© | Dror Bar-Natan: Classes: 2015-16: Math 475 - Problem Solving Seminar: | (5) |
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**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).

- 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 $b=f(a)$. - 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

**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

**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.