Find values of BC2+CA2+AB2 and locus of BC midpoint
Consider two coplanar circles of radii R and r (R > r) with the same center. Let P be a fixed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C. The perpendicular l to BP at P meets the smaller circle again at A. (If l is tangent to the circle at P then A = P.) (i) Find the set of values of BC2 + CA2 + AB2. (ii) Find the locus of the midpoint of BC.
Topic: Geometria piana Metodo: Metodo delle coordinate Area: Geometria Fonte: apri PDF p.1
Find values of BC2+CA2+AB2 and locus of BC midpoint
Consider two coplanar circles of radii R and r (R > r) with the same center. Let P be a fixed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C. The perpendicular l to BP at P meets the smaller circle again at A. (If l is tangent to the circle at P then A = P.) (i) Find the set of values of BC2 + CA2 + AB2. (ii) Find the locus of the midpoint of BC.
when can elements be labeled 0/1 evenly
Let n be a positive integer and let A1, A2, … , A2n+1 be subsets of a set B. Suppose that (a) Each Ai has exactly 2n elements, (b) Each Ai ∩Aj (1 ≤i < j ≤2n + 1) contains exactly one element, and (c) Every element of B belongs to at least two of the Ai. For which values of n can one assign to every element of B one of the numbers 0 and 1 in such a way that Ai has 0 assigned to exactly n of its elements?
Topic: Combinatoria Metodo: parita Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1
when can elements be labelled 0/1 evenly
Let n be a positive integer and let A1, A2, … , A2n+1 be subsets of a set B. Suppose that (a) Each Ai has exactly 2n elements, (b) Each Ai ∩Aj (1 ≤i < j ≤2n + 1) contains exactly one element, and (c) Every element of B belongs to at least two of the Ai. For which values of n can one assign to every element of B one of the numbers 0 and 1 in such a way that Ai has 0 assigned to exactly n of its elements?
Count n<=1988 with recursively-defined f(n)=n
A function f is defined on the positive integers by f(1)
1, f(3) = 3, f(2n)
f(n), f(4n + 1)
2f(2n + 1) −f(n), f(4n + 3)
3f(2n + 1) −2f(n), for all positive integers n. Determine the number of positive integers n, less than or equal to 1988, for which f(n) = n.
29th International Mathematical Olympiad Canberra, Australia Day II
Topic: successioni, Teoria dei Numeri Metodo: Ricorsione Abilita: Riconoscimento di pattern Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1
Count n<=1988 with recursively-defined f(n)=n
A function f is defined on the positive integers by f(1) = 1, f(3) = 3, f(2n) = f(n), f(4n + 1) = 2f(2n + 1) −f(n), f(4n + 3) = 3f(2n + 1) −2f(n), for all positive integers n. Determine the number of positive integers n, less than or equal to 1988, for which f(n) = n.
29th International Mathematical Olympiad Canberra, Australia Day II
Inequality solution set is intervals of total length 1988
Show that set of real numbers x which satisfy the inequality 70 X k=1 k x −k ≥5 4 is a union of disjoint intervals, the sum of whose lengths is 1988.
Topic: Algebra Metodo: Analisi per casi Abilita: Manipolazione algebrica Area: Algebra e Analisi Fonte: apri PDF p.2
Inequality solution set is intervals of total length 1988
Show that set of real numbers x which satisfy the inequality 70 X k=1 k x −k ≥5 4 is a union of disjoint intervals, the sum of whose lengths is 1988.
prove area S>=2T
ABC is a triangle right-angled at A, and D is the foot of the altitude from A. The straight line joining the incenters of the triangles ABD, ACD intersects the sides AB, AC at the points K, L respectively. S and T denote the areas of the triangles ABC and AKL respectively. Show that S ≥2T.
Topic: Geometria piana, Disuguaglianze Metodo: Disuguaglianze classiche Abilita: Ragionamento geometrico Area: Algebra e Analisi, Geometria Fonte: apri PDF p.2
proves area S>=2T
ABC is a right-angled triangle at A, and D is the foot of the altitude from A. The straight line joining the incenters of the triangles ABD, ACD intersects the sides AB, AC at the points K, L respectively. S and T denote the areas of the triangles ABC and AKL respectively. Show that S ≥2T.
(a2+b2)/(ab+1) integer is a perfect square (Vieta jumping)
Let a and b be positive integers such that ab + 1 divides a2 + b2. Show that a2 + b2 ab + 1 is the square of an integer.
Topic: Teoria dei Numeri Metodo: Principio di estremalita Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.2
*(a2+b2)/(ab+1) integer is a perfect square (Vietnam jumping) *
Let a and b be positive integers such that ab + 1 divides a2 + b2. Show that a2 + b2 ab + 1 is the square of an integer.