| 31. Let R be the reals. f:R2 → R is such that f(x, y) is a polynomial in y for each fixed x, and a polynomial in x for each fixed y. Is f(x, y) a polynomial in x and y? What if we replace R by the rationals Q? | |
| 32. n is a positive integer. Prove that we cannot find an integer m and an integer r > 1 such that n(n + 1)(n + 2) = mr. | |
| 33. K is a polygon in the plane with perimeter length P and enclosing an area A. Prove that we can find a disk radius > A/P which lies inside K. | |
| 34. A chooses an integer from {0, 1, 2, ... , 15} and must answer yes or no to each of 7 questions from B. He must answer at least 6 of the questions truthfully. Devise a set of 7 questions to allow B to determine the chosen integer. | |
| 35. R is a bounded region of the plane. Prove that we can find a point P and three lines through P which divide R into 6 pieces of equal area. | |
| 36. R is a bounded region of the plane. Prove that we can find a point P such that there is no line through P dividing R into R' and R'' with area R' equal to twice area R''. | |
| 37. x1, x2, x3, ... are positive reals. Prove that √(∑i>0 xi) + √(∑i>1 xi) + √(∑i>2 xi) + ... ≥ √(∑i>0 i2xi). | |
| 38. Prove that for any prime p we can find an integer n such that n8 = 16 (mod p). | |
| 39. All points of the plane are colored red, blue or green. Prove that we can find two points a distance 1 apart with the same color. | |
| 40. All points of the plane are colored red or blue. Prove that either we can find two red points a distance x apart for every x > 0, or we can find two blue points a distance x apart for every x > 0. |
Donald J Newman, A problem seminar (Springer, problem books in mathematics, 1982). Highly recommended. It comes with short hints, giving the key idea and solutions which are concise but motivated (in other words, they often explain how you might have found the solution yourself).
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John Scholes
jscholes@kalva.demon.co.uk
25 Sep 1999