Difference between revisions of "KGS math club"

Revision as of 12:11, 9 February 2011

A group of people on Kiseido Go Server Mathematics room.

The meaning of this page is to collect the problems posed there and save hints and solution suggestions. In order to write something, I'm afraid you need to register to the AoPS wiki first. After that you're good to go.

Adding problems should be quite straightforward with the copy-paste template in the wiki source. Please add <math>-tags (or dollar signs, it seems) where required, e.g. $f''(x)$ . Still, if you don't, somebody else will; all additions are appreciated.

KGS math problems
Added	Author	Problem	Solutions

@@ Line 13: / Line 13: @@
 !style="background-color:rgb(220,230,255);" | Solutions
-|- valign="top"
+<!-- TEMPLATE, COPY-PASTE-FILL-IN: date, author, problem, solver; n = problem number, m = solution number, then click the red link and copypaste the solution, save. There. If anyone can do the wiki table formatting more elegantly, be my guest; after all,   this    is    wikiiii.
-|20.2.2007
-|style="background-color:rgb(220,230,255);" | StoneTiger
-|Does any member of the sequence <math>1, 4, 20, 80, ...</math> generated by <cmath>x(n) = 6x(n-1) - 12x(n-2) + 8x(n-3)</cmath> ever have a factor in common with <math>2007</math>?
-|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_2_1|sigmundur]]
 |- valign="top"
-|21.6.2008
+|11.8.2010
-|style="background-color:rgb(220,230,255);" | amkach
+|style="background-color:rgb(220,230,255);" | ghej
-|Consider the two player game that begins with an even length sequence of positive integers.  Each player, in turn, removes either the first or last of the remaining integers, ending when all the integers have been removed.  A player's score is the sum of the integers that they removed; the winner is the player with the higher score (with a tie if equal scores).  Show that Player One has a non-losing strategy, i.e., can always force a tie or a win.
+|For the curve x^2 + x y + y^2 = 1, find the tangent that passes through the point (0,2).
-|style="background-color:rgb(220,230,255);" | [[KGS math club/hints_1_1|hints]] [[KGS math club/solution_1_1|solution]] [[KGS math club/solution_1_2|solution2]]
+|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_10_1| solution ]]
 |- valign="top"
-|30.6.2008
+|19.8.2009
-|style="background-color:rgb(220,230,255);" | amkach
+|style="background-color:rgb(220,230,255);" | royu
-|For <math>n \geq 2</math>, consider the <math>n-</math>dimensional hypercube with side length <math>4</math> centered at the origin of <math>n-</math>space. Place inside of it <math>2^n</math> <math>n-</math>dimensional hyperspheres of radius <math>1</math>, centered at each of the points <math>(\pm1, \pm1, ..., \pm1)</math>. These hyperspheres are tangent to the hypercube and to each other.
+| You have the set {9, 99, 999, ...}.  Show that given any natural number n not divisible by 2 or 5, n divides at least one element of the set.
+|style="background-color:rgb(220,230,255);" | [[KGS math club/hints_9_1| hint ]]   [[KGS math club/solution_9_1| solution ]]
-Then place an <math>n-</math>dimensional hypersphere, centered at the origin, of size so that it is tangent to each of the <math>2^n</math> hyperspheres of radius <math>1</math>.  In which dimensions <math>n</math> is this central hypersphere contained within the hypercube?
-|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_3_1| solution ]]
 |- valign="top"
-|1.7.2008
+| 19.8.2009
-|style="background-color:rgb(220,230,255);" | quimey
+|style="background-color:rgb(220,230,255);" | bourbaki
-|Assume <math>m</math> and <math>n</math> are integers and can be expressed as sum of <math>2</math> squares (i.e, exists <math>a,b,c,d</math> integers with <math>m=a^2+b^2, n=c^2+d^2)</math>. Show <math>m*n</math> can be written as sum of <math>2</math> squares. And the same but with <math>4</math> squares.
+| Suppose A and B are n x n matrices with real entries such that either A or B commutes with C = AB - BA.  Prove that C is nilpotent, i.e. C^k = 0 for some integer k
-|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_4_1| solution ]]
+|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_8_1| solution ]]
 |- valign="top"
-|6.7.2008
+| 18.7.2009
-|style="background-color:rgb(220,230,255);" | amkach
+|style="background-color:rgb(220,230,255);" | taoyan
-|Prove or disprove: If P(x) is a polynomial (with non-zero degree) of one real variable and a and b satisfy <math>P^{(n)}(a) = P^{(n)}(b)</math> for all integers n > 0 (i.e., <math>P(a) = P(b), P'(a) = P'(b), P''(a) = P''(b)</math>, etc.), then a = b
+| How many times do the clock hands (hour and minute) overlap between 11:59:59 before lunch  and 00:00:01 at night?
-|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_5_1| solution ]]
+|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_7_1| solution ]]
 |- valign="top"
@@ Line 51: / Line 45: @@
 |style="background-color:rgb(220,230,255);" | [[KGS math club/solution_6_1| solution ]]
+|- valign="top"
+|6.7.2008
+|style="background-color:rgb(220,230,255);" | amkach
+|Prove or disprove: If P(x) is a polynomial (with non-zero degree) of one real variable and a and b satisfy <math>P^{(n)}(a) = P^{(n)}(b)</math> for all integers n > 0 (i.e., <math>P(a) = P(b), P'(a) = P'(b), P''(a) = P''(b)</math>, etc.), then a = b
+|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_5_1| solution ]]
 |- valign="top"
-| 18.7.2009
+|1.7.2008
-|style="background-color:rgb(220,230,255);" | taoyan
+|style="background-color:rgb(220,230,255);" | quimey
-| How many times do the clock hands (hour and minute) overlap between 11:59:59 before lunch  and 00:00:01 at night?
+|Assume <math>m</math> and <math>n</math> are integers and can be expressed as sum of <math>2</math> squares (i.e, exists <math>a,b,c,d</math> integers with <math>m=a^2+b^2, n=c^2+d^2)</math>. Show <math>m*n</math> can be written as sum of <math>2</math> squares. And the same but with <math>4</math> squares.
-|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_7_1| solution ]]
+|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_4_1| solution ]]
 |- valign="top"
-| 19.8.2009
+|30.6.2008
-|style="background-color:rgb(220,230,255);" | bourbaki
+|style="background-color:rgb(220,230,255);" | amkach
-| Suppose A and B are n x n matrices with real entries such that either A or B commutes with C = AB - BA.  Prove that C is nilpotent, i.e. C^k = 0 for some integer k
+|For <math>n \geq 2</math>, consider the <math>n-</math>dimensional hypercube with side length <math>4</math> centered at the origin of <math>n-</math>space. Place inside of it <math>2^n</math> <math>n-</math>dimensional hyperspheres of radius <math>1</math>, centered at each of the points <math>(\pm1, \pm1, ..., \pm1)</math>. These hyperspheres are tangent to the hypercube and to each other.
-|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_8_1| solution ]]
+Then place an <math>n-</math>dimensional hypersphere, centered at the origin, of size so that it is tangent to each of the <math>2^n</math> hyperspheres of radius <math>1</math>.  In which dimensions <math>n</math> is this central hypersphere contained within the hypercube?
+|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_3_1| solution ]]
 |- valign="top"
-|19.8.2009
+|21.6.2008
-|style="background-color:rgb(220,230,255);" | royu
+|style="background-color:rgb(220,230,255);" | amkach
-| You have the set {9, 99, 999, ...}.  Show that given any natural number n not divisible by 2 or 5, n divides at least one element of the set.
+|Consider the two player game that begins with an even length sequence of positive integers.  Each player, in turn, removes either the first or last of the remaining integers, ending when all the integers have been removed.  A player's score is the sum of the integers that they removed; the winner is the player with the higher score (with a tie if equal scores).  Show that Player One has a non-losing strategy, i.e., can always force a tie or a win.
-|style="background-color:rgb(220,230,255);" | [[KGS math club/hints_9_1| hint ]]   [[KGS math club/solution_9_1| solution ]]
+|style="background-color:rgb(220,230,255);" | [[KGS math club/hints_1_1|hints]] [[KGS math club/solution_1_1|solution]] [[KGS math club/solution_1_2|solution2]]
 |- valign="top"
-|11.8.2010
+|20.2.2007
-|style="background-color:rgb(220,230,255);" | ghej
+|style="background-color:rgb(220,230,255);" | StoneTiger
-|For the curve x^2 + x y + y^2 = 1, find the tangent that passes through the point (0,2).
+|Does any member of the sequence <math>1, 4, 20, 80, ...</math> generated by <cmath>x(n) = 6x(n-1) - 12x(n-2) + 8x(n-3)</cmath> ever have a factor in common with <math>2007</math>?
-|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_10_1| solution ]]
+|style="background-color:rgb(220,230,255);" | [[KGS math club/solution_2_1|sigmundur]]
-<!-- TEMPLATE, COPY-PASTE-FILL-IN: date, author, problem, solver; n = problem number, m = solution number, then click the red link and copypaste the solution, save. There. If anyone can do the wiki table formatting more elegantly, be my guest; after all,   this    is    wikiiii.
 |- valign="top"

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Difference between revisions of "KGS math club"

Revision as of 12:11, 9 February 2011