Tuymaada math competition

The International Olympiad '"Tuymaada" is an annual mathematics competition for secondary school students held in Republic of Saka, Russia. The contestants compete individually. The participating teams (national and local teams) can have up to three students. . The contest is held in two days of competitions, in July.
Here are a few interesting math problems from Tuymaada:

1. Positive integers a < b are given. Prove that among every b
consecutive positive integers there are two numbers whose product
is divisible by ab.

2. What minimum number of colours is sufficient to colour all positive real
numbers so that every two numbers whose ratio is 4 or 8 have different colours?

3. Several knights are arranged on an infinite chessboard.
No square is attacked by more than one knight (in particular, a square
occupied by a knight can be attacked by one knight but not by two). Sasha
outlined a 14x16 rectangle. What maximum number of knights can
this rectangle contain?

4. Prove that there exists a positive c such that for every positive
integer N among any N positive integers not exceeding 2N there
are two numbers whose greatest common divisor is greater than cN.

5. Seven different odd primes are given. Is it possible that the
difference of 8th powers of every two of them is divisible by each of the
remained numbers?

6. 100 boxers of different strength participate in the Boxing Championship
of Dirtytrickland. Each of them fights each other once. Several boxers
formed a plot: each of them put a leaden horseshoe in his boxing-glove
during one of his fights. When just one of two boxers has a horseshoe,
he wins; otherwise, the stronger boxer wins. It turned out after
the championship that three boxers won more fights than any of the three
strongest participants.
What is the minimum possible number of plotters?

7. Organizers of a mathematical congress found that if they
accomodate any participant in a single room the rest can be accomodated
in double rooms so that two persons living in every double room know each
other.
Prove that every participant can organize a round table on graph theory
for himself and even number of other people so that each participant of
the round table knows both his neighbours.

8.Several rooks stand in the squares of the table shown in the figure.
The rooks beat all the squares (we suppose that a rook beats the square
it stands in). Prove that one can remove several rooks so that not more
than 11 rooks are left and these rooks still beat all the squares.

Tournament of Towns problems and solutions

The International Mathematics Tournament of Towns is a mathematics problem solving competition in which towns throughout the world can participate on an equal basis.

Students participate in their own towns which involves minimal transport and administrative costs.

The Tournament is conducted each year in two stages - Autumn and Spring (northern hemisphere time). The southern hemisphere academic year coincides with this structure.

Each stage has two papers, an "O" level and an "A" level, which are spaced roughly one week apart. The A level paper is more difficult, but offers more points. Students and their towns may participate in either stages or levels, or in all levels and stages.

The Tournament is open to all high school students, with the highest age of students being about 17 years old.

Students are awarded points for their best three questions in each paper, and their annual score is based on their best score in any of the four papers for the year.

There are two versions of each paper, known as the Senior and Junior papers. Students in Years 10 and 11 (the final two years of high school in the Russian nomenclature) are classified as Senior participants and therefore attempt the Senior paper. So that Year 10 students are not disadvantaged their scores are multiplied by 5/4. Younger students, in Years 9 and below, attempt the Junior paper. To ensure that the scoring is fair to all levels of students, Year 8 students have their scores multiplied by 4/3, Year 7 students have their scores multiplied by 3/2 and Year 6 students and below have their scores multiplied by 2.

A town's score is based on the average of its best N students, where the population of the town is N hundred thousand (see below for definition of town). There is a minimum of N=5. If a population is less than 500,000 then the score is multiplied by an appropriate compensatory factor.

Students who exceed a certain minimum score are awarded a Diploma by the Russian Academy of Sciences. Local organising committees also present their own awards. The Tournament is managed by a central committee in Moscow, which is a subcommittee of the Russian Academy of Sciences.

You can find problems and solutions here.

Tournament of Towns 2007 problems and solutions junior A-level

International mathematics Tournament of Towns problems and solutions:
Fall 2007
Junior A-level problems:
1.Let ABCD be a rhombus.Let K be a point on the line CD, other than C or D, such that AB = BK.Let P be a point of intersection of BD with the perpendicular bisector of BC.Prove that A, K and P are collinear.
2. (a) Each of Peter and Basil thinks of 3 positive integers.For each pair of his numbers, Peter writes down the greatest common divisor of the 2 numbers.For each pair of his numbers, Basil writes down the least common multiple of the 2 numbers.If both Peter and Basil write down the same 3 numbers prove that these 3 numbers are equal to each other.
(b) Can the analogous result be proved if each of Peter and Basil thinks of 4 positive integers instead?
3.Michael is at the centre of a circle of radius 100 meters. Each minute he will announce the direction in which he will be moving.Catharine can leave it as is or change it to the opposite direction. Then Michael moves exactly 1 metre in direction determined by Catherine.Does Michael have a strategy which guarantees that he can get out of the circle, even though Catherine will try to stop him?
4.Two players take turns entering a symbol in an empty cell of a 1 x n chessboard where n is an integer greater that 1. Aaron always enters the symbol x and Betty always enters the symbol O.
Two identical symbols may not occupy adjacent cells.A player without a move loses a game.If Aaron goes first, which player has a winning strategy?
5.Attached to each of a number of objects is a tag which states the correct mass of the object.The tags have fallen of and have been replaced on the objects at random.We wish to determine if by fact all tags are in fact correct.We may use exactly once a horizontal lever which is supported at its middle. The objects can be hung from the lever at any point on either side of support. The lever either stays horizontal or tilts to one side. Is this task always possible?
6.The audience arranges n coins in a row.The sequence of heads and tails is chosen arbitrary. The audience also chooses a number between 1 and n inclusive. Then the assistant turns one of the coins over and magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by audience.
(a) Prove that if this is possible for some n then it is also possible for 2n.
(b) Determine all n for which this is possible.
7.For each letter in the English alphabet, William assigns an English word which contains that letter.His first document consists only of the word assigned to the letter A.In each subsequent document, he replaces each letter of the preceding document by its assigned word. The fortieth document begins with "`Till whatsoever start that guides my moving".Prove that this sentence reappears later in this document.
Solutions

Tournament of Towns 2007 problems and solutions junior O-level

International mathematics Tournament of Towns problems and solutions:
Fall 2007
Junior O-level problems:
1.Black and white checkers are placed on an 8 x 8 chessboard, with at the most one checker on each cell.What is the maximum level of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?
2.Initially , the number 1 and an non-integral number x are written on a blackboard.In each step we can choose 2 numbers on a blackboard, not necessarily different, and write their sum or their difference on the blackboard.We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard.Is it possible to write x^2 on a blackboard in a finite number of moves?
3.D is the midpoint on the side BC of triangle ABC.E and F are points on CA and AB respectively, such that BE is perpendicular to CA and CF is perpendicular to AB.If DEF is an equilateral triangle, does it follow that ABC is also equilateral?
4.Each cell of a 29x29 table contains one of the integers 1,2,3...29 and each of these integers appears 29 times.The sum of all the numbers above the main diagonal is equal to 3 times the sum of all the numbers below this diagonal.Determine the number in central cell of the table.
5.The audience chooses 2 of 5 cards numbered from 1 to 5 respectively.The assistant of a magician chooses 2 of remaining 3 cards, and asks the member of audience to take them to magician, who is in another room.The 2 cards are presented to magician in arbitrary order.By an arrangement with the assistant beforehand, the magician is able to deduce witch 2 cards the audience has chosen only from the 2 cards he receives.Explain how this may be done.

Solutions

Coins

Here are some interesting problems I encountered during the last week:
1.Several identical coins are placed on the table without overlapping. Prove that there is a coin which touches no more than three other coins.
2.Several coins are placed on the table without overlapping.Prove that there is a coin that touches no more than five other coins.
3.Several coins are placed on the table without overlapping.Prove that one can slide one of the coins to the edge of table without moving any other coin.
In next problem one can apply the method of small perturbations which is related to the method of extreme case (see previous post):
4.A straight line intersects a polygon exactly 2001 times.Prove that there is a straight line which is not parallel to any side of polygon and has more than 2001 common points with it.
5.There are 100 chords in a circle such that any two of them intersect.Is it always possible to draw one more chord such that it intersects all of them?

How to solve problems? - Extreme case

Here are some of my tips, and some problems from MathClub:
Sometimes in a solution of a problem it is crucial to consider extreme objects: The largest number, the closest point or vertex, a degenerate circle, any other limit case.
The minimal counterexample method is a combination of extreme case and a proof by contradiction: Assuming that some statement A is not true, there exists minimal (in some sense) counterexample.If we can somehow "decrease" it, then we get a contradiction.
Problems:
1.Prove that any polyhedron has at least 2 faces with the same number of sides.
2.A traveller went from his city A to the city B of his country, furthest away from A.Then from B he went to the city C, which is furthest away from B, and so on.Prove that if C and A are different cities, then the traveller will never return to A (all distances between cities of this country are different).
3.Numbers are placed on a chessboard.It is given that any number is equal to an arithmetical mean of its neighbours by a side.Prove that all numbers are equal.
4.Prove that for any integer n>1, 1 + 1/2 + 1/3 +...+1/n is not an integer.
5.Let us consider a point P inside a convex polygon and drop the perpendiculars from P to each side or its extension.Prove that at least one of the perpendiculars meets a corresponding side.
6. A six digit number is called lucky if 7 divides the sum of its digits.Are there any 2 consecutive lucky numbers?

Triangle problems

This is a very tricky problem:
1.Triangle ABC is given.Divide it by a broken line BDEFG into 5 triangles of equal area.
This one is from MathBattle:
2.Point E is given on the diameter AC of the circle.Draw a chord BD through E in such way that an area of the quadrilateral ABCD is maximal.
3.Find the maximal area of the right-angled triangle.

Proof by contradiction, and a few problems

It is known that m^2 + n^2 + m is divisible by mn for some positive integers m and n.Prove that m is a perfect square.
Proof to this problem is proof by contradiction.If you can`t solve this one, try solving one of these:
1.Prove that there is no largest real number.
2.Prove that there is no largest prime number.
3.5 children have picked up 9 flowers.Prove that at least 2 of them picked up the same number of flowers.
4.Prove that (n-1)(n)(n+1) is divisible by 6.
5.Prove that (n-2)(n-1)(n)(n+1)(n+2) is divisible by 2*3*4*5.
6.Prove that if n divides (n-1)! + 1 then n is a prime.

3! = 1*2*3

How to solve problems?

Here are some very good tips from Math club about problem solving:
If problem is hard, try to solve simpler problem related to it.This often gives you a clue to the original problem.
Try to consider a special case and generalize the idea of it`s solution afterwards.
Break the problem down into the smaller ones.
Generalize the problem.
Reduce the problem to simpler one.

Interesting problems

1.Here`s conversation between liars and truthtellers:
B to A:You are a liar.
C to B:It`s you who is a liar.
D to C:Both of them are liars.
D to C:And you are a liar too.
Can you tell who is who?
2.There are 2 ropes.It is known that each of them burns out completely in exactly 1 hour.
How one can measure 45 minutes using these ropes and a box of matches, if pace of burning is not uniform?
These 2 problems are very good examples of good problems.

Analysis

2n+1 people are on the party, and for every n people there is one who knows them all.Prove that one man (or woman:)) knows all 2n+1 people.
This is very hard problem, and although it seams easy it can be quite tricky.
Good luck trying to solve it.

JBMO geometry problem

This one`s from Junior Balkan Math Olympiad.It`s quite easy to find solution.
Given a (acute angled)trianle ABC and the tangent at A to the circumcircle intersects BC at P.
M is the midpoint of AP. BM intersects the circumcircle again at R and PR intersects the circumcircle again at S prove that AP || CS.

Geometry problems

Here are 3 geometry problems, and they`re not very hard.
1.Determine angle x
2.Determinate angle x
3.Determinate angle x if AB=BC






First 2 are easy, third is a bit harder .

1st Geometry problem

Here`s another one...
If AP/BP=q,
what is the value of AC/BC?
Notice that angle cao = angle oap and angle cbo = angle obp.

Seven millenium problems

In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each.
This is one of the coolest ways of getting rich.
Solve one of these and ...
Birch and Swinnerton-Dyer Conjecture
Hodge Conjecture
P vs NP Problem
Poincaré Conjecture
Riemann Hypothesis
Yang-Mills and Mass Gap

7th one is solved, so it`s possible.
More info on claymath.org.