MathDB

2

Part of 2016 Tournament Of Towns

Problems(3)

No roots but roots

Source: Tournament of Towns Spring 2016 Junior A-level

2/23/2017
Do there exist integers aa and bb such that : (a) the equation x2+ax+b=0x^2 + ax + b = 0 has no real roots, and the equation ⌊x2⌋+ax+b=0\lfloor x^2 \rfloor + ax + b = 0 has at least one real root? (2 points) (b) the equation x2+2ax+bx^2 + 2ax + b = 0 has no real roots, and the equation ⌊x2⌋+2ax+b=0\lfloor x^2 \rfloor + 2ax + b = 0 has at least one real root? 3 points (By ⌊k⌋\lfloor k \rfloor we denote the integer part of kk, that is, the greatest integer not exceeding kk.) Alexandr Khrabrov
floor functionalgebraquadratics
All lines touch same circle

Source: Tournament of Towns oral round p2

3/21/2016
On plane there is fixed ray ss with vertex AA and a point PP not on the line which contains ss. We choose a random point KK which lies on ray. Let NN be a point on a ray outside AKAK such that NK=1NK=1. Let MM be a point such that NM=1,M∈PKNM=1,M \in PK and M!=K.M!=K. Prove that all lines NMNM, provided by some point KK, touch some fixed circle.
geometryLocuscircles
Tiling with dominoes

Source: Tournament of Towns, 2016 Fall Tour, A Senior, Problem #2

4/22/2017
A natural number is written in each cell of an 8×88 \times 8 board. It turned out that for any tiling of the board with dominoes, the sum of numbers in the cells of each domino is different. Can it happen that the largest number on the board is no greater than 3232?
(N. Chernyatyev)
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
combinatorics