MathDB

3

Part of 1993 Bundeswettbewerb Mathematik

Problems(2)

infinitely many pairs of squares with 2 properties

Source: 1993 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

11/20/2022
There are pairs of square numbers with the following two properties: (1) Their decimal representations have the same number of digits, with the first digit starting is different from 00 . (2) If one appends the second to the decimal representation of the first, the decimal representation results another square number. Example: 1616 and 8181; 1681=4121681 = 41^2. Prove that there are infinitely many pairs of squares with these properties.
number theoryPerfect SquaresPerfect Square
Bundeswettbewerb Mathematik 1993 Problem 2.3

Source: Bundeswettbewerb Mathematik 1993 Round 2

10/10/2022
In the triangle ABCABC, let A′A' be the intersection of the perpendicular bisector of ABAB and the angle bisector of ∠BAC\angle BAC and define B′,C′B', C' analogously. Prove that
a) The triangle ABCABC is equilateral if and only if A′=B′.A' =B'.
b) If A′,B′A', B' and C′C' are distinct, we have ∠B′A′C′=90∘−12∠BAC.\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.
geometryperpendicular bisectorangle bisectorTriangle