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Part of 2004 Tournament Of Towns
Problems(7)
Concurrent lines in a triangle
Source: Tournament of Towns Spring 2004 Junior O #1
5/22/2014
In triangle the bisector of angle , the perpendicular to side from its midpoint, and the altitude from vertex , intersect in the same point. Prove that the bisector of angle , the perpendicular to side from its midpoint, and the altitude from vertex also intersect in the same point.
geometry unsolvedgeometry
TOT 2004 Spring - Junior A-Level p1 power of 2 sum of arithm. progression
Source:
2/25/2020
The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.
Arithmetic ProgressionSumpower of 2number theory
TOT 2004 Spring - Senior O-Level p1 collinear wanted
Source:
2/25/2020
Segments and of the broken line are equal and are tangent to a circle with centre at the point . Prove that the point of contact of this circle with , the point and the intersection point of and are collinear.
geometrycollinear
10 consecutive numbers is divisible by 10
Source: Tournament of towns, Junior B-Level paper, Fall 2004
12/25/2004
Is it possible to arrange numbers from 1 to 2004 in some order so that the sum of any 10 consecutive numbers is divisble by 10?
number theory unsolvednumber theory
Rational triangles
Source: Tournament of towns, Junior A-Level paper, Fall 2004
12/25/2004
Let us call a triangle rational if each of its angles is a rational number when measured in degrees. Let us call a point inside triangle rational if joining it to the three vertices of the triangle we get three rational triangles. Show that any acute rational triangle contains at least three distinct rational points.
geometrygeometric transformationreflectionincentercircumcirclegeometry unsolved
Mutually inverse functions
Source: Tournament of towns, Senior A-Level paper, Fall 2004
12/25/2004
Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)
functiongeometryalgebra solvedalgebra
Three circles passing through X...
Source: Tournament of towns, Senior B-Level paper, Fall 2004
12/25/2004
Three circles pass through point X. Their intersection points (other than X) are denoted A, B, C. Let A' be the second point of intersection of line AX and the circle circumscribed around triangle BCX, and define similarly points B', C'. Prove that triangles ABC', AB'C, and A'BC are similar.
geometrygeometry solved