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Locus of Mobile points on Circle and Square
Kunihiko_Chikaya   1
N 6 minutes ago by Mathzeus1024
Source: 2012 Hitotsubashi University entrance exam, problem 4
In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane.

(1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$,
find the equation of the locus of the point $R$.

(2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.
1 reply
Kunihiko_Chikaya
Feb 28, 2012
Mathzeus1024
6 minutes ago
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Mathematical Olympiad Finals 2013
parkjungmin   0
31 minutes ago
Mathematical Olympiad Finals 2013
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parkjungmin
31 minutes ago
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Complicated FE
XAN4   2
N Apr 24, 2025 by cazanova19921
Source: own
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
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XAN4
Apr 23, 2025
cazanova19921
Apr 24, 2025
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XAN4
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XAN4
#1 h  Apr 23, 2025, 11:53 AM
Y by
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
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jasperE3
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jasperE3
#2 h  Apr 23, 2025, 8:32 PM
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pogress
This post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:34 PM
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cazanova19921
552 posts
cazanova19921
#3 h  Apr 24, 2025, 3:03 AM
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XAN4 wrote:
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.

I suppose you have a proof to the result you claimed since it’s your own FE ( I know you don’t).


General solution : let $a$ be an additive function from $\mathbb{R}$ to itself and set $g=\exp \circ a \circ \log $, then $f=g+\frac1{g}$
This post has been edited 1 time. Last edited by cazanova19921, Apr 24, 2025, 3:03 AM
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