MathDB

2

Part of 2005 District Olympiad

Problems(6)

triangle, midpoint and interior angle bisector

Source:

3/5/2005
Let ABCABC be a triangle and let MM be the midpoint of the side ABAB. Let BDBD be the interior angle bisector of ∠ABC\angle ABC, D∈ACD\in AC. Prove that if MD⊥BDMD \perp BD then AB=3BCAB=3BC.
geometrygeometric transformationreflectionangle bisector
space geometry, perpendicular squares

Source: RMO District 2005, 8th Grade, Problem 2

3/5/2005
Let ABCDABCD and ABEFABEF be two squares situated in two perpendicular planes and let OO be the intersection of the lines AEAE and BFBF. If AB=4AB=4 compute: a) the distance from BB to the line of intersection between the planes (DOC)(DOC) and (DAF)(DAF); b) the distance between the lines ACAC and BFBF.
geometry3D geometrypyramid
Romania District Olympiad 2005 - Grade XI

Source:

4/10/2011
Let f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} a continuous function such that for any a,b∈Ra,b\in \mathbb{R}, with a<ba<b such that f(a)=f(b)f(a)=f(b), there exist some c∈(a,b)c\in (a,b) such that f(a)=f(b)=f(c)f(a)=f(b)=f(c). Prove that ff is monotonic over R\mathbb{R}.
functionreal analysisreal analysis unsolved
classical triangle geometry - probably already posted

Source: RMO District 2005, 9th Grade, Problem 2

3/5/2005
Let ABCABC be a triangle inscribed in a circle of center OO and radius RR. Let II be the incenter of ABCABC, and let rr be the inradius of the same triangle, O≠IO\neq I, and let GG be its centroid. Prove that IG⊥BCIG\perp BC if and only if b=cb=c or b+c=3ab+c=3a.
geometryincenterinradiussymmetrygeometry proposed
determine a real function of two integer variables

Source: RMO District 2005, 10th Grade, Problem 2

3/5/2005
Find the functions f:Z×Z→Rf:\mathbb{Z}\times \mathbb{Z}\to\mathbb{R} such that a) f(x,y)⋅f(y,z)⋅f(z,x)=1f(x,y)\cdot f(y,z) \cdot f(z,x) = 1 for all integers x,y,zx,y,z; b) f(x+1,x)=2f(x+1,x)=2 for all integers xx.
functionalgebra proposedalgebra
the usual integral limit equation with linear solution

Source: RMO District 2005, 12th Grade, Problem 2

3/5/2005
Let f:[0,1]→Rf:[0,1]\to\mathbb{R} be a continuous function and let {an}n\{a_n\}_n, {bn}n\{b_n\}_n be sequences of reals such that lim⁡n→∞∫01∣f(x)−anx−bn∣dx=0. \lim_{n\to\infty} \int^1_0 | f(x) - a_nx - b_n | dx = 0 . Prove that: a) The sequences {an}n\{a_n\}_n, {bn}n\{b_n\}_n are convergent; b) The function ff is linear.
calculusintegrationfunctionlimitgeometryrectangleanalytic geometry