MathDB

3

Part of 1997 All-Russian Olympiad

Problems(5)

$DE$ bisects sides $AB$ and $AC$

Source: ARMO 1997, 10.7

4/20/2013
The incircle of triangle ABCABC touches sides ABAB;BCBC;CACA at MM;NN;KK, respectively. The line through AA parallel to NKNK meets MNMN at DD. The line through AA parallel to MNMN meets NKNK at EE. Show that the line DEDE bisects sides ABAB and ACAC of triangle ABCABC. M. Sonkin
geometryAsymptoteparallelogramgeometry proposed
box completely covered by rectangles

Source: ARMO 1997, 9.3

4/20/2013
The lateral sides of a box with base a×ba\times b and height cc (where aa; bb;c c are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if cc is odd, then the number of possible coverings is even. D. Karpov, C. Gukshin, D. Fon-der-Flaas
geometryrectanglecombinatorics proposedcombinatorics
Show that \angle MKN =\pi/2

Source: ARMO 1997, 10.3

4/20/2013
Two circles intersect at AA and BB. A line through AA meets the first circle again at CC and the second circle again at DD. Let MM and NN be the midpoints of the arcs BCBC and BDBD not containing AA, and let KK be the midpoint of the segment CDCD. Show that ∠MKN=π/2\angle MKN =\pi/2. (You may assume that CC and DD lie on opposite sides of AA.) D. Tereshin
geometry proposedgeometry
m + n = gcd(m; n)^2 ...

Source: ARMO 1997, 10.7

4/20/2013
Find all triples mm; nn; ll of natural numbers such that m+n=gcd(m;n)2m + n = gcd(m; n)^2; m+l=gcd(m;l)2m + l = gcd(m; l)^2; n+l=gcd(n;l)2n + l = gcd(n; l)^2: S. Tokarev
number theory proposednumber theory
Show that the tetrahedron is regular

Source: ARMO 1997, 11.7

4/20/2013
A sphere inscribed in a tetrahedron touches one face at the intersection of its angle bisectors, a second face at the intersection of its altitudes, and a third face at the intersection of its medians. Show that the tetrahedron is regular. N. Agakhanov
geometry3D geometrytetrahedronsphere