MathDB

4

Part of 2002 Tournament Of Towns

Problems(7)

cyclic and circumscribed

Source: Tournament of Towns,Spring 2002, Junior O Level, P4

5/13/2014
Quadrilateral ABCDABCD is circumscribed about a circle Γ\Gamma and K,L,M,NK,L,M,N are points of tangency of sides AB,BC,CD,DAAB,BC,CD,DA with Γ\Gamma respectively. Let S≡KM∩LNS\equiv KM\cap LN. If quadrilateral SKBLSKBL is cyclic then show that SNDMSNDM is also cyclic.
geometry proposedgeometry
lamps go on and off

Source: Tournament of Towns,Spring 2002, Junior A Level, P4

5/13/2014
There are nn lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which nn one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time?
combinatorics proposedcombinatorics
Already in a derangement

Source: Tournament of Towns,Spring 2002, Senior A Level, P4

5/14/2014
The spectators are seated in a row with no empty places. Each is in a seat which does not match the spectator's ticket. An usher can order two spectators in adjacent seats to trade places unless one of them is already seated correctly. Is it true that from any initial arrangement, the spectators can be brought to their correct seats?
countingderangementcombinatorics proposedcombinatorics
Number of ways

Source: Tournament of Towns,Spring 2002, Senior O Level, P4

5/14/2014
In how many ways can we place the numbers from 11 to 100100 in a 2×502\times 50 rectangle (divided into 100100 unit squares) so that any two consecutive numbers are always placed in squares with a common side?
geometryrectanglecombinatorics proposedcombinatorics
Winning strategy for last digit

Source: Tournament of Towns, Fall 2002, Junior O Level, P4

5/15/2014
20022002 cards with numbers 1,2,…,20021,2,\ldots ,2002 written on them are put on a table face up. Two players A,BA,B take turns to pick up a card until all are gone. AA goes first. The player who gets the last digit of the sum of his cards larger than his opponent wins. Who has a winning strategy and how should one play to win?
combinatorics proposedcombinatorics
This might be Jensen!!

Source: Tournament of Towns, Fall 2002, Senior O Level, P4

5/17/2014
x,y,z∈(0,π2)x,y,z\in\left(0,\frac{\pi}{2}\right) are given. Prove that: xcos⁡x+ycos⁡y+zcos⁡zx+y+z≤cos⁡x+cos⁡y+cos⁡z3 \frac{x\cos x+y\cos y+z\cos z}{x+y+z}\le \frac{\cos x+\cos y+\cos z}{3}
inequalitiestrigonometryinequalities proposed
Orthocentre's unique property

Source: Tournament of Towns, Fall 2002, Junior A Level, P4

5/17/2014
Point PP is chosen in the plane of triangle ABCABC such that ∠ABP\angle{ABP} is congruent to ∠ACP\angle{ACP} and ∠CBP\angle{CBP} is congruent to ∠CAP\angle{CAP}. Show PP is the orthocentre.
geometrycircumcirclepower of a pointradical axisgeometry proposed