MathDB

3

Part of 2002 Tournament Of Towns

Problems(6)

one zero means two zeroes

Source: Tournament of Towns,Spring 2002, Junior O Level, P3

5/13/2014
Show that if the last digit of the number x2+xy+y2x^2+xy+y^2 is 00 (where x,y∈Nx,y\in\mathbb{N} ) then last two digits are zero.
modular arithmeticalgebrapolynomialquadraticsnumber theory proposednumber theory
Maximising the area

Source: Tournament of Towns,Spring 2002, Junior A Level, P3

5/13/2014
Let EE and FF be the respective midpoints of BC,CDBC,CD of a convex quadrilateral ABCDABCD. Segments AE,AF,EFAE,AF,EF cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of ΔBAD\Delta BAD.
geometrygeometry proposed
Weighting once more

Source: Tournament of Towns,Spring 2002, Senior O Level, P3

5/14/2014
There are 66 pieces of cheese of different weights. For any two pieces we can identify the heavier piece. Given that it is possible to divide them into two groups of equal weights with three pieces in each. Give the explicit way to find these groups by performing two weightings on a regular balance.
combinatorics proposedcombinatorics
Hard problems that most of the students fail to solve

Source: Tournament of Towns, Fall 2002, Junior O Level, P3

5/15/2014
[*] A test was conducted in class. It is known that at least 23\frac{2}{3} of the problems were hard. Each such problems were not solved by at least 23\frac{2}{3} of the students. It is also known that at least 23\frac{2}{3} of the students passed the test. Each such student solved at least 23\frac{2}{3} of the suggested problems. Is this possible? [*] Previous problem with 23\frac{2}{3} replaced by 34\frac{3}{4}. [*] Previous problem with 23\frac{2}{3} replaced by 710\frac{7}{10}.
combinatorics proposedcombinatorics
Several straight lines

Source: Tournament of Towns, Fall 2002, Senior O Level, P3

5/17/2014
Several straight lines such that no two are parallel, cut the plane into several regions. A point AA is marked inside of one region. Prove that a point, separated from AA by each of these lines, exists if and only if AA belongs to an unbounded region.
geometry proposedgeometry
arcs and parallel sides

Source: Tournament of Towns, Fall 2002, Junior A Level, P3

5/17/2014
The vertices of a 50-gon50\text{-gon} divide a circumference into 5050 arcs, whose lengths are 1,2,…501,2,\ldots 50 in some order. It is known that any two opposite arcs (corresponding to opposite sides) differ by 2525. Prove that the polygon has two parallel sides.
combinatorics proposedcombinatorics