As the Kubiks head homeward, away from the beach in the family van, Jerry decides to take a different route away from the beach than the one they took to get there. The route involves lots of twists and turns, prompting Hannah to wonder aloud if Jerry's "shortcut" will save any time at all. Michael offers up a problem as an analogy to his father's meandering: "Suppose dad drives around, making right-angled turns after $\textit{every}$ mile. What is the farthest he could get us from our starting point after driving us $500$ miles assuming that he makes exactly $300$ right turns?" "Sounds almost like an energy efficiency problem," notes Hannah only half jokingly. Hannah is always encouraging her children to think along these lines. Let $d$ be the answer to Michael's problem. Compute $\lfloor d\rfloor$.

If a right triangle is drawn in a semicircle of radius $1/2$ with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area?

Suppose all the pairs of a positive integers from a finite collection \[A=\{a_{1}, a_{2}, \cdots \}\] are added together to form a new collection \[A^{*}=\{a_{i}+a_{j}\;\; \vert \; 1 \le i < j \le n \}.\] For example, $A=\{ 2, 3, 4, 7 \}$ would yield $A^{*}=\{ 5, 6, 7, 9, 10, 11 \}$ and $B=\{ 1, 4, 5, 6 \}$ would give $B^{*}=\{ 5, 6, 7, 9, 10, 11 \}$. These examples show that it's possible for different collections $A$ and $B$ to generate the same collections $A^{*}$ and $B^{*}$. Show that if $A^{*}=B^{*}$ for different sets $A$ and $B$, then $|A|=|B|$ and $|A|=|B|$ must be a power of $2$.

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying \[\forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \quad \text{and} \quad \lim_{x \to 0} g(x)=1.\] Find the derivative of $f$ in an arbitrary point $x.$

For which positive integer $n$ is the quantity $\frac{n}{3} + \frac{40}{n}$ minimized? [i]Proposed by Eugene Chen[/i]

Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.

Let be a twice-differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the properties that: $ \text{(i) supp} f''=f\left(\mathbb{R}\right) $ $ \text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R}\quad\forall x\in\mathbb{R}\quad f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right) $ Prove that: [b]a)[/b] any such $ g $ is injective. [b]b)[/b] $ f $ is of class $ C^{\infty } , $ and for any natural number $ n, $ any real number $ x $ and any such $ g, $ $$f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) . $$ [i]Laurențiu Panaitopol[/i]

Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}\]

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]

Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$. (Here $\sec''$ means the second derivative of $\sec$).