MathDB

7

Part of 2001 Tournament Of Towns

Problems(4)

Find the Strategy

Source: ToT - 2001 Spring Junior A-Level #7

8/17/2011
Alex thinks of a two-digit integer (any integer between 1010 and 9999). Greg is trying to guess it. If the number Greg names is correct, or if one of its digits is equal to the corresponding digit of Alex’s number and the other digit differs by one from the corresponding digit of Alex’s number, then Alex says “hot”; otherwise, he says “cold”. (For example, if Alex’s number was 6565, then by naming any of 64,65,66,5564, 65, 66, 55 or 7575 Greg will be answered “hot”, otherwise he will be answered “cold”.)
(a) Prove that there is no strategy which guarantees that Greg will guess Alex’s number in no more than 18 attempts. (b) Find a strategy for Greg to find out Alex’s number (regardless of what the chosen number was) using no more than 2424 attempts. (c) Is there a 2222 attempt winning strategy for Greg?
number theory unsolvednumber theory
Chips in a Box

Source: ToT - 2001 Spring Senior A-Level #7

8/17/2011
Several boxes are arranged in a circle. Each box may be empty or may contain one or several chips. A move consists of taking all the chips from some box and distributing them one by one into subsequent boxes clockwise starting from the next box in the clockwise direction.
(a) Suppose that on each move (except for the first one) one must take the chips from the box where the last chip was placed on the previous move. Prove that after several moves the initial distribution of the chips among the boxes will reappear. (b) Now, suppose that in each move one can take the chips from any box. Is it true that for every initial distribution of the chips you can get any possible distribution?
combinatorics unsolvedcombinatorics
First Digit of 4

Source: ToT - 2001 Fall Junior A-Level #7

8/17/2011
It is given that 23332^{333} is a 101-digit number whose first digit is 1. How many of the numbers 2k2^k, 1≤k≤3321\le k\le 332 have first digit 4?
number theory unsolvednumber theory
Common Interior Points

Source: ToT - 2001 Fall Senior A-Level #7

8/17/2011
The vertices of a triangle have coordinates (x1,y1)(x_1,y_1), (x2,y2)(x_2,y_2) and (x3,y3)(x_3,y_3). For any integers hh and kk, not both 0, both triangles whose vertices have coordinates (x1+h,y1+k),(x2+h,y2+k)(x_1+h,y_1+k),(x_2+h,y_2+k) and (x3+h,y3+k)(x_3+h,y_3+k) has no common interior points with the original triangle.
(a) Is it possible for the area of this triangle to be greater than 12\tfrac{1}{2}?
(b) What is the maximum area of this triangle?
analytic geometrygeometrygeometric transformationabsolute valuegeometry unsolved