Sum-closed subset average is at least (n+1)/2
M is the midpoint of BC and O is the point on the line AM such that OB is perpendicular to AB;
Topic: Combinatoria Metodo: Principio di estremalita Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1
Sum-closed subset average is at least (n+1)/2
M is the midpoint of BC and O is the point on the line AM such that OB is perpendicular to AB;
OQ perpendicular EF iff QE=QF
Q is an arbitrary point on the segment BC different from B and C;
Topic: Geometria piana Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1
The following table shows the results of the evaluation:
Q is an arbitrary point on the segment BC different from B and C;
Count numbers with three binary 1s; surjectivity of f
E lies on the line AB and F lies on the line AC such that E, Q, F are distinct and collinear. Prove that OQ is perpendicular to EF if and only if QE = QF. 3. For any positive integer k, let f(k) be the number of elements in the set {k + 1, k + 2, … , 2k} whose base 2 representation has precisely three 1s. • (a) Prove that, for each positive integer m, there exists at least one positive integer k such that f(k) = m. • (b) Determine all positive integers m for which there exists exactly one k with f(k) = m.
Topic: Combinatoria, Teoria dei Numeri Metodo: Analisi per casi, Conteggio combinatorio Abilita: Conteggio sistematico Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1
Count numbers with three binary 1s; surjectivity of f
E lies on the line AB and F lies on the line AC such that E, Q, F are distinct and collinear. Prove that OQ is perpendicular to EF if and only if QE = QF. 3. For any positive integer k, let f(k) be the number of elements in the set {k + 1, k + 2, … , 2k} whose base 2 representation has precisely three 1s. • (a) Prove that, for each positive integer m, there exists at least one positive integer k such that f(k) = m. • (b) Determine all positive integers m for which there exists exactly one k with f(k) = m.
Find all (m,n) with (n^3+1)/(mn-1) integer
Determine all ordered pairs (m, n) of positive integers such that n3 + 1 mn −1 is an integer.
Topic: Teoria dei Numeri Metodo: congruenze Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1
Find all (m,n) with (n^3+1)/(mn-1) integer
Determine the ordered pairs (m, n) of positive integers such that n3 + 1 mn −1 is an integer.
Find all f on (-1,inf) with functional and monotonicity conditions
Let S be the set of real numbers strictly greater than −1. Find all functions f : S →S satisfying the two conditions:
Topic: successioni Abilita: generalizzazione Area: Algebra e Analisi Fonte: apri PDF p.1
Find all f on (-1,inf) with functional and monotonicity conditions
Let S be the set of real numbers strictly greater than −1. Find all functions f : S →S satisfying the two conditions:
prime-product membership condition for infinite prime sets
- Show that there exists a set A of positive integers with the following property: For any infinite set S of primes there exist two positive integers m ∈A and n /∈A each of which is a product of k distinct elements of S for some k ≥2.
Topic: Teoria dei Numeri, Combinatoria Metodo: Conteggio combinatorio Abilita: generalizzazione Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1
prime-product membership condition for infinite prime sets
- Show that there exists a set A of positive integers with the following property: For any infinite set S of primes there exist two positive integers m ∈A and n /∈A each of which is a product of k distinct elements of S for some k ≥2.