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 © | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: (7) Next: Blackboards for Thursday January 15 Previous: Blackboards for Tuesday January 13

# Handout for January 15, "Draw a Figure" (2)

Reading. Section 1.2 of Larson's textbook.

Next Quiz. A subset of the problems here and problems 1.2.5-1.2.10 in Larson's.

Note. From next week and on, our TA Gaurav Patil will be available in front of our classroom, MP 134, every Tuesday from 2:55PM until the beginning of class, to answer questions regarding the marking of quizes.

Problem 1. 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 2 (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 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. Prove

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

Problem 5 (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 6 (Larson's 7.1.14). In a convex quadrilateral, prove that the sum of the lengths of the diagonal lies between the perimeter and half the perimeter.

Problem 7 (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 8. 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.