MathDB

4

Part of 2014 Saudi Arabia BMO TST

Problems(3)

Prove that DIMP is cyclic

Source: Saudi Arabia BMO TST Day I Problem 4

8/3/2014
Let ABCABC be a triangle with ∠B≤∠C\angle B \le \angle C, II its incenter and DD the intersection point of line AIAI with side BCBC. Let MM and NN be points on sides BABA and CACA, respectively, such that BM=BDBM = BD and CN=CDCN = CD. The circumcircle of triangle CMNCMN intersects again line BCBC at PP. Prove that quadrilateral DIMPDIMP is cyclic.
geometryincentercircumcirclegeometry unsolved
Labeling a self intersecting polygon

Source: Saudi Arabia BMO TST Day II Problem 4

8/3/2014
Let nn be an integer greater than 22. Consider a set of nn different points, with no three collinear, in the plane. Prove that we can label the points P1, P2,…,PnP_1,~ P_2, \dots , P_n such that P1P2…PnP_1P_2 \dots P_n is not a self-intersecting polygon. (A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex )
inequalitiestriangle inequalitygeometry unsolvedgeometry
Injective function f

Source: Saudi Arabia BMO TST Day III Problem 4

8/3/2014
Let f:N→Nf :\mathbb{N} \rightarrow\mathbb{N} be an injective function such that f(1)=2, f(2)=4f(1) = 2,~ f(2) = 4 and f(f(m)+f(n))=f(f(m))+f(n)f(f(m) + f(n)) = f(f(m)) + f(n) for all m,n∈Nm, n \in \mathbb{N}. Prove that f(n)=n+2f(n) = n + 2 for all n≥2n \ge 2.
functionalgebra unsolvedalgebra