## 近谷邦彦 さんがコメントした問題詳細

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2443761#p2443761
2011/09/21 14:26

kunny
2011/09/21 17:43

Prove that any multivariate polynomials, also with real coefficients, whose values over the reals are nonnegative may be written as a sum of two multivariate polynomials, also with real coefficients, whose values over the reals are nonnegative.
2011/09/22 09:04

I imagined the identity $(A+B)^2+(A-B)^2=2(A^2+B^2)$.
2011/09/22 13:14

Let $a,\ b,\ c,\ x,\ y,\ z\in{\mathbb{Z}}$,

$(a^3+b^3+c^3-3abc)(x^3+y^3+z^3-3xyz)$

$=(a+b+c)(a+b\omega +c\omega ^ 2)(a+b\omega ^2+c\omega)$

$*(x+y+z)(x+y\omega +z\omega ^ 2)(x+y\omega ^ 2 +z\omega)$

$=....$
2011/09/23 00:39

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=432965

My Memo

1982 東北大 J^2=-E
200? 早稲田/理工

197?東工大

Norm

N(p^2-pq+q^2)=1 197? 東北大

2011/09/24 20:35

Take a look here.

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2448884#p2448884
2011/09/26 14:24

kunny
2011/09/27 15:21

おそらく, モノグラフ(科学新興社)の公式集§74あたりが参考になるかと思います。

http://www.foruma.co.jp/books/print/pri_7527/monograph.pdf
2011/10/05 23:15

http://john.fremlin.de/schoolwork/logic/logic.pdf

2011/10/06 03:03

Consider the triangle with vertices at $(0,0),(1,1),(-1,2)$, with area $\frac32$. Suppose there is a smaller triangle. Consider the smallest side of the triangle. If it is at least length 2, then its smallest possible area is $\sqrt3$ when it is equilateral, but is more than 1.5. So it's smallest side is of length 1 or $\sqrt2$. Suppose it is of length 1, and the 2 vertices are at $({0,0),(1,0)}$. The third vertex must lie between the lines $x=0$ and $x=1$, which is not possible. Now suppose the smallest side is of length $\sqrt2$. Then let 2 of its vertices be at $(0,0),(1,1)$. Again, the third vertex must lie between $y=-x-1$ and $y=2-x$, so it must lie on $y=1-x$. Also the third vertex cannot be $(0,1)$ or $(0,1)$ otherwise there will be a right angle. Thus the smallest possible area is 1.5.}\$
2011/10/14 10:17

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=38&t=438050
2011/10/15 07:49

1980 津田塾大 羊と牧草地、羊の可動面積
2011/10/21 17:28

たまたま、問題を見て,類題？がうかんだだけです。
2011/10/22 00:43

$\left(x-\frac{1}{2}\right)^2+\frac {4}{3}y^2=1$ を描く。

この面積を求め, $\frac{\sqrt{3}}{2}\pi$を得ました。

2011/10/23 19:34

See here:

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2482222#p2482222
2011/10/24 05:25

2011/10/25 13:21

My Memo

197? 北大/理類

1980 日本女子大

198?, 199?　名大

200? 理科大

200? 東大実践　写像 説明付

2011/10/25 14:30

(1) $f(x)$の逆関数$g(x)$を求めよ。

(2) 方程式$f(x)=g(x)$を解け。

1987 同志社大/経済
2011/10/27 10:07