MathDB

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Part of 2011 Junior Balkan Team Selection Tests - Romania

Problems(4)

finite number of balanced numbers

Source: 2011 Romania JBMO TST 1.1

6/1/2020
Call a positive integer balanced if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number 385=5⋅7⋅11385 = 5 \cdot 7 \cdot 11 is balanced, while 275=52⋅11275 = 5^2 \cdot 11 is not. Prove that there exist only a finite number of balanced numbers.
number theorydecimal representationDigits
x_1 \le |x_2−x_3|, x_2 \le |x_3−x_4| , ... , x_{n-2}\le |x_{n-1}-x_n|

Source: 2011 Romania JBMO TST 2.1

6/1/2020
Determine a) the smallest number b) the biggest number n≥3n \ge 3 of non-negative integers x1,x2,...,xnx_1, x_2, ... , x_n, having the sum 20112011 and satisfying: x1≤∣x2−x3∣,x2≤∣x3−x4∣,...,xn−2≤∣xn−1−xn∣,xn−1≤∣xn−x1∣x_1 \le | x_2 - x_3 | , x_2 \le | x_3 - x_4 | , ... , x_{n-2} \le | x_{n-1} -x_n | , x_{n-1} \le | x_n - x_1 | and xn≤∣x1−x2∣x_n \le | x_1 - x_2 | .
inequalitiesalgebra
in prime factorization at least one of the prime factors has exponent 2

Source: 2011 Romania JBMO TST 3.1

6/1/2020
It is said that a positive integer n>1n > 1 has the property (pp) if in its prime factorization n=p1a1⋅...⋅pjajn = p_1^{a_1} \cdot ... \cdot p_j^{a_j} at least one of the prime factors p1,...,pjp_1, ... , p_j has the exponent equal to 22. a) Find the largest number kk for which there exist kk consecutive positive integers that do not have the property (pp). b) Prove that there is an infinite number of positive integers nn such that n,n+1n, n + 1 and n+2n + 2 have the property (pp).
number theoryprime factorizationprimesconsecutive
\tau (n) = n/3, number of its positive factors

Source: 2011 Romania JBMO TST 4.1

6/1/2020
For every positive integer nn let τ(n)\tau (n) denote the number of its positive factors. Determine all n∈Nn \in N that satisfy the equality τ(n)=n3\tau (n) = \frac{n}{3}
factorspositivenumber theory