Download Space Olympics Tasks 2001

The international space Olympiad, 2001
year.
8th form.
1. (2). Write down numbers 1 in all cells of the table 3 4 so that all
7 sums of the numbers in rows and sums in columns were various.
2. (2). Let AD be the median of a triangle ABC. Find angles of a
triangle ABC, if ADC  120, DAB  60.
3. (3). The points D and E are taken respectively on sides AB and BC
of a triangle ABC, and points G and F are taken on side AC so that
DEFG is a rectangle. Prove that if BF = BG, then triangle ABC is
isosceles.
4. (4). The numbers a1, a2 ,..., a2001 are obtained by permutation of
numbers 1,2, …, 2001. Prove that the product
a1 1a2  2   ...  a2001  2001 is even.
5. (5). Prove that if number 2 m  3n is divisible by 17 for some
positive integers m and n, then number 9m  5n is also divisible by
17.
6. (5). In a planetary system N the one-direction flights are carried out
between any two planets. Prove that from any planet it is possible to
reach any other, by making no more than one transfer.
The international space Olympiad, 2001
year.
9th form.
1. (2). Find 19 sequential positive integers such that the sum of first 10
of them is equal to the sum of 9 remaining.
2. (2). The real numbers a, b, c, d are such, that a  b  c  d.
Compare pairwise numbers x  a  bc  d , y  a  cb  d ,
z  a  d b  c.
3. (3). The circle passing through midpoints of sides of a triangle ABC,
touches one of its sides. Prove that the triangle ABC is isosceles.
4. (4). The real numbers a, b, c, d are such, that a2  b2  1,
c2  d 2  1, ac  bd  0. Find the value of expression ab  cd .
5. (5). Prove, that if for some integers m and n the number 25m  3n is
divisible by 83, then the number 3m  7n is also divisible by 83.
6. (5). The space station consists of modules located in one plane.
Thus each module is connected by passages to three others. An
astronaut performs a tour of station by the following rule: passing
through the module he turns to the right, if he turned to the left after
leaving the previous module and he turns to the left if he turned to
the right after leaving the previous module. Prove that leaving any
module he will enter it again after a while.
The international space Olympiad, 2001
year.
10th form.
1. (2). Find 12 sequential positive integers such that the sum of first 7
numbers is equal to the sum of 5 remaining numbers.
2. (2). The real numbers a, b, c and d, where 0  a  c, are such, that
the equation x 2  ax  b  0 has two equal roots, and the equation
x 2  cx  d  0 also has two equal roots. Prove that the equation
x 2  c  a x  d  b   0 has no real roots.
3. (3). The altitudes of an isosceles triangle ABC (AB = BC) intersect
in a point O. Prove, that the circumcircle of the triangle ABO,
touches the line AC.
4. (4). 20 astronauts have completed training course at the base. Every
day training is performed by any three of them. Some days it has
appeared that two astronauts trained together at least once. Prove
that there are at least two astronauts that met on training at least 2
times.
5. (5). Let's call a positive integer interesting, if it is the product of
exactly two prime divisors (not necessarily various). What is the
greatest amount of sequential interesting numbers?
6. (5). There are 37 different points with integer coordinates selected
in a space. Any three of them do not lie on one line. Prove, that it is
possible to choose three points A, B and C from them so, that the
medians of the triangle ABC intersect in the point with integer
coordinates.
The international space Olympiad, 2001
year.
11th form.
1. (2). How many ways are there to place two rooks on a chessboard so
that they did not threaten each other?
2. (2). Prove that the sum of cubes of three sequential positive integers
is not a product of two prime numbers.
3. (3). Find angles  ,  ,  of a triangle ABC, if the following equations
are satisfyied:
3 1
sin   sin  sin  
 ,
4 2
3
cos   cos  cos  
.
4
4. (4). It is known, that the numbers a2  b2 , a 3  b 3 and a 4  b 4 are
all rational. Prove that the number a  b is rational too.
5. (5). The edge of a cube is equal to 1. On each edge and on one of
main diagonals (of length 3 ) directions are chosen. Let S be the
sum of 13 obtained vectors. Find the smallest value of | S | .
6. (5). In a planetary system N the one-direction traffic is established
between each pair of planets. Prove that it is possible to change a
direction of travel between one pair of planets so that from any
planet it was possible to reach another.
INTERNATIONAL SPACE OLYMPIAD
PHYSICS
18-28 October 2001
8-th form
1. Speed of the Earth around the Sun is 30 km/s. What distance does the Earth travel for 1 h?
2. How many times is the period of the Earth’s rotation around the Sun greater than the period of
rotation around its own axis?
3. The vessel of 1 m3 is filled with equal masses of water and oil. Find the volume taken by water.
Density of oil is 800 kg/m3.
4. Mass of a cork life-raft is 3·5 kg. They put a bar of steel on the raft. What is the maximum
volume of the bar so that the raft will not sink? Density of cork is 240 kg/m3, steel- 7800 kg/m3.
5. One third (by mass) of mixture of two solid fuels is a fuel with a specific heat of combustion of
4107 J/kg. Two thirds of the mixture is a fuel with a specific heat of combustion of 2107 J/kg.
Find specific heat of combustion of the mixture.
6. Water flowing through the pipe of 1 см2 in section with speed of 0·5 m/s cools an engine. While
passing through, the water is heated up by 10°C. Find the energy the engine transmits to water
in 1 second. (Specific heat capacity of water is 4200 J/kg/K density of water is 1000 kg/m³).
INTERNATIONAL SPACE OLYMPIAD
PHYSICS
18-28 October 2001
9-th form
1. How many rotations around its axis does the Earth make during one cycle of travelling
around the Sun?
2. The interplanetary automatic station "Mars-1" began its flight with speed of 12 km/s. The
speed of "Mars-1" has decreased by the end of the first million km down to 3 km/s because
of the Earth gravitation. Find acceleration of flight, considering it constant.
3. Aristotle weighed an empty leather bag and then the same bag filled with air to prove the
weightlessness of the air. Both results were identical. Why was Aristotle conclusion about
weightlessness of the air wrong?
4. When the switch is in position 1 ammeter shows 1 A current, when in position 2
- 4 A current. Find resistance of each conductor, if the voltage on terminals of
the circuit is 12V.
5. A 60 W electric lamp was put in the transparent calorimeter containing 0·6 kg of water. The
temperature of water increased by 4°C over 5 minutes. What part of the energy consumed by
the lamp, has the calorimeter emitted to the environment? Specific
heat capacity of water 4200 J/kg/°C.
6. A rocket launched from the surface of a planet, is moving vertically up
with constant acceleration equal to the free-fall acceleration of the
planet. At the certain moment the engines of the rocket were cut off
and 10 s after the launch the rocket fell on the ground. For how many seconds did were the
engines working?
INTERNATIONAL SPACE OLYMPIAD
PHYSICS
18-28 October 2001
10-th form
1. By what factor is the pressure of gas in the cylinder changed, if the gas volume is reduced,
by advancing the piston by 1/3 of the cylinder length? Temperature of gas is constant.
2. Two trains go towards each other with speeds by 36 km/h and 54 km/h. The passenger in the
first train notices, that the second train passes by for 6 seconds. Find the length of the second
train.
3. Find the average density of a planet, where period of rotation is T, and the weight of a body
on its equator is % less, than on a pole.
4. The cycle 1-2-3-1 is given in P-V co-ordinates. Plot this cycle in -T
co-ordinates. The process 3-1 is isothermal.
5. The chain is passed through the pulley block, so that a part of it lies on
the table, and a part is on the floor. After the chain was released, it began
to move. Find the speed of the chain when motion became uniform.
Height of the table is h. Friction to be neglected.
6. A small body of mass m was dragged up the hill with constant speed by
means of force, directed in each point on a tangent to trajectory. Find the
work done by this force, if height of the hill is h, length of its base is l and
the friction factor is .
INTERNATIONAL SPACE OLYMPIAD
PHYSICS
18-28 October 2001
11-th form
1. 6 1itre (dm³) of ideal gas is being isothermally compressed so, that the pressure of the gas is increased
by a factor of 3 times. How much will its volume change?
2. What speed will a rocket, which is initially at rest in outer space, get to if the mass of the gases ejected
(instantly) is 1/5 of the initial mass of the rocket and the gas speed is 1 km/h?
3. An astronaut with the mass of 100 kg stands on a surface of a spherical asteroid with radius of 1 km and
holds in his hands a stone with the mass of 10 kg. What is the minimum speed relatively to the surface
of the asteroid, which the astronaut can throw the stone, without risking becoming a satellite of the
asteroid himself? Density of the asteroid is 5000 kg/m3
4. A charge q creates the electric field strength 200 N/C in point A and 100 N/c in point B. Find strength
of the field in point C.
5. There are two pistons with masses m1 and m2 in a long smooth
heat-proof pipe with no external pressure. There is a gas with
monoatomic molecules of volume Vo and pressure Po between
the pistons. The mass of the gas is much less than the mass of
pistons. The pistons are released and start moving. Find their
maximum velocity.
6. Metal rod with a mass of m and a length L is hung on two light wires of length
magnetic field, with magnetic flux density directed vertically down. Clips
the wires with a capacitor of capacity C, which is charged up to voltage U. Find
maximum angle of the rod deviation from the equilibrium position after the
discharging over very short time. Resistance of the rod and wires to be neglected
l each in of
connect
the
capacitor
Thoughtless journalist.
Journalist B. wrote an article for newspaper. He made some mistakes in it. He involved central heroes,
places and events. Editor-in-chief read the article and made up a correction list: wrong expressions in left and
correct expressions in right. He wanted to insert right corrections instead of wrong corrections in text.
Editor-in-chief wasn't interested by any order of the changes.
Write a program to correct B's text using a correction list
The input data: B's article contain Russian and/or English words.
The article consists of no more then
100 lines. The length of a line is less then 256 characters. The article is in the file ARTICLE.DAT.
The output data.
The checking text must be written into the file OK.DAT
For example:
The file ARTICLE.DAT contains this text:
The dog is sitting in the tree. The cat said "Bow-Wow".
The correction list contains:
Cat-dog
Dog-cat
Then reformed text must be:
The cat is sitting in the tree. The dog said "Bow-Wow". Maximum score 35%
Special Tic-Tac Game
Two players play a special Tic-Tac game on a grid 3x3 squares. Players move by turns. Each player can
write "X" or "O" into an empty square. A player fills the line (vertical, horizontal, or diagonal) with the same
sign (either only "X" or only "O") the first is a winner.
Write a program playing this game and winning if possible. The program always makes the first move.
The program asks for the opponent's moves and then moves itself until the field is filled completely or a line
with the same sign is formed.
The program must diagnose the game result: win, loss or draw.
Input. The program asks an opponent to move as follows: first line - "X" or "O"
second line - checks coordinates: horizontal position (1-3) and vertical position (1-3)
Output. The program writes to the console program's moves as follows:
“X” or "O", space, horizontal position (I -3), space and vertical position (I -3).
Maximum score 20%
The sun and the moon
The sun rises in Sunmorning time and it sets in Sunevening time. The whole of sun's disk is to be seen
above the horizon between Sunmorning and Sunevening. It moves uniformly through an arc. It starts in the
east and finishes in the west. The sun is at highest point at noon (zenith). The bottom border of the sun touches
with horizon at the crack of dawn (Sunmorning) and at sunset (Sunevening).
The moon rises in Lunaevening time and it sets in Lunamorning time. The whole of moon's disk is to be seen
above horizon between Lunaevening and Lunamorning. It moves uniformly through an arc. It starts in the
east and finishes in the west. The moon is at highest