IMO 1997 problema 1
Let d be any positive integer not equal to 2, 5, or 13. Show that one can find distinct a, b in the set {2, 5, 13, d} such that ab −1 is not a perfect square.
Topic: Combinatoria, Geometria analitica Metodo: parita, congruenze Abilita: Conteggio sistematico Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.62
This is the case in the Member States.
Let d be any positive integer not equal to 2, 5, or 13. Show that one can find distinct a, b in the set {2, 5, 13, d} such that ab −1 is not a perfect square.
IMO 1997 problema 2
A triangle A1A2A3 and a point P0 are given in the plane. We define As = As−3 for all s ≥4. We construct a set of points P1, P2, P3, … , such that Pk+1 is the image of Pk under a rotation with center Ak+1 through angle 120°clockwise (for k = 0, 1, 2, … ). Prove that if P1986 = P0, then the triangle A1A2A3 is equilateral.
Topic: Geometria piana Metodo: Tecniche trigonometriche Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.62
This is the case for the Commission.
A triangle A1A2A3 and a point P0 are given in the plane. We define As = As−3 for all s ≥4. We construct a set of points P1, P2, P3, … , such that Pk+1 is the image of Pk under a rotation with center Ak+1 through angle 120°clockwise (for k = 0, 1, 2, … ). Prove that if P1986 = P0, then the triangle A1A2A3 is equilateral.
IMO 1997 problema 3
To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers x, y, z respectively and y < 0 then the following operation is allowed: the numbers x, y, z are replaced by x+y, −y, z +y respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to and end after a finite number of steps.
27th International Mathematical Olympiad Warsaw, Poland Day II July 10, 1986
Topic: Disuguaglianze, Algebra Metodo: manipolazione algebrica Abilita: Manipolazione algebrica Area: Algebra e Analisi Fonte: apri PDF p.62
This is the case for the European Union.
To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers x, y, z respectively and y < 0 then the following operation is allowed: the numbers x, y, z are replaced by x+y, −y, z +y respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
27th International Mathematical Olympiad Warsaw, Poland Day II July 10, 1986
IMO 1997 problema 4
Let A, B be adjacent vertices of a regular n-gon (n ≥5) in the plane having center at O. A triangle XY Z, which is congruent to and initially conincides with OAB, moves in the plane in such a way that Y and Z each trace out the whole boundary of the polygon, X remaining inside the polygon. Find the locus of X.
Topic: Combinatoria, Geometria analitica Metodo: Analisi per casi, congruenze Abilita: Conteggio sistematico Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.62
This is the case for the European Union.
Let A, B be adjacent vertices of a regular n-gon (n ≥5) in the plane having center at O. A triangle XY Z, which is congruent to and initially conincides with OAB, moves in the plane in such a way that Y and Z each trace out the entire boundary of the polygon, X remaining inside the polygon. Find the locus of X.
IMO 1997 problema 5
Find all functions f, defined on the non-negative real numbers and taking nonnegative real values, such that: (i) f(xf(y))f(y) = f(x + y) for all x, y ≥0, (ii) f(2) = 0, (iii) f(x) ̸= 0 for 0 ≤x < 2.
Topic: Teoria dei Numeri Abilita: Riconoscimento di pattern Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.62
This is the case in the Member States.
Find all functions f, defined on the non-negative real numbers and taking non-negative real values, such that: (i) f(xf(y)) f(y) = f(x + y) for all x, y ≥0, (ii) f(2) = 0, (iii) f(x) = 0 for 0 ≤x < 2.
IMO 1997 problema 6
One is given a finite set of points in the plane, each point having integer coordinates. Is it always possible to color some of the points in the set red and the remaining points white in such a way that for any straight line L parallel to either one of the coordinate axes the difference (in absolute value) between the numbers of white point and red points on L is not greater than 1?
28th International Mathematical Olympiad Havana, Cuba Day I July 10, 1987
Topic: Teoria dei Numeri, Combinatoria Metodo: Induzione Abilita: Riconoscimento di pattern Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.62
This is the case for the European Union.
One is given a finite set of points in the plane, each point having integer coordinates. Is it always possible to color some of the points in the set red and the remaining points white in such a way that for any straight line L parallel to either of the coordinate axes the difference (in absolute value) between the numbers of white point and red points on L is not greater than 1?
28th International Mathematical Olympiad Havana, Cuba Day I July 10, 1987
IMO 1962 problema 7
The tetrahedron has the following property: there exist five spheres, each tangent to the edges , , , , , , or to their extensions.
(a) Prove that the tetrahedron is regular.
(b) Prove conversely that for every regular tetrahedron five such spheres exist.
Topic: Geometria solida Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.7
This is the case for the European Union.
The tetrahedron has the following property: there exist five spheres, each tangent to the edges , , , , , , or to their extensions.
(a) Prove that the tetrahedron is regular.
(b) Prove conversely that for every regular tetrahedron five such spheres exist.