Tournament of Towns, 1981-1982, Spring

• Question

  Find all numbers that are divisible by 30 and have exactly 30 distinct divisors (enter the number of such numbers to check your answer)

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Prime Numbers -> Prime Factorization

• Question

  In a quadrilateral, the lengths of all diagonals and all sides are less than 1. Prove that the quadrilateral can be covered by a circle with a radius of 0.9.

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  Topics:

  Geometry -> Plane Geometry -> Circles Minimum and Maximum Problems / Optimization Problems

• Question

  In a certain country, there are more than 101 cities. The capital is connected by flight routes to 100 cities, and every city other than the capital is connected by flight routes to exactly 10 cities. It is given that from any city, it is possible to reach any other city (possibly not by a direct route). Prove that it is possible to close half of the flight routes leading to the capital such that the possibility of reaching any city from any other city is preserved.

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  Topics:

  Combinatorics -> Graph Theory Logic -> Reasoning / Logic Proof and Example

• Question

  Given the sequence `1 , 1/2 ,1/3 ,1/4 ,1/5,...`, does there exist an arithmetic sequence composed of terms from the aforementioned sequence?

  1. Of length 5

  2. Of any length

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Arithmetic -> Fractions Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM)