MathDB

11.2

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

<ACB=? BL _|_AC, AL _|_DK , rectangle (I Soros Olympiad 1994-95 R1 11.2)

Source:

7/31/2021
Given a rectangle ABCDABCD with AB>BCAB> BC. On the side CDCD, take a point LL such that BLBL and ACAC are perpendicular. Let KK be the intersection point of segments BLBL and ACAC. It is known that segments ALAL. and DKDK are perpendicular. Find ∠ACB.\angle ACB.
anglesperpendiculargeometryrectangle
sin x <= sin (x+1)<=. ..<= sin (x+4) (I Soros Olympiad 1994-99 Round 2 11.2)

Source:

5/26/2024
Find the smallest positive xx for which holds the inequality sin⁡x≤sin⁡(x+1)≤sin⁡(x+2)≤sin(x+3)≤sin⁡(x+4).\sin x \le \sin (x+1)\le \sin (x+2)\le sin (x+3)\le \sin (x+4) .
algebrainequalitiestrigonometry
partition of set of all finite ordered sets of 0 and 1

Source: I Soros Olympiad 1994-95 Ukraine R2 11.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/6/2024
The set of all finite ordered sets of 00 and 1 1 is somehow partitioned into two disjoint classes. Prove that any infinite sequence of 00 and 11 can be cut into non-intersecting finite parts such that all of these parts (except perhaps the first) belong to the same class.
combinatorics