MathDB

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Part of 2017 All-Russian Olympiad

Problems(4)

Planes and cities

Source: All Russian Olympiad 2017,Day1,grade 9,P1

5/3/2017
In country some cities are connected by oneway flights( There are no more then one flight between two cities). City AA called "available" for city BB, if there is flight from BB to AA, maybe with some transfers. It is known, that for every 2 cities PP and QQ exist city RR, such that PP and QQ are available from RR. Prove, that exist city AA, such that every city is available for AA.
combinatorics
Quotients

Source: All Russian Olympiad 2017,Day2,grade 9,P5

5/3/2017
There are n>3n>3 different natural numbers, less than (n−1)!(n-1)! For every pair of numbers Ivan divides bigest on lowest and write integer quotient (for example, 100100 divides 77 =14= 14) and write result on the paper. Prove, that not all numbers on paper are different.
number theory
Parabolas

Source: All Russian Olympiad 2017,Day1,grade 10,P1

5/3/2017
f1(x)=x2+p1x+q1,f2(x)=x2+p2x+q2f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2 are two parabolas. l1l_1 and l2l_2 are two not parallel lines. It is knows, that segments, that cuted on the l1l_1 by parabolas are equals, and segments, that cuted on the l2l_2 by parabolas are equals too. Prove, that parabolas are equals.
algebraconicsparabola
Rational cosin and sin

Source: All Russian Olympiad 2017,Day1,grade 11,P1

5/1/2017
S=sin⁡64x+sin⁡65xS=\sin{64x}+\sin{65x} and C=cos⁡64x+cos⁡65xC=\cos{64x}+\cos{65x} are both rational for some xx. Prove, that for one of these sums both summands are rational too.
number theorytrigonometry