Search found 79 matches
- Sun Nov 16, 2014 8:10 pm
- Forum: Number Theory
- Topic: Numbers from blackboard
- Replies: 3
- Views: 3213
Numbers from blackboard
Consider the following game. In every position of this game, if in a blackboard there are two unequal numbers, remove the least and replace it with the ratio of their product and difference. The game terminates if we get two equal numbers. Suppose you started with two positive integers in the blackb...
- Sun Nov 16, 2014 7:43 pm
- Forum: Number Theory
- Topic: a square form
- Replies: 1
- Views: 2343
Re: a square form
Let $4xy - (x+y) = z^2$ with z nonnegative. Then $x = \dfrac {y+z^2} {4y-1}$ is a positive integer. Here $4y-1$ is an odd positive integer >1; with not all prime divisor of form 4k+1 for positive integer k; else 4y-1 would itself have this form. Let p be a prime divisor of 4y-1 of form 4k-1 for pos...
- Sun Nov 16, 2014 7:18 pm
- Forum: Geometry
- Topic: collinearity from russia
- Replies: 2
- Views: 4087
Re: collinearity from russia
My solution: Let $S,T$ be circles through $(A,B),(A,C)$, respectively; with centers $X,W$. Then $BC$ is their common external tangent. Also $I$, the intersection of $BC$ and $XW$; is their external center of similitude. So letting the second point of intersections of $IA$ with $S,T$ be $R,U$; $RB$ i...
- Sat Nov 15, 2014 11:46 pm
- Forum: Geometry
- Topic: collinearity from russia
- Replies: 2
- Views: 4087
collinearity from russia
Let triangle $ABC$ has intersection of $A$-tangent with side $ BC$ to be V. Suppose X,W are points in this plane so that both of $BX,CW$ are perpendicular to $ BC $ and $BX = AX, CW = AW$. Show $X,W,V$ collide.
- Fri Nov 14, 2014 10:49 pm
- Forum: Number Theory
- Topic: a square form
- Replies: 1
- Views: 2343
a square form
Determine all perfect squares of form $4xy-(x+y)$ with $x,y$ positive integers.
- Fri Nov 14, 2014 12:13 am
- Forum: Combinatorics
- Topic: Distribute the balls
- Replies: 2
- Views: 2848
Re: Distribute the balls
We denote by $S(n)$ to be the number of ways $5$ balls of same colour can be kept in $n$ boxes, with possibly some boxes empty. Now note that $S(n)^2$ is the number of ways five red and five blue balls can be kept in $n$ boxes in this way; considering disjoint events. So our desired result is $S(3)^...
- Sun Nov 09, 2014 4:06 pm
- Forum: Number Theory
- Topic: Perfect Square ratio
- Replies: 5
- Views: 5850
Perfect Square ratio
Determine all ordered pair of positive integers $(x,y)$ so that
$\frac{x^2+2y^2}{2x^2+y^2}$ is the square of an integer.
$\frac{x^2+2y^2}{2x^2+y^2}$ is the square of an integer.
- Sat Nov 08, 2014 8:34 pm
- Forum: Geometry
- Topic: A Self Posed Geo Prob
- Replies: 2
- Views: 2595
Re: A Self Posed Geo Prob
Triangle $OPQ$ has foot of perpendiculars from $P,Q$ to be $M,N$. Letting the other foot from $O$ be $K$, and orthocenter $H$; the desired radical axis is $OHK$ and the diameter of $OMN$ is simply $OH$. Isn't that?
- Fri Nov 07, 2014 10:10 pm
- Forum: Number Theory
- Topic: NT from Vietnam 2005
- Replies: 3
- Views: 3377
Re: NT from Vietnam 2005
Well, for the first question; a factorial(such as $x!$ and $y!$) is always $\geq 1$. So $n$ can't be $0$.
For the second question, I haven't said $x,y,n>0$. Here $0$ and $1$ has same factorial $1$; so I have said
"So let for advantage, $x;y;n>0$.
For the second question, I haven't said $x,y,n>0$. Here $0$ and $1$ has same factorial $1$; so I have said
"So let for advantage, $x;y;n>0$.
- Fri Nov 07, 2014 2:49 pm
- Forum: Number Theory
- Topic: Pythagorean FE(Self-made)
- Replies: 0
- Views: 1932
Pythagorean FE(Self-made)
Find all surjective function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that
if the triple $(f(a),f(b),f(c))$ is pythagorean in some order; then the triple $(a,b,c)$ is also pythagorean in some order.
if the triple $(f(a),f(b),f(c))$ is pythagorean in some order; then the triple $(a,b,c)$ is also pythagorean in some order.