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 © | Dror Bar-Natan: Classes: 2015-16: Math 475 - Problem Solving Seminar: (5) Next: Blackboards for Tuesday January 19 Previous: Blackboards for Thursday January 14

# Handout for January 19, "Draw a Figure"

Reading. Section 1.2 of Larson's textbook.

Next Quiz. A subset of the problems here and problems 1.2.6-1.2.10 in Larson's.

Problem 1 (Larson's 1.2.2). A particle moving on a straight line starts at rest and attains a velocity $v_0$ after traversing a distance $s_0$. If the motion is such that the acceleration is never increasing, find the maximum time for the transverse.

Problem 2 (Young's Inequality).

1. Let $f\colon{\mathbb R}\to{\mathbb R}$ be an increasing continuous function satisfying $f(0)=0$, and let $f^{-1}$ be the inverse function of $f$. Then for every $a,b\geq 0$,
$\displaystyle ab\leq\int_0^a f(x)dx+\int_0^b f^{-1}(y)dy,$
and equality hold iff $b=f(a)$.
2. Prove that if $a$ and $b$ are non-negative real numbers and $p$ and $q$ are positive real numbers such that $\frac1p+\frac1q = 1$, then
$\displaystyle ab\leq\frac{a^p}{p} + \frac{b^q}{q}.$

Problem 3 (Larson's 1.2.3). If $a$ and $b$ are positive integers with no common factor, show that

$\displaystyle \left\lfloor\frac{a}{b}\right\rfloor + \left\lfloor\frac{2a}{b}\right\rfloor + \left\lfloor\frac{3a}{b}\right\rfloor + \cdots + \left\lfloor\frac{(b-1)a}{b}\right\rfloor = \frac{(a-1)(b-1)}{2}.$

Problem 4. One hundred indistinguishable ants are dropped on a hoop of diameter $1$. Each ant is traveling either clockwise or counterclockwise with a constant speed of $1$ meter per minute. When two ants meet, they bounce off each other and reverse directions. Will the ants ever return to their original configuration? After how many minutes?

Problem 5. Cars $A$, $B$, $C$, and $D$ travel in the Sahara Desert (an infinite boundless flat plane), each one at a constant speed and direction (though these constants are not all the same). It is given that cars $A$ and $B$ meet - namely, arrive to the same location at the same time (and let's pretend that they simply continue driving through each other without crashing). Likewise it is given that cars $A$ and $C$ meet, cars $A$ and $D$ meet, cars $B$ and $C$ meet, and cars $B$ and $D$ meet. Do cars $C$ and $D$ necessarily meet?

Problem 6. Prove

$\displaystyle \sum_{k=0}^n\binom{n}{k}^2 = \binom{2n}{n}.$

Problem 7 (Larson's 1.2.5). Two poles, with heights $a$ and $b$, are a distance $d$ apart (on level ground). A bird wishes to fly from the top of the first pole to the top of the second pole, touching the ground in between at some point $P$. Where should $P$ be located so that the bird trajectory will be the shortest?

Problem 8 (Larson's 7.4.19). Noting that the function $f(x)=\sqrt{x}$ is concave, show that if $a,b,c$ are positive and satisfy $c>a\cos^2\theta+b\sin^2\theta$, then $\sqrt{c}>\sqrt{a}\cos^2\theta+\sqrt{b}\sin^2\theta$.

Problem 9. Use the back of this page to draw a figure of something interesting. The best figures will be placed somewhere on this class' web site.