MathDB

5

Part of 2011 Tournament of Towns

Problems(9)

2011 ToT Spring Junior O p5 a dragon gave a captured knight 100 coins

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3/4/2020
A dragon gave a captured knight 100100 coins. Half of them are magical, but only dragon knows which are. Each day, the knight should divide the coins into two piles (not necessarily equal in size). The day when either magic coins or usual coins are spread equally between the piles, the dragon set the knight free. Can the knight guarantee himself a freedom in at most (a) 5050 days? (b) 2525 days?
combinatoricsgamegame strategy
Geometry problem

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10/3/2016
ADAD and BEBE are altitudes of an acute triangle ABCABC. From DD, perpendiculars are dropped to ABAB at GG and ACAC at KK. From EE, perpendiculars are dropped to ABAB at FF and BCBC at HH. Prove that FGFG is parallel to HKHK and FK=GHFK = GH.
geometry
2011 ToT Spring Senior O p5 100 towns in a country, some joined by roads

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3/4/2020
In a country, there are 100100 towns. Some pairs of towns are joined by roads. The roads do not intersect one another except meeting at towns. It is possible to go from any town to any other town by road. Prove that it is possible to pave some of the roads so that the number of paved roads at each town is odd.
combinatorics
2011 ToT Spring Senior A p5 tangent arcs lead to tangent arcs

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3/4/2020
In the convex quadrilateral ABCD,BCABCD, BC is parallel to ADAD. Two circular arcs ω1\omega_1 and ω3\omega_3 pass through AA and BB and are on the same side of ABAB. Two circular arcs ω2\omega_2 and ω4\omega_4 pass through CC and DD and are on the same side of CDCD. The measures of ω1,ω2,ω3\omega_1, \omega_2, \omega_3 and ω4\omega_4 are α,β,β\alpha, \beta,\beta and α\alpha respectively. If ω1\omega_1 and ω2\omega_2 are tangent to each other externally, prove that so are ω3\omega_3 and ω4\omega_4.
arcstangent circles
2011 ToT Fall Junior O p5 pedestrian and cyclist, meet a cart and a car

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3/22/2020
On a highway, a pedestrian and a cyclist were going in the same direction, while a cart and a car were coming from the opposite direction. All were travelling at different constant speeds. The cyclist caught up with the pedestrian at 1010 o'clock. After a time interval, she met the cart, and after another time interval equal to the first, she met the car. After a third time interval, the car met the pedestrian, and after another time interval equal to the third, the car caught up with the cart. If the pedestrian met the car at 1111 o'clock, when did he meet the cart?
algebracombinatorics
2011 ToT Fall Junior A p5 (a + b + c + d) -(a + c)(b + d) >= 1

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3/22/2020
Given that 0<a,b,c,d<10 < a, b, c, d < 1 and abcd=(1−a)(1−b)(1−c)(1−d)abcd = (1 - a)(1 - b)(1 - c)(1 - d), prove that (a+b+c+d)−(a+c)(b+d)≥1(a + b + c + d) -(a + c)(b + d) \ge 1
inequalitiesalgebra
2011 ToT Fall Senior O p5 10 lines in general position, sum of angles

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3/22/2020
In the plane are 1010 lines in general position, which means that no 22 are parallel and no 33 are concurrent. Where 22 lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these 4545 angles?
Sumcombinatorial geometrycombinatoricslinesangles
2011 ToT Fall Senior A p5 good and special integers, k digits

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3/22/2020
We will call a positive integer good if all its digits are nonzero. A good integer will be called special if it has at least kk digits and their values strictly increase from left to right. Let a good integer be given. At each move, one may either add some special integer to its digital expression from the left or from the right, or insert a special integer between any two its digits, or remove a special number from its digital expression.What is the largest kk such that any good integer can be turned into any other good integer by such moves?
number theoryDigits
2011 Tournament of Towns, oral round, p5

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3/28/2011
Find all positive integers a,ba,b such that b619b^{619} divides a1000+1a^{1000}+1 and a619a^{619} divides b1000+1b^{1000}+1.
number theory unsolvednumber theory