MathDB

5

Part of 2002 Tournament Of Towns

Problems(8)

Weighting coins

Source: Tournament of Towns,Spring 2002, Junior O Level, P5

5/13/2014
[*] There are 128128 coins of two different weights, 6464 each. How can one always find two coins of different weights by performing no more than 77 weightings on a regular balance? [*] There are 88 coins of two different weights, 44 each. How can one always find two coins of different weights by performing two weightings on a regular balance?
combinatorics proposedcombinatorics
A dissection once more

Source: Tournament of Towns,Spring 2002, Junior A Level, P5

5/13/2014
An acute triangle was dissected by a straight cut into two pieces which are not necessarily triangles. Then one of the pieces were dissected by a straight cut into two pieces and so on. After a few dissections it turns out the pieces were all triangles. Is it possible they were all obtuse?
geometry proposedgeometry
Forming an equilateral hexagon

Source: Tournament of Towns,Spring 2002, Senior A Level, P5

5/14/2014
Let AA1,BB1,CC1AA_1,BB_1,CC_1 be the altitudes of acute ΔABC\Delta ABC. Let Oa,Ob,OcO_a,O_b,O_c be the incentres of ΔAB1C1,ΔBC1A1,ΔCA1B1\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1 respectively. Also let Ta,Tb,TcT_a,T_b,T_c be the points of tangency of the incircle of ΔABC\Delta ABC with BC,CA,ABBC,CA,AB respectively. Prove that TaOcTbOaTcObT_aO_cT_bO_aT_cO_b is an equilateral hexagon.
geometrytrigonometryrhombusgeometry proposed
A query about existence

Source: Tournament of Towns,Spring 2002, Senior O Level, P5

5/14/2014
Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)
geometry3D geometryprismgeometry proposed
Angle and a point

Source: Tournament of Towns, Fall 2002, Junior O Level, P5

5/15/2014
An angle and a point AA inside it is given. Is it possible to draw through AA three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?
geometry proposedgeometry
Infinite sequence contains an even number

Source: Tournament of Towns, Fall 2002, Senior O Level, P5

5/17/2014
An infinite sequence of natural number {xn}n≥1\{x_n\}_{n\ge 1} is such that xn+1x_{n+1} is obtained by adding one of the non-zero digits of xnx_n to itself. Show this sequence contains an even number.
floor functionnumber theory proposednumber theory
convex N-gon

Source: Tournament of Towns, Fall 2002, Junior A Level, P5

5/17/2014
A convex N-gonN\text{-gon} is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each NN find the maximal difference between the numbers of black and white triangles.
combinatorics proposedcombinatorics
Two circles

Source: Tournament of Towns, Fall 2002, Senior A Level, P5

5/17/2014
Two circles Γ1,Γ2\Gamma_1,\Gamma_2 intersect at A,BA,B. Through BB a straight line ℓ\ell is drawn and ℓ∩Γ1=K,ℓ∩Γ2=M  (K,M≠B)\ell\cap \Gamma_1=K,\ell\cap\Gamma_2=M\;(K,M\neq B). We are given ℓ1∥AM\ell_1\parallel AM is tangent to Γ1\Gamma_1 at QQ. QA∩Γ2=R  (≠A)QA\cap \Gamma_2=R\;(\neq A) and further ℓ2\ell_2 is tangent to Γ2\Gamma_2 at RR. Prove that:
[*]ℓ2∥AK\ell_2\parallel AK [*]ℓ,ℓ1,ℓ2\ell,\ell_1,\ell_2 have a common point.
projective geometrygeometrycyclic quadrilateralgeometry proposed