Sum of k times permutations with k fixed points equals n!
Let pn(k) be the number of permutations of the set {1, … , n}, n ≥1, which have exactly k fixed points. Prove that n X k=0 k · pn(k) = n!. (Remark: A permutation f of a set S is a one-to-one mapping of S onto itself. An element i in S is called a fixed point of the permutation f if f(i) = i.)
Topic: Combinatoria Metodo: Doppio conteggio Abilita: Conteggio sistematico Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1
Sum of k times permutations with k fixed points equals n!
Let pn(k) be the number of permutations of the set {1, … , n}, n ≥1, which have exactly k fixed points. Prove that n x k=0 k · pn(k) = n!. (Note: A permutation f of a set S is a one-to-one mapping of S onto itself. An element i in S is called a fixed point of the permutation f if f(i) = i.)
Quadrilateral AKNM and triangle ABC have equal areas
In an acute-angled triangle ABC the interior bisector of the angle A intersects BC at L and intersects the circumcircle of ABC again at N. From point L perpendiculars are drawn to AB and AC, the feet of these perpendiculars being K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal areas.
Topic: Geometria piana Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1
Quadrilateral AKNM and triangle ABC have equal areas
In an acute-angled triangle ABC the interior bisector of the angle A intersects BC at L and intersects the circumcircle of ABC again at N. From point L perpendiculars are drawn to AB and AC, the feet of these perpendiculars being K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal areas.
Bounded integer combination of unit-norm reals is small
Let x1, x2, … , xn be real numbers satisfying x2 1 + x2 2 + · · + x2 n = 1. Prove that for every integer k ≥2 there are integers a1, a2, … , an, not all 0, such that |ai| ≤k −1 for all i and |a1x1 + a1x2 + · · + anxn| ≤(k −1)√n kn −1 .
28th International Mathematical Olympiad Havana, Cuba Day II July 11, 1987
Topic: Algebra Metodo: Principio dei cassetti Area: Algebra e Analisi Fonte: apri PDF p.1
Bounded integer combination of unit-norm reals is small
Let x1, x2, … , xn be real numbers satisfying x2 1 + x2 2 + · · + x2 n = 1. Prove that for every integer k ≥2 there are integers a1, a2, … An, not to 0, such that k −1 for all i and a1x1 + a1x2 + · · + anxn t ≤(k −1) √n kn −1.
28th International Mathematical Olympiad Havana, Cuba Day II July 11, 1987
No function on naturals with f(f(n))=n+1987
Prove that there is no function f from the set of non-negative integers into itself such that f(f(n)) = n + 1987 for every n.
Topic: successioni Metodo: monovarianti Area: Algebra e Analisi Fonte: apri PDF p.2
This is a list of the countries of the European Economic Area.
Prove that there is no function f from the set of non-negative integers into itself such that f(f(n)) = n + 1987 for every n.
n points with irrational distances and rational triangle areas
Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
Topic: Geometria piana, Teoria dei Numeri Metodo: Metodo delle coordinate Area: Aritmetica e Teoria dei Numeri, Geometria Fonte: apri PDF p.2
n points with irrational distances and rational triangle areas
Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
Prime-generating quadratic extends primality to larger range
Let n be an integer greater than or equal to 2. Prove that if k2 + k + n is prime for all integers k such that 0 ≤k ≤ p n/3, then k2 + k + n is prime for all integers k such that 0 ≤k ≤n −2.
Topic: Teoria dei Numeri Metodo: congruenze Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.2
Prime-generating quadratic extends primality to larger range
Let n be an integer greater than or equal to 2. Prove that if k2 + k + n is prime for all integers k such that 0 ≤k ≤ p n/3, then k2 + k + n is prime for all integers k such that 0 ≤k ≤n −2.