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Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Find all positive integers $m$ and $n$ so that for any $x$ and $y$ in the interval $[m, n]$, the value of $5/x + 7/y$ will also be in $ [m, n]$. I evaluated the inequalities into $(ym-5)(xm-7) < 35$ $(yn-5)(xn-7) > 35$ but can't really think of what to do next. inequality share | cite | improve this question edited Jul 31 at 13:04 asked Jul 31 at 10:14 SuperMage1 567 1 9 Are $x,y$ assumed to be positive? – Dr. Sonnhard Graubner Jul 31 at 10:16 yes they are000 – SuperMage1 Jul 31 at 10:17 You need $12nle n$, which is a bit difficult for non-zero numbers. Where did you get this question? – Macavity Jul 31 at 12:38 pmo.ph/wp-content/uploads/2014/08/16th-PMO-Area.pdf – SuperMage1 Jul 31 at 13:02 Its a...

Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite In G2-continuous splines(connected curves), people say the curvature is the same at the junction point. In case of C2-continuous splines it is clear because; $ kappa := fracc' times c'' c' $ , and C2-continuity requires C1-continuity at the same time by construction. Then $ c'_1(t_0)=c'_2(t_0), ,, and ,, c''_1(t_0) = c''_2(t_0) ,, holds, , thus, $ $ ,, kappa_0(t_0) = kappa_1(t_0) ,, must ,, hold. $ But in case of G2-continuity, the identity of the curvature is not obvious. Rather than just saying that's the definition, is there a clear derivation to show it? Thanks :) differential-geometry algebraic-geometry euclidean-geometry analytic-geometry share | cite | improve this question edited Jul 31 at 10:39 John Ma 37.5k 9 36 69 asked Jul 31 at 10:17 Robin 139 6 add a comment  |  up vote 0 down vote favorite In ...

Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite On the picture, we have a point on the circle $(a, b)$ parametrized using the arc length and distance to diameter. As you can see, $a = rtheta$. I want to write polar coordinates, $(r, theta)$ in terms of $(a, b)$. As you can see, if we palce this circle to origin, this is quite like transformation from cartesian to polar coordinates but with one ceveat: $a$ is not $x$ coordinate value but arc length. geometry euclidean-geometry share | cite | improve this question asked Jul 31 at 10:21 meguli 358 2 9 add a comment  |  up vote 0 down vote favorite On the picture, we have a point on the circle $(a, b)$ parametrized using the arc length and distance to diameter. As you can see, $a = rtheta$. I want to write polar coordinates, $(r, theta)$ in terms of $(a, b)$. As you can see, if we palce this circle to origin, this is quite like transformation from ...