MathDB

1

Part of 2008 Romania National Olympiad

Problems(6)

Acute triangle with perpendicular lines

Source: RMO, Grade 7, Problem 1

4/30/2008
Let ABC ABC be an acute angled triangle with ∠B>∠C \angle B > \angle C. Let D D be the foot of the altitude from A A on BC BC, and let E E be the foot of the perpendicular from D D on AC AC. Let F F be a point on the segment (DE) (DE). Show that the lines AF AF and BF BF are perpendicular if and only if EF\cdot DC \equal{} BD \cdot DE.
trigonometrygeometrycircumcirclegeometric transformation
A tetrahedron with side lengths positive integers

Source: RMO 2008, Grade 8, Problem 1

4/30/2008
A tetrahedron has the side lengths positive integers, such that the product of any two opposite sides equals 6. Prove that the tetrahedron is a regular triangular pyramid in which the lateral sides form an angle of at least 30 degrees with the base plane.
geometry3D geometrytetrahedronpyramid
Function

Source: Romania NMO 2008, 9 form, Problem 1

4/30/2008
Find functions f:N→N f: \mathbb{N} \rightarrow \mathbb{N}, such that f(x^2 \plus{} f(y)) \equal{} xf(x) \plus{} y, for x,y∈N x,y \in \mathbb{N}.
functioninductionalgebra proposedalgebra
Circumcenters of triangles coincide

Source: RMO 2008, Grade 10, Problem 1

4/30/2008
Let ABC ABC be a triangle and the points D∈(BC) D\in (BC), E∈(CA) E\in (CA), F∈(AB) F\in (AB) such that \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}. Prove that if the circumcenters of the triangles DEF DEF and ABC ABC coincide then ABC ABC is equilateral.
geometrycircumcirclegeometric transformationreflectionperpendicular bisectorgeometry proposed
Nice nondecreasing function

Source: RMO 2008, 11th Grade, Problem 1

4/30/2008
Let f:(0,∞)→R f : (0,\infty) \to \mathbb R be a continous function such that the sequences {f(nx)}n≥1 \{f(nx)\}_{n\geq 1} are nondecreasing for any real number x x. Prove that f f is nondecreasing.
functionreal analysisreal analysis unsolved
Darboux property

Source: RMO 2008, Grade 12, Problem 1

4/30/2008
Let a>0 a>0 and f:[0,∞)→[0,a] f: [0,\infty) \to [0,a] be a continuous function on (0,∞) (0,\infty) and having Darboux property on [0,∞) [0,\infty). Prove that if f(0)\equal{}0 and for all nonnegative x x we have xf(x)≥∫0xf(t)dt, xf(x) \geq \int^x_0 f(t) dt , then f f admits primitives on [0,∞) [0,\infty).
functionintegrationcalculusderivativelimitreal analysisreal analysis unsolved