MathDB

4

Part of 1998 Tournament Of Towns

Problems(8)

TOT 1998 Spring AJ4 always tell the truth or always lie, on a circle

Source:

5/11/2020
A traveller visited a village whose inhabitants either always tell the truth or always lie. The villagers stood in a circle facing the centre of the circle, and each villager announced whether the person standing to his right is a truth-teller. On the basis of this information, the traveller was able to determine what fraction of the villagers were liars. What was this fraction?
(B, Frenkin)
combinatoricsgamegame strategy
TOT 1998 Spring OJ4 sum of product of digits of all 3-digit numbers

Source:

5/11/2020
For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result?
(G Galperin)
3-digitSumProductDigitsnumber theory
TOT 1998 Spring OS4 x^2 + y^2 +z^2= X and $|x| + |y| +|z| = D

Source:

5/11/2020
For some positive numbers A,B,CA, B, C and DD, the system of equations {x2+y2=A∣x∣+∣y∣=B\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases} has mm solutions, while the system of equations {x2+y2+z2=X∣x∣+∣y∣+∣z∣=D\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases} has nn solutions. If m>n>1m > n > 1, find mm and nn.
( G Galperin)
algebrasystem of equations
exists points such that BNC and DNA are equilateral, <AMB=<CMD=120^o

Source: Tournament of Towns Spring 1998 Seniors A p4

5/11/2020
A point MM is found inside a convex quadrilateral ABCDABCD such that triangles AMBAMB and CMDCMD are isoceles (AM=MB,CM=MDAM = MB, CM = MD) and ∠AMB=∠CMD=120o\angle AMB= \angle CMD = 120^o . Prove that there exists a point N such that trianglesBNC BNC and DNADNA are equilateral.
(I.Sharygin)
Equilateralgeometryisosceles
Lies of candidates

Source: ToT Junior O Level Autumn 1998

10/5/2008
Twelve candidates for mayor participate in a TV talk show. At some point a candidate said: "One lie has been told." Another said: "Now two lies have been told". "Now three lies," said a third. This continued until the twelfth said: "Now twelve lies have been told". At this point the moderator ended the discussion. It turned out that at least one of the candidates correctly stated the number of lies told before he made the claim. How many lies were actually told by the candidates?
combinatorics proposedcombinatorics
TOT 1998 Autumn AJ4 no 9 diagonals of a regular 25 -gon are concurent

Source:

5/11/2020
All the diagonals of a regular 2525-gon are drawn. Prove that no 99 of the diagonals pass through one interior point of the 2525-gon.
(A Shapovalov)
combinatorial geometrycombinatoricsdiagonalsconcurrent
TOT 1998 Autumn AS4 12 places at a round table for members of Jury,

Source:

5/11/2020
Twelve places have been arranged at a round table for members of the Jury, with a name tag at each place . Professor K. being absent-minded instead of occupying his place, sits down at the next place (clockwise) . Each of the other Jury members in turn either occupies the place assigned to this member or, if it has been already occupied, sits down at the first free place in the clockwise order. The resulting seating arrangement depends on the order in which the Jury members come to the table. How many different seating arrangements of this kind are possible?
(A Shapovalov)
combinatorics
TOT 1998 Autumn OS4 max product x_i when prod x_i=prod (1-x_i), x_i in (0,1)

Source:

5/11/2020
Among all sets of real numbers {x1,x2,...,x20}\{ x_1 , x_2 , ... , x_{20} \} from the open interval (0,1)(0, 1 ) such that x1x2...x20=(1−x1)(1−x2)...(1−x20)x_1x_2...x_{20}= ( 1 - x_1 ) ( 1 -x_2 ) ... (1 - x_{20} ) find the one for which x1x2...x20x_1 x_2... x_{20} is maximal.
(A Cherniatiev)
inequalitiesalgebramax