1

Part of 2016 IFYM, Sozopol

Problems(5)

Problem 1 of First round - Magician and assistant

Source: VII International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

8/29/2019

A participant is given a deck of thirteen cards numerated from 1 to 13, from which he chooses seven and gives them to the assistant. Then the assistant chooses three of these seven cards and the participant – one of the remaining six in his hand. The magician then takes the chosen four cards (arranged by the participant) and guesses which one is chosen from the participant. What should the magician and assistant do so that the magic trick always happens?

combinatoricsgame strategy

Problem 1 of Second round - Triangle inequality

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

8/31/2019

Find all functions

f: \mathbb{R}^+\rightarrow \mathbb{R}^+

with the following property:

a,b,

and

c

are lengths of sides of a triangle, if and only if

f(a),f(b),

and

f(c)

are lengths of sides of a triangle.

functionalgebratriangle inequalityinequalities

Problem 1 of Third round - Convex decagons

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/1/2019

The numbers from 1 to

n

are arranged in some way on a circle. What’s the smallest value of

n

, for which no matter how the numbers are arranged there exist ten consecutively increasing numbers

A_1<A_2<A_3…<A_{10}

such that

A_1 A_2…A_{10}

is a convex decagon?

combinatorial geometrycombinatorics

Problem 1 of Fourth round - Airline companies

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/3/2019

There are

2^{2n+1}

towns with

2n+1

companies and each two towns are connected with airlines from one of the companies. What’s the greatest number

k

with the following property: We can close

k

of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

combinatoricsgraph

Problem 1 of Finals - Game strategy with lines and points

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/19/2019

We are given a set

P

of points and a set

L

of straight lines. At the beginning there are 4 points, no three of which are collinear, and

L=\emptyset

. Two players are taking turns adding one or two lines to

L

, where each of these lines has to pass through at least two of the points in

P

. After that all intersection points of the lines in

L

are added to

P

, if they are not already part of it. A player wins, if after his turn there are three collinear points from

P

, which lie on a line that isn’t from

L

. Find who of the two players has a winning strategy.

game strategycombinatorial geometry