Download Space Olympics Tasks 97

THE COMPUTER SCIENCE IN SPACE TRIP - 1997
TASKS FOR THE PARTICIPANTS
While you fulfill any of 6 tasks try to increase reliability of your programs
(provide inadmissible, boundary and surplus data).
TASK 1. “Time-Table”. 6 marks
There are 3 launching pads at the spaceport: for cargo modules, passenger ships
and communication satellites. All the above mentioned vehicles can be launched from
the cargo launching pad, and the two last vehicles can be launched from the passenger
launching pad. The complex which is launched in parts consists of N1 cargo modules,
N2 passenger ships and N3 communication satellites. The duration of preparation and
launch is the same for all parts and pads and equal T. Write the program, which:
1) calculates a minimum time Tk of the launch of the whole complex into space,
2) makes the “optimal” time-table of launches of the parts for Tk (a table with columns
“Time (multiple of T)” and 3 lines “Pads”),
3) calculates a number of “optimal” time-tables,
4) prints all variants of “optimal” time-tables.
TASK 2. “DECODING of THE MESSAGE”. 5 marks
Your orbital station has received a coded message of length less than 256
symbols about a meeting with American astronauts. You remember the formula of
coding – the cyclic shift: N of the code = (N of the message + M) in absolute value
27, where N is a number of a symbol (number of the American capital letter in the
alphabet from 1 up to 26, and the number of a “space” blank is 0); M - shift, whole
number M from 1 up to 26. Write the program for decoding of the message if you have
forgotten M, but you have guessed, that there should be the word “MEETING” in the
message consisting of more than one word.
TASK 3. “WINNING STRATEGICS”. 5 marks
While waiting for the docking with American spaceship at 12:00, two Russian
owners of the orbital station have decided to while away the time by a simple game. The
first player names minutes of a stretch from 0 to 59. They play in turn. The next player
names a later time, increasing one or the other: hours or minutes. The player who names
11:59, wins. Find winning strategics of the first player and write the program to realise
it.
TASK 4. “ANIMATION OF DOCKING”. 3 marks
Represent on your computer screen a manual docking of American spaceship
with Russian orbital station in space as a cartoon film.
TASK 5. “ELIMINATION OF DEPRESSURISATION”. 1 mark
NASA has pointed out N possible depressurised places of the external covering
of the orbital station; each of them has the probability of depressurisation Pi, i=1… N.
Write the program of optimal choice (Pi1Pi2...PiN) of a sequence for investigation of
indicated N places.
TASK 6. “SEARCH FOR THE IDENTICAL ONES”. 4 marks
3 astronauts took pictures of views in space by their cameras. After their return
to the Earth they have decided to choose the best picture of several similar shots (the
same view) in order to make a photo from it. Write the program of search for identical
views and indicate their numbers on each of 3 photo films x [1: p], y [1: q], z [1: r].
Instruction. Your files and txt-comments (your surname and name, instructions about
input-output) to record under the name: “the number of your class - number of your
computer - number of the task” with appropriate expansion in the subdirectory under
the name: “the number of your class - number of your computer”.
INTERNATIONAL SPACE OLYMPICS-97
PHYSICS
16-25 October 1997,
Korolev
11-th form
1. A merry-go-round spins at the angular speed . A man stands on the
Merry-go-round at a distance r from the axis. It rains and the drops fall vertically at a
speed v. How should the man hold an umbrella to make the best cover?
2.Estimate the acceleration of a free fall on a planet, where a man can go onto water in
waterproof shoes. The coefficient of the surface tension of water 73 mN/m.
3. A plane flies in the circle of radius r with constant speed v. What angle do the plane’s
wings make with the horizon? Mind that the force of the air pressure is perpendicular to
the surface of wings.
4.You approach the last carriage of a train at the moment when it starts moving with
acceleration 0.3 m/s2. The only open door is 60 m from you. What constant speed you
should have to catch the train?
5. A satellite of volume 103m3, is filled with air under normal conditions (P=1 atm,
t=00C). A meteorite makes a hole of 1 cm2. Estimate, what time it takes for the pressure
inside satellite to fall by 1 %. (The temperature of air is constant).
6. Atom of Positronium consists of an electron and positron (the positron mass is equal
to the mass m of the electron, the charge of the positron is +e, where -e is the charge of
the electron). The positron and electron rotate in the circle of radius r around their center
of mass. What minimal energy is necessary to give to the electron in order to make
positron decay?
Note: potential energy of two charges q1 and q2 with distance r between is kq1q2/r.
10-th form
1. A merry-go-round spins at the angular speed . A man stands on the
Merry-go-round at a distance r from the axis. It rains and the drops fall vertically at a
speed v. How should the man hold an umbrella to make the best cover?
2. The satellite stays all the time zenith over one and the same point of the Earth surface.
The radius of the Earth is 6400 km, g=9.8 m/s2. Find the radius of the satellite orbit.
3. A force measuring device shows that the marble block hanged on a fine thread weighs
1.6 N. What will the device show if the half of the block is dipped into water? Marble
density is 2,7 .103 kg/m3.
4. The atmosphere of a planet has the constant density and consists of gas with molar
mass . The thickness h of the atmosphere is far less then radius R of the planet. The
mass of the planet is M. Find the temperature on the surface of the planet.
5. A helicopter of mass m hovers in the air. What is the power of the engine of
helicopter, if the speed of a downward - directed air jet is v?
6. A half of rare gas flows out from a vessel through the hole during 10 sec. How long
does it take for the rare gas to flow out at the same temperature if all the dimensions of
the vessel (including the hole size) are enlarged in three times?
9-th form
1. What amount of heat will a wire spiral (resistance 15 Om) give off during 10 minutes
if the current is 2 A?
2. In order to heat a brick to 650C and the same mass of water to 130C the same
amount of heat is needed. Find the specific heat of the brick. The specific heat of water
is 4,2. 103 J/kg0 C.
3. A force measuring device shows that the marble block hanged on a fine thread weighs
1.6 N. What will the device show if the half of the block is dipped into water? Marble
density is 2,7 .103 kg/m3.
4. There are three electric lamps: one of power 50 W and two other of 25 W made for
voltage of 110 V. They must be used for voltage of 220 V - net in the way that each
lamp works in normal regime. Show the circuit and find the current of each lamp.
5. The speedometer scale is 15 cm long. It measures the car speed from 0 up to 150
km/h. Find the speed of speedometer pointer if the car goes with acceleration 2 m/s2.
6. You approach the last carriage of a train at the moment when it starts moving with
acceleration 0.3 m/s2. The only open door is 60 m from you. What constant speed you
should have to catch the train?
8-th form
1. What amount of heat is given off during burning 600 g of petrol?
(q=4.6 . 107 J/kg).
2. In order to heat a brick to 650C and the same mass of water to 130C the same
amount of heat is needed. Find the specific heat of the brick. The specific heat of water
is 4,2. 103 J/kg0 C.
3. A force measuring device shows that the marble block hanged on a fine thread weighs
1.6 N. What will the device show if the half of the block is dipped into water? Marble
density is 2,7 .103 kg/m3.
4. A train car moving at a speed of 72 km/h, was punctured by a bullet flying
perpendicular to the car direction. It has turned out that one hole was displaced from
another by 6 cm. The car width is 2.7 m. Find the velocity of the bullet.
5. After leaving a raft, a boy swims facing into the stream during 5 minutes and three
times faster than the stream. Then the boy turns back, swims with the same effort and
catches up his raft. What time did the boy swim after the turn?
6. Density of an alloy of gold and silver (mass 600 g) is 14 . 103 kg/m3. Find the mass
of gold in the alloy. Gold density is 19,3 . 103 kg/m3, silver density is 10,5 . 103 kg/m3.
MATHEMATICS
11-th form
1. A1,a2,a3,... is a sequence that satisfies the requirements: an+1= 2an + an2, n N,
a1 =9. Find the number of symbols in decimal representation of the number a10.
2. The altitudes AA1, BB1 and CC1 of the triangle ABC intersect at point O.
Prove that ABC is a regular triangle if OA+OC=BO.
3. O is a center of the circle of the polygon M. Prove that if the line 1 divides M
into two polygons with equal perimeters then 1 passes through point O.
4. Find the minimal value of the term x1. x2 + x1. x3 + ...x1996 . x1997 (the term
consists of the sum of products of all pairs of numbers x1, ... , x11997 ) if each of the
numbers x1, ... , x1997 can be equal to 0.1 or -1.
5. Find all nonnegative solutions of the sistem
x+y-xy2z2 = 0
y+z-yz2x2 = 0
z+x-zx2y2 = 0
6.A planet system N consists of n (n>4) planets. Prove that it is possible to
establish a direct spaceship communication between pairs of the planets, so that one can
get from any planet to another planet using maximum 2 spaceships.
10-th form
1. Solve the system:
x+y+xyz=2
y+z+2xyz=2
z+x+3xyz=2
2. The bisectors AA1 and CC1 of triangle ABC intersect in point O. Prove that
AB=BC of the areas of the triangles AOC1 and COA1 are equal.
3. Can you find such coefficients a>c>0 and d>b>0 for which the equationts
x4+ax+b=0 and x4+cx+d=0 have a common root?
4. Prove that two triangles with the sides a, b, c and a1, b1, c1 are similar if the
equality aa1+bb1+cc1= (a+b+c)(a1+b1+c1) is valid.
5. The sequence a0, a1, a2,...of positive integers satisfies conditions an+1 = 11an 10an-1, n N, and a1>a0. Find the minimal possible number of symbols in decimal
representation of the number a1997.
6. There are 13 planets in the planet system N and n space companies launching
spaceships between the planets. Each company launches spaceships to only 4 planets
(some pairs of planets can be connected by launches of different companies. Find the
minimal value of n if it is known that there are launches between every pair of planets.
9-th form
1. Find any permutation a1, ..., a21 of the numbers 1,2, ..., 21 in the circle for
which one of the sums an+an-1 or an+an+1 (a22=a1) divides by 5 for every n=1,...,21.
2. AA1 and CC1 are medians of triangle ABC. Prove that AB=BC if
BAA1=BCC1.
3.Can we find such a squared polynomial y=ax2+bx+c with integer coefficients
that has the value 1 for x=3 and value 11 for x =7?
4. One can make one of the following two operations with number A written on
the blackboard:
(i) change A by 2A;
(ii) change A by A+1.
Find the minimal number of these operations that can change number O to
number 1997.
5. The longest altitude AM of triangle ABC is equal to median BM. Prove that
ABC < 600.
6. There are 7 planets in the planet system N and n space companies launching
spaceships between the planets. Each company launches spaceships to only 4 planets
(some pairs of planets can be connected by launches of different companies. Find the
minimal value of n if it is known that there are launches between every pair of planets.
8-th form
1.Find any permutation a1, ...a10 of the numbers 1,2,...,10 in the circle for which
the sum of every pair of neighbouring numbers (an+an+1, n=a, ... 9 or a10+a1) does not
divide by 3.
2. D and E are the points on the side AC of triangle ABC so that AD=DE=EC.
Prove that AB=BC if BD=BE.
3. Find all the values of parameter a that satisfy the following condition: lines
x+2y=3, 2x+3y=a, 3x+4y=a+1 pass through one point.
4. In accordance with the instruction each super agent of the intelligence service
must spy after one of the other super agents and follow the scheme:
agent 001 spies after that agent who spies after the agent 002, the agent 002 spies after
that agent who spies after the agent 003 etc.; the last agent spies after that agent who
spies after the agent 001.
Can the chief of the IS find the required scheme if there are a) 7, b) 8 super
agents in the IS?
5. There is a rectangle 1x97 with squares a1, ...a97 and two coins: white in the
square as, black in the square a97. Two players move their coins (first - white, second black coin) one after the other: white coin from an to an+1 or an+2. Black coin from am to
am-1 or am-2 . By the rule both coins can’t by in one square and can’t jump one over
the other. If a player can’t move his coin he loses the game. Who would win in this
game? The answer must be explained.
6. In the planet sistem N spaceships go between some pairs of planets(in both
directions). We know that there are 21 launches from the Central planet, 1 launch
from the Remote planet and 20 launches from each of the other planets. Prove that using
these launches one can get from the Central planet to the Remote planet.