6.
雙曲線\(\Gamma\)：\(xy=k\)，\(k<0\)，點\(P(2,2)\)，過\(P\)作\(\Gamma\)兩切線，切點為\(A\)、\(B\)點，若三角形\(\Delta PAB\)是正三角形，求\(k=\)
。
12.
小明在森林中迷了路，若繼續往前走經過5分鐘後會回到原地，若返回走則有一半的機會於5分鐘後回到原地，另一半的機會於10分鐘走出森林；假設小明向前走的機率為0.6，問小明能夠走出森林所花費的時間期望值為
。
相關問題
https://math.pro/db/viewthread.php?tid=784&page=1#pid1475
14.
There are two distinguishable flagpoles, and there are 19 flags, of which 10 are identical blue flags, and 9 are identical green flags. Let \(N\) be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when \(N\) is divided by 1000.
(2008AIMEII，連結有解答
https://artofproblemsolving.com/ ... Problems/Problem_12)
17.
多項式\(f(x)=x^{130}-1\)，\(g(x)=x^4-x^3+2x^2-x+1\)，求\(f(x)\)除以\(g(x)\)的餘式為
。
若\((x^{2000}-1)\)除以\((x^4+x^3+2x^2+x+1)\)之餘式為\(ax^3+bx^2+cx+d\)，則實數\(a+b+c+d\)之值=
。(最簡分數)
(99中興高中，
https://math.pro/db/viewthread.php?tid=1013&page=2#pid2533)
19.
Four regular hexagons surround a square with side length 1, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as \(m \sqrt{n} + p\), where \(m\), \(n\), and \(p\) are integers and \(n\) is not divisible by the square of any prime. What is \(m+n+p\)?
(2022AMC12B，連結有解答
https://artofproblemsolving.com/ ... Problems/Problem_25)
