MathDB

1

Part of MathLinks Contest 1st

Problems(7)

0111 n major cities , connected by railroads 1st edition Round 1 p1

Source:

5/9/2021
In a country there are nn major cities, n≥4n \ge 4, connected by railroads, such that each city is directly connected to each other city. Each railroad company in that country has but only one train, which connects a series of cities, at least two, such that the train doesn’t pass through the same city twice in one shift. The companies divided the market such that any two cities are directly1^1 connected only by one company. Prove that among any n+1n+1 companies, there are two which have no common train station or there are two cities that are connected by two trains belonging to two of these n+1n+1 companies.
1^1 directly connected means that they are connected by a railroad, without no other station between them
combinatorics1st edition
0141 combo geo 1st edition Round 4 p1

Source:

5/9/2021
Given are 40044004 distinct points, which lie in the interior of a convex polygon of area 11. Prove that there exists a convex polygon of area 12003\frac{1}{2003}, included in the given polygon, such that it does not contain any of the given points in its interior.
combinatoricsgeometry1st edition
0121 number theory 1st edition Round 2 p1

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5/9/2021
Let AA be a finite set of positive integers. Prove that there exists a finite set BB of positive integers such that A⊂BA \subset B and ∏x∈Bx=∑x∈Bx2\prod_{x \in B} x =\sum_{x \in B}x^2
number theory1st edition
0161 number theory 1st edition Round 6 p1

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5/9/2021
Let a,ma, m be two positive integers, a≠10ka \ne 10^k, for all non-negative integers kk and d1,d2,...,dmd_1, d_2, ... , d_m random decimal1^1 digits with d1>0d_1 > 0. Prove that there exists some positive integer nn for which the representation in the decimal base of the number ana^n begins with the digits d1,d2,...,dmd_1, d_2, ... , d_m in this order.
1^1 lesser or equal with 99
number theory1st edition
0131 2003 circus flees, each vertex of a convex polygon 1st edition Round 3 p1

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5/9/2021
A pack of 20032003 circus flees are deployed by their circus trainer, named Gogu, on a sufficiently large table, in such a way that they are not all lying on the same line. He now wants to get his Ph.D. in fleas training, so Gogu has thought the fleas the following trick: we chooses two fleas and tells one of them to jump over the other one, such that any flea jumps as far as twice the initial distance between him and the flea over which he is jumping. The Ph.D. circus committee has but only one task to Gogu: he has to make the his flees gather around on the table such that every flea represents a vertex of a convex polygon. Can Gogu get his Ph.D., no matter of how the fleas were deployed?
combinatorics1st edition
0151 geometry 1st edition Round 5 p1

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5/9/2021
In a triangle ABCABC, ∠B=70o\angle B = 70^o, ∠C=50o\angle C = 50^o. A point MM is taken on the side ABAB such that ∠MCB=40o\angle MCB = 40^o , and a point NN is taken on the side ACAC such that ∠NBC=50o\angle NBC = 50^o. Find ∠NMC\angle NMC.
geometry1st edition
0171 inequalities 1st edition Round 7 p1

Source:

5/9/2021
Prove that for every positive numbers x,y,zx, y, z the following inequality holds: 4x2+4x(y+z)+(y−z)2<4y2+4y(z+x)+(z−x)2+4z2+4z(x+y)+(x−y)2.\sqrt{4x^2 + 4x(y + z) + (y - z)^2} <\sqrt{4y^2 + 4y(z + x) + (z - x)^2}+\sqrt{4z^2 + 4z(x + y) + (x - y)^2}.
inequalities1st edition