MathDB

7

Part of 2012 239 Open Mathematical Olympiad

Problems(2)

Guessing the number

Source: 239 2012 S7

7/30/2020
Vasya conceived a two-digit number aa, and Petya is trying to guess it. To do this, he tells Vasya a natural number kk, and Vasya tells Petya the sum of the digits of the number kaka. What is the smallest number of questions that Petya has to ask so that he can certainly be able to determine Vasya’s number?
Circumscribed quadrilateral

Source: 239 2012 J7

7/30/2020
A circumscribed quadrilateral ABCDABCD is given. It is known that ∠ACB=∠ACD\angle{ACB} = \angle{ACD}. On the angle bisector of ∠C\angle{C}, a point EE is marked such that AE⊥BDAE \bot BD. Point FF is the foot of the perpendicular line from point EE to the side BCBC. Prove that AB=BFAB = BF.
geometry