MathDB

1

Part of 1993 IberoAmerican

Problems(2)

8th ibmo - mexico 1993/q1.

Source: Spanish Communities

5/7/2006
A number is called capicua if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: 8,23432,64468, 23432, 6446. Let x1<x2<⋯<xi<xi+1,⋯x_1<x_2<\cdots<x_i<x_{i+1},\cdots be the sequence of all capicua numbers. For each ii define yi=xi+1−xiy_i=x_{i+1}-x_i. How many distinct primes contains the set {y1,y2,…}\{y_1,y_2, \ldots\}?
number theory proposednumber theory
8th ibmo - mexico 1993/q4.

Source: Spanish Communities

5/7/2006
Let ABCABC be an equilateral triangle and Γ\Gamma its incircle. If DD and EE are points on the segments ABAB and ACAC such that DEDE is tangent to Γ\Gamma, show that ADDB+AEEC=1\frac{AD}{DB}+\frac{AE}{EC}=1.
trigonometrygeometry proposedgeometry