MathDB

3

Part of 2013 Iran MO (2nd Round)

Problems(2)

Height intersects circumcircle and 2 equal segments

Source: Iran second round- 2013- P3

5/4/2013
Let MM be the midpoint of (the smaller) arc BCBC in circumcircle of triangle ABCABC. Suppose that the altitude drawn from AA intersects the circle at NN. Draw two lines through circumcenter OO of ABCABC paralell to MBMB and MCMC, which intersect ABAB and ACAC at KK and LL, respectively. Prove that NK=NLNK=NL.
geometrycircumcircletrapezoidparallelogramtrigonometrygeometric transformationreflection
recursive sequence defined by floor function

Source: Iran second round- 2013- P6

5/4/2013
Let {an}n=1∞\{a_n\}_{n=1}^{\infty} be a sequence of positive integers for which an+2=[2anan+1]+[2an+1an]. a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. Prove that there exists a positive integer mm such that am=4a_m=4 and am+1∈{3,4}a_{m+1} \in\{3,4\}.
Note. [x][x] is the greatest integer not exceeding xx.
functioninequalitiesalgebra proposedalgebraRecurrence