MathDB

1

Part of 2003 Tournament Of Towns

Problems(5)

Dollars, Purses, and Pockets

Source: Tournament of Towns Spring 2003 - Junior O-Level - Problem 1

6/14/2011
20032003 dollars are placed into NN purses, and the purses are placed into MM pockets. It is known that NN is greater than the number of dollars in any pocket. Is it true that there is a purse with less than MM dollars in it?
combinatorics proposedcombinatorics
The match on the equation ax^2 + bx + c = 0 - always win

Source:

6/14/2011
Johnny writes down quadratic equation ax2+bx+c=0.ax^2 + bx + c = 0. with positive integer coefficients a,b,ca, b, c. Then Pete changes one, two, or none “++” signs to “−-”. Johnny wins, if both roots of the (changed) equation are integers. Otherwise (if there are no real roots or at least one of them is not an integer), Pete wins. Can Johnny choose the coefficients in such a way that he will always win?
quadraticsalgebra
Show that R / r > a / h in any triangular pyramid ABCD

Source: Tournament of Towns Spring 2003 - Senior A-Level - Problem 1

6/15/2011
A triangular pyramid ABCDABCD is given. Prove that Rr>ah\frac Rr > \frac ah, where RR is the radius of the circumscribed sphere, rr is the radius of the inscribed sphere, aa is the length of the longest edge, hh is the length of the shortest altitude (from a vertex to the opposite face).
geometry3D geometrypyramidspheregeometry unsolved
Arranging numbers in squares to get sum 120.

Source: ToT 2003 JO-1

6/26/2011
There is 3×4×53 \times 4 \times 5 - box with its faces divided into 1×11 \times 1 - squares. Is it possible to place numbers in these squares so that the sum of numbers in every stripe of squares (one square wide) circling the box, equals 120120?
geometryrectanglecombinatorics unsolvedcombinatorics
Every two terms are relatively prime in AP of 100 integers

Source: ToT 2003-JA-1

6/18/2011
An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?
arithmetic sequencenumber theoryrelatively primeprime numbersnumber theory proposed