Found problems: 2
2017 F = ma, 25
25) A planet orbits around a star S, as shown in the figure. The semi-major axis of the orbit is a. The perigee, namely the shortest distance between the planet and the star is 0.5a. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is its speed $v_2$ when it passes the perigee?
A) $v_2 = \frac{3}{\sqrt{5}}v_1$
B) $v_2 = \frac{3}{\sqrt{7}}v_1$
C) $v_2 = \frac{2}{\sqrt{3}}v_1$
D) $v_2 = \frac{\sqrt{7}}{\sqrt{3}}v_1$
E) $v_2 = 4v_1$
2016 USA Team Selection Test, 1
Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$.
Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$.
[i]Proposed by Maria Monks Gillespie[/i]