MathDB

2

Part of 2007 Tournament Of Towns

Problems(10)

Delete digits (ToT 2007-Spring-O2)

Source:

9/2/2011
Two 20072007-digit numbers are given. It is possible to delete 77 digits from each of them to obtain the same 20002000-digit number. Prove that it is also possible to insert 77 digits into the given numbers so as to obtain the same 20142014-digit number.
Find area of triangle MND (ToT 2007-Spring-A2)

Source:

9/2/2011
K,L,MK, L, M and NN are points on sides AB,BC,CDAB, BC, CD and DADA, respectively, of the unit square ABCDABCD such that KMKM is parallel to BCBC and LNLN is parallel to ABAB. The perimeter of triangle KLBKLB is equal to 11. What is the area of triangle MNDMND?
geometryperimeter
Polynomial has three roots in (0,2), show that -2 <p+q r< 0

Source: Tournament of Towns 2007 - Spring - Senior O-Level - P2

9/2/2011
The polynomial x3+px2+qx+rx^3 + px^2 + qx + r has three roots in the interval (0,2)(0,2). Prove that −2<p+q+r<0-2 <p + q + r < 0.
algebrapolynomialalgebra proposed
Is F necessarily a circle?

Source: Tournament of Towns 2007 - Spring - Senior A-Level - P2

9/3/2011
A convex figure FF is such that any equilateral triangle with side 11 has a parallel translation that takes all its vertices to the boundary of FF. Is FF necessarily a circle?
geometrygeometric transformationcombinatorics proposedcombinatorics
Is it possible to write x^2 on the blackboard?(ToT2007-Fall)

Source:

9/3/2011
Initially, the number 11 and a non-integral number xx are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write x2x^2 on the blackboard in a finite number of moves?
calculusintegration
Peter and Basil playing a game

Source: Tournament of Towns 2007 - Fall - Junior A-Level - P2

9/3/2011
(a) Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other.
(b) Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?
greatest common divisorleast common multiplecombinatorics unsolvedcombinatorics
Almost Right Angle Triangles

Source: Fall 2007 Tournament of Towns Junior P-Level #2

3/31/2015
Let us call a triangle “almost right angle triangle” if one of its angles differs from 90∘90^\circ by no more than 15∘15^\circ. Let us call a triangle “almost isosceles triangle” if two of its angles differs from each other by no more than 15∘15^\circ. Is it true that that any acute triangle is either “almost right angle triangle” or “almost isosceles triangle”?
(2 points)
geometry
Prove that circumradii of triangles are the same

Source: Tournament of Towns 2007 - Fall - Senior A-Level - P2

9/4/2011
Let K,L,MK, L, M and NN be the midpoints of the sides AB,BC,CDAB, BC, CD and DADA of a cyclic quadrilateral ABCDABCD. Let PP be the point of intersection of ACAC and BDBD. Prove that the circumradii of triangles PKL,PLM,PMNPKL, PLM, PMN and PNKPNK are equal to one another.
geometrycyclic quadrilateralgeometry proposed
Is it possible to write x^2 or xy on the blackboard?

Source: Tournament of Towns 2007 - Fall - Senior O-Level - P2

9/4/2011
Initially, the number 11 and two positive numbers xx and yy are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write on the blackboard, in a finite number of moves, the number a) x2x^2; b) xyxy?
combinatorics unsolvedcombinatorics
Missing Multiplication

Source: Fall 2007 Tournament of Towns Senior P-Level #2

3/31/2015
A student did not notice multiplication sign between two three-digit numbers and wrote it as a six-digit number. Result was 7 times more that it should be. Find these numbers.
(2 points)
algebra