Math 289 – Winter 2006 – Problem Set 5
Due February 15, 2006
1.(10) There is public poll on a website “http://www.misterpoll.com/1941119654.html”
about the most important method of mathematical proof. The number of votes for each
option is displayed in percents (after the rounding to the closest integer). After a poll of a
student Kurt G¨odel for ”Proof By Physics Intuition” the number of votes for this option
remained at 7%. What is the smallest possible number of people who voted for this option
(including Mr. G¨odel)? You may assume that everybody voted only once for a single
option.
2.(10) Prove that for every integer n,n > 3, which is not divisible by three it is possible
to cut a chessboard n×nto one square 1 ×1 and rectangles 3 ×1.
3.(10) We will call a natural number “a magic number” if it is possible to write it as a
sum of two three-digit numbers with the same digits but written in the reverse order. An
example of a magic number is 1,413 since 1,413 = 756 + 657. The smallest possible magic
number is clearly 202. Find the total number of magic numbers. Also, show that the sum
of all magic numbers is 187,000.
4.(15) Let a, b, c and dare four real numbers which satisfy a+d=b+c.
a) Prove the inequality
(a−b)(c−d)+(a−c)(b−d)≥0.
b) Prove the inequality
(a−b)(c−d)+(a−c)(b−d)+(d−a)(b−c)≥0.
5.(20) Prove that for any four real numbers p, q, r, s, where q6= 1 and s6= 1 it holds:
Quadratic equations
x2+px +q= 0 , x2+rx +s= 0
have a common real root and their remaining roots are reciprocal if and only if
pr = (q+ 1)(s+ 1) a p(q+ 1)s=r(s+ 1)q .
(Here we count a double root of a quadratic equation twice.)
6.(20) Find for which integer mthere exist exactly 215 subsets Xof the set {1,2,3, . . . , 47}
with the following property: Number mis the smallest element of the set Xand for every
x∈Xeither x+m∈Xor x+m > 47.
7.(20) Let us consider arithmetic progressions of real numbers (xi)∞
i=1 and (yi)∞
i=1 which
have the same first term and satisfy (for some k > 1) the following equalities:
xk−1yk−1= 42 xkyk= 30 , xk+1 yk+1 = 16 .
Find all such progressions for which is the index klargest possible.
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