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[ G.Polya ] PART IV PROBLEMS, HINTS, SOLUTIONS --------------------------------------------------------------------------- This last part offers the reader additional opportunity for practice. The problems require no more preliminary knowledge than the reader could have acquired from a good high- school curriculum. Yet they are not too easy and not mere routine problems; some of them demand originality and ingenuity12)footnote:Except Problem 1 widely known, but too amusing to miss) all the problems are taken from the Stanford university Competitive Examinations in Mathematics (there are a few minor changes). Some of the problems were formerly published in The American Mathematical Monthly and/or The California mathematics coun= council bulletin. In the latter periodical also some solutions were published by the author; they appear appropriately rearranged in the sequel.) The hints offer indications leading to the result, mostly by quoting an appropriate sentence from the list; to a very attentive reader ready to pick up suggestions they may reveal the key idea of the solution. The solutions bring not only the answer but also the procedure leading to the answer, although, of course, the reading has to supply some of the details.,. Some solutions try to open up some further outlook by a few words placed at the end. The reader who has earnestly tried to solve the problem has the best chance to profit by the hint and the solution. If he obtains the result by his own means, eh may learn something by comparing his method with the method given in print. I f, after a serious effort,t he is incline3ed to give up, the hint may supply him with the missing idea. If even the hint does not help, he may look at the solution, try to isolate the key idea, put the look aside, and then try to work out the solution. PROBLEMS 1. A bear, starting from point P, walked one mile due south. Then he changed direction and walked one mile due east. Then he turned again to the left and walked one mile due north, and arrived exactly at the point P he started from. What was the colour of the bear? 2. Bob wants a piece of land, exactly level, which has four boundary lines. Two boundary lines run exactly north-south, the two others exactly east-west, and each boundary line measures exactly 100 feet. Can bob buy such a piece of land in the U.S.? 3. Bob has 10 pockets and 44 silver dollars. He wants to put his dollars into his pockets so distributed that each pocket contains a different number of dollars. Can he do so? 4. To number the pages of a bulky volume, the printer used 2989 digits. How many pages has the volume? 5. Among grandfather's papers was bill was found: 72 Turkeys $_67.9_ The first and last digit of the number that obviously represented the total priced of those fowls are replaced here by blanks, for they have faded and are now illegible. What are the two faded digits and what was the price of one turkey? 6. Given a regular hexagon and a point in its plane. Draw a straight line through the given point that divides the given hexagon into two parts of equal area. 7. Given a square. Find the locus of the points from which the square is seen under such an angle (a) of 90 degrees b) of 45 degrees. (Let P be a point outside the square, but in the same plane. The smallest angle with vertex P con- training the square is the "angle under which the square is seen" from P.) Sketch clearly both loci and give a full description. 8. Call axis" of a solid a straight line joining two points of the surface of the solid and such that the solid rotated about this line through an angle which is greater than 0 degrees and less than 360 degrees coincides with itself. Find the axes of a cube. Describe9 clearly the location of the axes, find the angle of the rotation associated with each . Assuming that the edge of each cube is of unit length, compute the arithmetic mean of the lengths of the axes. 9. In a tetrahedron (which is not necessarily regular) two opposite edges have the same length a and they are perpendicular to each other. moreover they are each per perpendicular to a line of length b which joins their mid-= points. Express the volume of the tetrahedron in terms of a and b, and prove your answer. 10. The vertex of a pyramid opposite the base is called the a apex. A) let us call pyramid "isosceles" if its apex is at the same3e distance from all vertices of the base adopting this definition, prove that the base of an isosceles pyramid is inscribed in a circle the centre of which is the foot of the pyramid's altitude. b) now let us call a pyramid "isosceles" if its apex is at the same (perpendicular) distance from all sides of the base. Adopting this definition *I(different form the foregoing) prove that the base of an isosceles pyramid is circumscribed about a circle the centre of which is the foot of the pyramid's altitude. 11. Find x,y,u, and v, satisfying the system of four equations: x + 7y + 3v + 5u = 16 8x + 4y + 6v + 2u = -16 2x + 8y + 4v + 8u = 16 5x + 3y + 7v + u = -16 (this may look long and boring : look for a short cut.) 12. Bob, peter and Pual travel together. Peter and Paul are good hikers, each walk p miles per hour. Bob has a bad food and drives a small car in which two people can ride, but not three; the car covers c miles per hour. The three friends adopted the following scheme they start together, Pal rides in the car with bob, peter walks. After awhile, bob drops Paul, who walks on, bob returns to pick up peter, and then bob and peter ride in the car till they overtake Paul. At this point they change: Paul rides and peter walks just as they st started and the whole procedure is repeated as often as necessary. a) how much progress (how many miles) does the company make per hour? b) though which fraction oft he travel time does the car carry just one man? c) check the extreme cases p =0 and p=c 13. three numbers are in arithmetic progression, three other numbers in geometric progression,m adding the corresponding-- responding terms of these progressions successively we obtain 85, 76, and 84 respectively, and , adding all three terms of the arithmetic progression, we obtain 126. Find the terms of both progressions. 14. Determine m so that the equation in x ^4 - (3m+2)x^2 +m^2 = 0 has four real roots in arithmetic progression. 15. The length of the perimeter of a right triangle is 60 inches and the length of the altitude perpendicular to the hypotenuse is 12 inches. Find the sides. 16. From the peak of a mountain you see two points A and B, in the plain. The lines of vision, directed to these points, include the angle y. The inclination of the first line of vision to a horizontal plane is a, that of the second line B. it is known that the points A and B are on the same level and that the distance between them is c. Express the elevation x of the peak above the common level of A and B in terms of the angles a, b, y, and the distance c. 17. Observe that the value of 1/2! = 2/3! + 3/4! + ... _+ n/(n+1)! is 1/2,5/6,23/24, for n=1,2,3, respectively guess the general law (by observing more values if necessary) and prove your guess 18. Consider the table 1 = 1 3 + 5 = 8 7 + 9 + 11 = 27 13 + 15+17 + 19 = 64 21+ 23 +25 +27 +29 =125 Guess s the general law suggested by these examples, ex press it in suitable mathematical notation, and prove it. 19. The side of a regular hexagon is of length n(n is an integer). BG equidistant parallels to is sides the hexagon is divided into T equilateral triangles each of which has sides of length 1. Let V denote the number of vertices appearing in this division, and L the number of bound- boundary lines of length 1. (a boundary line belongs to one or two triangles, a vertex to two or more triangles.) When n = 1, which is the simplest case, T=6, V=7,L=12. consider the general case and express T,V, and L in terms of n(Guessing is good, proving is better.)) 20. In how many ways can you change one dollar? (The way of changing is determined if it is known how many coins of each kind-cents, nickels, dimes, quarters half dollars are used.)