MathDB

3

Part of 2017 All-Russian Olympiad

Problems(4)

Dwarfes and river

Source: All Russian Olympiad 2017,Day1,grade 9,P3

5/3/2017
There are 100100 dwarfes with weight 1,2,...,1001,2,...,100. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?
number theorycombinatorics
An isosceles triangle

Source: All Russian 2017,grade 9,day 2,P7

5/1/2017
In the scalene triangle ABCABC,∠ACB=60\angle ACB=60 and Ω\Omega is its cirumcirle.On the bisectors of the angles BACBAC and CBACBA points A′A^\prime,B′B^\prime are chosen respectively such that AB′∥BCAB^\prime \parallel BC and BA′∥ACBA^\prime \parallel AC.A′B′A^\prime B^\prime intersects with Ω\Omega at D,ED,E.Prove that triangle CDECDE is isosceles.(A. Kuznetsov)
circumcircleangle bisectorparallelgeometryIsosceles Triangle
Stones in heaps

Source: All Russian Olympiad 2017,Day1,grade 10,P3

5/3/2017
There are 3 heaps with 100,101,102100,101,102 stones. Ilya and Kostya play next game. Every step they take one stone from some heap, but not from same, that was on previous step. They make his steps in turn, Ilya make first step. Player loses if can not make step. Who has winning strategy?
combinatorics
Numbers on board [All-Russian 2017, Day 1, G11, P3]

Source: All Russian Olympiads 2017, Day1, Grade 11, Problem 3

5/1/2017
There are nn positive real numbers on the board a1,…,ana_1,\ldots, a_n. Someone wants to write nn real numbers b1,…,bnb_1,\ldots,b_n,such that: bi≥aib_i\geq a_i If bi≥bjb_i \geq b_j then bibj\frac{b_i}{b_j} is integer. Prove that it is possible to write such numbers with the condition b1⋯bn≤2n−12a1⋯an.b_1 \cdots b_n \leq 2^{\frac{n-1}{2}}a_1\cdots a_n.
number theoryalgebra