MathDB

2

Part of 2008 Tournament Of Towns

Problems(8)

TT2008 Junior O-Level - P2

Source:

9/4/2010
Twenty-fi ve of the numbers 1,2,⋯ ,501, 2, \cdots , 50 are chosen. Twenty- five of the numbers51,52,⋯ ,100 51, 52, \cdots, 100 are also chosen. No two chosen numbers diff er by 00 or 5050. Find the sum of all 5050 chosen numbers.
combinatorics proposedcombinatorics
TT2008 Junior A-Level - P2

Source:

9/4/2010
Each of 44 stones weights the integer number of grams. A balance with arrow indicates the di fference of weights on the left and the right sides of it. Is it possible to determine the weights of all stones in 44 weighings, if the balance can make a mistake in 11 gram in at most one weighing?
combinatorics unsolvedcombinatorics
TT2008 Senior O-Level - P2

Source:

9/4/2010
Solve the system of equations (n>2)(n > 2)
 x1+x2+x3+⋯+xn=x2+x3+x4+⋯+xn+x1=⋯=xn+x1+x2+⋯+xn−1,\begin{array}{c}\ \sqrt{x_1}+\sqrt{x_2+x_3+\cdots+x_n}=\sqrt{x_2}+\sqrt{x_3+x_4+\cdots+x_n+x_1}=\cdots=\sqrt{x_n}+\sqrt{x_1+x_2+\cdots+x_{n-1}} \end{array}, x1−x2=1.x_1-x_2=1.
quadraticsalgebrasystem of equationsalgebra proposed
TT2008 Senior A-Level - P2

Source:

9/4/2010
Space is dissected into congruent cubes. Is it necessarily true that for each cube there exists another cube so that both cubes have a whole face in common?
geometry3D geometryanalytic geometrygeometry proposed
2008 ToT Spring Junior O P2 10 equal segments, interset at ratio 3:4

Source:

3/7/2020
There are ten congruent segments on a plane. Each intersection point divides every segment passing through it in the ratio 3:43:4. Find the maximum number of intersection points.
ratiocombinatorial geometrycombinatoricssegments
2008 ToT Spring Junior A P2 if AK = AO and KM = MC, then AM = KB

Source:

2/26/2020
A line parallel to the side ACAC of triangle ABCABC cuts the side ABAB at KK and the side BCBC at MM. OO is the intersection point of AMAM and CKCK. If AK=AOAK = AO and KM=MCKM = MC, prove that AM=KBAM = KB.
geometryequal segments
2008 ToT Spring Senior O P2 2008(lcm of 1,2,...,m)=lcm of 1,2,...,n

Source:

3/7/2020
Can it happen that the least common multiple of 1,2,...,n1, 2,... , n is 20082008 times the least common multiple of 1,2,...,m1, 2, ... , m for some positive integers mm and nn ?
number theoryleast common multiple
2008 ToT Spring Senior A P2 game on the real line, strategy wanted

Source:

3/7/2020
Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number xx where 0<x<10 < x < 1. In each move, Brian chooses a positive number dd. Alice must move the checker to either x+dx + d or x−dx - d. If it lands on 00 or 11, Brian wins. Otherwise the game proceeds to the next move. For which values of xx does Brian have a strategy which allows him to win the game in a finite number of moves?
game strategygamecombinatorics