MathDB

4

Part of 2006 Tournament of Towns

Problems(8)

Sharing Piles of Nuts

Source: Spring 2006 Tournament of Towns Junior O-Level #4

4/15/2015
Anna, Ben and Chris sit at the round table passing and eating nuts. At first only Anna has the nuts that she divides equally between Ben and Chris, eating a leftover (if there is any). Then Ben does the same with his pile. Then Chris does the same with his pile. The process repeats itself: each of the children divides his/her pile of nuts equally between his/her neighbours eating the leftovers if there are any. Initially, the number of nuts is large enough (more than 3). Prove that a) at least one nut is eaten; (3 points) b) all nuts cannot be eaten. (3 points)
number theory
Digits of 2^n and 5^n

Source: Spring 2006 Tournament of Towns Junior A-Level #4

4/15/2015
Is there exist some positive integer nn, such that the first decimal of 2n2^n (from left to the right) is 55 while the first decimal of 5n5^n is 22?
(5 points)
number theory
A Cyclic Quadrilateral

Source: Spring 2006 Tournament of Towns Senior O-Level #4

9/9/2015
Quadrilateral ABCDABCD is a cyclic, AB=ADAB = AD. Points MM and NN are chosen on sides BCBC and CDCD respectfully so that ∠MAN=1/2(∠BAD)\angle MAN =1/2 (\angle BAD). Prove that MN=BM+NDMN = BM + ND.
(5 points)
geometrycyclic quadrilateral
equal angles starting with a fixed point on angle bisector

Source: 2006 ToT Spring Senior A P4

9/11/2018
In triangle ABCABC let XX be some fixed point on bisector AA′AA' while point B′B' be intersection of BXBX and ACAC and point C′C' be intersection of CXCX and ABAB. Let point PP be intersection of segments A′B′A'B' and CC′CC' while point QQ be intersection of segments A′C′A'C' and BB′BB'. Prove τhat ∠PAC=∠QAB\angle PAC = \angle QAB.
geometryequal anglesangle bisector
TOT 2006 Fall - Junior O-Level p4 cyclic wanted

Source:

2/25/2020
Given triangle ABC,BCABC, BC is extended beyond BB to the point DD such that BD=BABD = BA. The bisectors of the exterior angles at vertices BB and CC intersect at the point MM. Prove that quadrilateral ADMCADMC is cyclic. (4)
geometryCyclicangle bisector
TOT 2006 Fall - Senior O-Level p4 infinite geom./ arithmetic progression

Source:

2/25/2020
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)
geometric progressionArithmetic ProgressionSequenceIntegernumber theory
TOT 2006 Fall - Junior A-Level p4 sum of diameters of 3 in circles

Source:

2/25/2020
A circle of radius RR is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter QQ. Find the sum of diameters of circles inscribed into the three right triangles. (6)
geometryTangentsinradiushexagon
TOT 2006 Fall - Senior A-Level p4 prism split into set of pyramids

Source:

2/25/2020
Is it possible to split a prism into disjoint set of pyramids so that each pyramid has its base on one base of the prism, while its vertex on another base of the prism ? (6)
geometry3D geometryprismpyramid