Difference between revisions of "KGS math club"

Revision as of 10:43, 8 November 2009

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
20.2.2007	StoneTiger	Does any member of the sequence $1, 4, 20, 80, ...$ generated by $x(n) = 6x(n-1) - 12x(n-2) + 8x(n-3)$ ever have a factor in common with $2007$ ?	sigmundur
21.6.2008	amkach	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.	hints solution solution2
30.6.2008	amkach	For $n \geq 2$ , consider the $n-$ dimensional hypercube with side length $4$ centered at the origin of $n-$ space. Place inside of it $2^n$ $n-$ dimensional hyperspheres of radius $1$ , centered at each of the points $(\pm1, \pm1, ..., \pm1)$ . These hyperspheres are tangent to the hypercube and to each other. Then place an $n-$ dimensional hypersphere, centered at the origin, of size so that it is tangent to each of the $2^n$ hyperspheres of radius $1$ . In which dimensions $n$ is this central hypersphere contained within the hypercube?	solution
1.7.2008	quimey	Assume $m$ and $n$ are integers and can be expressed as sum of $2$ squares (i.e, exists $a,b,c,d$ integers with $m=a^2+b^2, n=c^2+d^2)$ . Show $m*n$ can be written as sum of $2$ squares. And the same but with $4$ squares.	solution
6.7.2008	amkach	Prove or disprove: If P(x) is a polynomial (with non-zero degree) of one real variable and a and b satisfy $P^{(n)}(a) = P^{(n)}(b)$ for all integers n > 0 (i.e., $P(a) = P(b), P'(a) = P'(b), P''(a) = P''(b)$ , etc.), then a = b	solution
27.7.2008	royu StoneTiger	You have a collection of 11 balls with the property that if you remove any one of the balls, the other 10 can be split into two groups of 5 so that each weighs the same. If you assume that all of the balls have rational weight, there is a cute proof that they all must weigh the same. Can you find a proof? Can you find a way to extend the result to the general case where the balls have real weights?	solution
18.7.2009	taoyan	How many times do the clock hands (hour and minute) overlap between 11:59:59 before lunch and 00:00:01 at night?	solution
19.8.2009	bourbaki	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	solution
19.8.2009	royu	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.	hint solution

@@ Line 16: / Line 16: @@
 |20.2.2007
 |style="background-color:rgb(220,230,255);" | StoneTiger
-|Does any member of the sequence 1, 4, 20, 80, ... generated by x(n) = 6x(n-1) - 12x(n-2) + 8x(n-3) ever have a factor in common with 2007?
+|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]]
@@ Line 28: / Line 28: @@
 |30.6.2008
 |style="background-color:rgb(220,230,255);" | amkach
-|For n >= 2, consider the n-dimensional hypercube with side length 4 centered at the origin of n-space. Place inside of it 2^n n-dimensional hyperspheres of radius 1, centered at each of the points (+-1, +-1, ..., +-1). These hyperspheres are tangent to the hypercube and to each other.
+|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.
-Then place an n-dimensional hypersphere, centered at the origin, of size so that it is tangent to each of the 2^n hyperspheres of radius 1.  In which dimensions n is this central hypersphere contained within the hypercube?
+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 ]]
@@ Line 36: / Line 36: @@
 |1.7.2008
 |style="background-color:rgb(220,230,255);" | quimey
-|Assume m and n are integers and can be expressed as sum of 2 squares (i.e, exists a,b,c,d integers with m=a*a+b*b, n=c*c+d*d). Show m*n can be written as sum of 2 squares. And the same but with 4 squares.
+|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_4_1| solution ]]

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Difference between revisions of "KGS math club"

Revision as of 10:43, 8 November 2009