MathDB

1993 Bundeswettbewerb Mathematik

Part of Bundeswettbewerb Mathematik

Subcontests

(4)
3
2

infinitely many pairs of squares with 2 properties

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.

Bundeswettbewerb Mathematik 1993 Problem 2.3

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.
1
2
2
2
4
2