e. f(xy) = f(x)f(y) for all x, y ∈ R). Show that if f does not map all the elements of R into the zero element of S and S is an integral domain, then f(1_{R}) = 1_{S}.
Q: Who wrote the novels Mrs Dalloway and To the Lighthouse? A: Click here for answer.
