MathDB

4

Part of 2004 Tournament Of Towns

Problems(6)

b is a multiple of a^2

Source: Tournament of Towns Spring 2004 Junior O #4

5/22/2014
A positive integer a>1a > 1 is given (in decimal notation). We copy it twice and obtain a number b=aa‾b = \overline{aa} which happens to be a multiple of a2a^2. Find all possible values of b/a2b/a^2.
modular arithmeticnumber theory unsolvednumber theory
TOT 2004 Spring - Junior A-Level p4 collinear, common tangent

Source:

2/25/2020
Two circles intersect in points AA and BB. Their common tangent nearer BB touches the circles at points EE and FF, and intersects the extension of ABAB at the point MM. The point KK is chosen on the extention of AMAM so that KM=MAKM = MA. The line KEKE intersects the circle containing EE again at the point CC. The line KFKF intersects the circle containing FF again at the point DD. Prove that the points A,CA, C and DD are collinear.
tangentcirclesgeometrycollinear
TOT 2004 Spring - Senior O-Level p4 a_1^2,a_2^2,a_3^2 in arithm. progression

Source:

2/25/2020
Arithmetical progression a1,a2,a3,a4,...a_1, a_2, a_3, a_4,... contains a12,a22a_1^2 , a_2^2 and a32a_3^2 at some positions. Prove that all terms of this progression are integers.
IntegersArithmetic ProgressionSequencenumber theory
Construct a square!

Source: Tournament of towns, Junior B-Level paper, Fall 2004

12/25/2004
We have a circle and a line which does not intersect the circle. Using only compass and straightedge, construct a square whose two adjacent vertices are on the circle, and two other vertices are on the given line (it is known that such a square exists).
geometry unsolvedgeometry
Vanya and Petya choosing numbers x and y...

Source: Tournament of towns, Junior A-Level paper, Fall 2004

12/25/2004
Vanya has chosen two positive numbers, x and y. He wrote the numbers x+y, x-y, x/y, and xy, and has shown these numbers to Petya. However, he didn't say which of the numbers was obtained from which operation. Show that Petya can uniquely recover x and y.
geometrygeometric transformationalgebra unsolvedalgebra
TOT 2004 Fall - Senior A-Level p4 locus of circumcenters

Source:

2/25/2020
A circle with the center II is entirely inside of a circle with center OO. Consider all possible chords ABAB of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle AIBAIB.
geometryLocusCircumcentercircumcircle