MathDB

Problem 6

Part of 1999 Mongolian Mathematical Olympiad

Problems(5)

show concyclic, common chords of intersecting circles

Source: Mongolia 1999 Grade 8 P6

5/4/2021
Two circles in the plane intersect at CC and DD. A chord ABAB of the first circle and a chord EFEF of the second circle pass through a point on the common chord CDCD. Show that the points A,B,E,FA,B,E,F lie on a circle.
geometry
3^n and 7^n both begin with 10 for some n

Source: Mongolia 1999 Grade 9 P6

5/4/2021
Show that there exists a positive integer nn such that the decimal representations of 3n3^n and 7n7^n both start with the digits 1010.
number theory
locus of point on triangle

Source: Mongolia 1999 Grade 10 P6

5/5/2021
A point MM lies on the side ACAC of a triangle ABCABC. The circle γ\gamma with the diameter BMBM intersects the lines ABAB and BCBC at PP and QQ, respectively. Find the locus of the intersection point of the tangents to γ\gamma at PP and QQ when point MM varies.
geometry
minimum length of sum of 1999 unit vectors with nonnegative coordinates

Source: Mongolia 1999 Teachers elementary level P6

5/6/2021
Find the minimum possible length of the sum of 19991999 unit vectors in the coordinate plane whose both coordinates are nonnegative.
vectorgeometryanalytic geometry
functional geometry, d(a,b)=n->d(f(a),f(b))=n for n=1 implies ∀n∈N

Source: Mongolia 1999 Teachers secondary level P6

5/6/2021
Let ff be a map of the plane into itself with the property that if d(A,B)=1d(A,B)=1, then d(f(A),f(B))=1d(f(A),f(B))=1, where d(X,Y)d(X,Y) denotes the distance between points XX and YY. Prove that for any positive integer nn, d(A,B)=nd(A,B)=n implies d(f(A),f(B))=nd(f(A),f(B))=n.
fefunctional equationfunctional geometrygeometry