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  1. B

    Strange combinatorics

    given 17 coins. a coin can either be red, blue or yellow. show there exists a three coins all of which are the same color.
  2. B

    Let E be an algebraic over F, F is perfect. Show that E is perfect

    let E be an algebraic over F where F is perfect. Show that E is perfect. :uhh:
  3. B

    Direct product

    Let k be a positive integer. define G_k = {x| 1<= x <= k with gcd(x,k)=1} prove that: a)G_k is a group under multiplication modulos k (i can do that). b)G_nm = G_n x G_m be defining an isomorphism.
  4. B

    Any subset

    i got this problem which is killin me :tongue: given a set {1,2,...,2n} choose any (n+1) element subset, show there exists an element which divides another element.
  5. B

    Simplifying a product of sin functions

    can someone please simplify? \sin \frac{\pi}{n} \sin \frac{2\pi}{n} ... \sin \frac{(n-1)\pi}{n}
  6. B

    Algebraic closure

    given a field F and two algebraic closures of F, are those two the isomorphic? and why doesnt this show that C and A (algebraic numbers) arent isomorphic?
  7. B

    Last question

    so embarrased askin so much :redface: On a table there are 14 cards. On each card there is a number between 1 to 1000. Show it is possible to divide the cards into two piles so that the total sums are the same.
  8. B

    Prove for all a,b,c>0: a/(b+c) + b/(a+c) + c/(a+b) >= 3/2 ?

    can somebody prove that for all a,b,c>0: a/(b+c) + b/(a+c) + c/(a+b) >= 3/2
  9. B

    Given an algebraic alpha be of degree n over F, show at most . . .

    let alpha be algebraic over F of degree n, show that there exists at most n isomorphisms mapping F(alpha) onto a subfield of bar F (this means the algebraic closure). thanx
  10. B

    5 points (last one i swear)

    im so embarrased askin so much :blushing: show that given 5 distinct lattice points in the plane (points with integer coordinates) there exists a line segment between both of them containing another lattice point on its interior.
  11. B

    Hard matrix prob

    i just cant figure this out. given a n x n matrix (with n>1) "A" such that all entries are integers and A is invertible such that A^{-1} also has integer entries. Let B be another matrix with integer coefficients so that: A+B, A+2B, A+3B, ... A+(n^2)B Are all invertible with integer...
  12. B

    Field problem

    help me its so hard working in the finite field Z_p show that the all the factors of polynomial x^{p^n}-x have degree "d" such that d|n. thanx