MathDB

2

Part of 2000 Tournament Of Towns

Problems(8)

area of BCKL wanted, trapezoid related

Source: Tournament Of Towns Spring 2000 Junior 0 Level p2

4/22/2020
In a quadrilateral ABCDABCD of area 11, the parallel sides BCBC and ADAD are in the ratio 1:21 :2 . KK is the midpoint of the diagonal ACAC and LL is the point of intersection of the line DKDK and the side ABAB. Determine the area of the quadrilateral BCKLBCKL .
(M G Sonkin)
geometrytrapezoid
TOT 2000 Spring AJ2 integer sidelengths in 2 // sides

Source:

5/10/2020
Two parallel sides of a quadrilateral have integer lengths. Prove that this quadrilateral can be cut into congruent triangles.
(A Shapovalov)
geometrycongruent trianglesparallelograminteger sides
TOT 2000 Spring OS2 8 cubes construct a 2x2x2 cube, dots count

Source:

5/11/2020
Each of a pair of opposite faces of a unit cube is marked by a dot. Each of another pair of opposite faces is marked by two dots. Each of the remaining two faces is marked by three dots. Eight such cubes are used to construct a 2×2×22\times 2 \times 2 cube. Count the total number of dots on each of its six faces. Can we obtain six consecutive numbers?
(A Shapovalov)
combinatoricscombinatorial geometry
OM = KN wanted, intersecting chords, circumcenters

Source: Tournament of Towns Spring 2000 Seniors A p2

5/11/2020
The chords ACAC and BDBD of a, circle with centre OO intersect at the point KK. The circumcentres of triangles AKBAKB and CKDCKD are MM and NN respectively. Prove that OM=KNOM = KN.
(A Zaslavsky )
geometryChordsequal segmentsCircumcentercircumcircle
TOT 2000 Autumn OJ2 isosceles wanted, parallelogram related

Source:

5/10/2020
ABCDABCD is parallelogram, MM is the midpoint of side CDCD and HH is the foot of the perpendicular from BB to line AMAM. Prove that BCHBCH is an isosceles triangle.
(M Volchkevich)
geometryparallelogramisosceles
TOT 2000 Autumn AJ2 inradius wanted, isosceles related

Source:

5/10/2020
In triangle ABC,AB=ACABC, AB = AC. A line is drawn through AA parallel to BCBC. Outside triangle ABCABC, a circle is drawn tangent to this line, to the line BCBC, to ABAB and to the incircle of ABCABC. If the radius of this circle is 11 , determine the inradius of ABCABC.
(RK Gordin)
geometryisoscelesinradius
TOT 2000 Autumn OS2 if ad - bc > 1 then at least one not divisible by ad - bc

Source:

5/11/2020
Positive integers a,b,c,da, b, c, d satisfy the inequality ad−bc>1ad - bc > 1. Prove that at least one of the numbers a,b,c,da, b, c, d is not divisible by ad−bcad - bc.
(A Spivak)
number theorydividesdivisible
TOT 2000 Autumn AS2 n points on surface of cube,vertices of regular n-gon

Source:

5/11/2020
What is the largest integer nn such that one can find nn points on the surface of a cube, not all lying on one face and being the vertices of a regular nn-gon?
(A Shapovalov)
geometry3D geometrycombinatorial geometrycombinatoricsregular polygoncube