© | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: | (7) |
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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
Problem 4. Prove
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.