MathDB

Fixed domination of coins forming values

Source: AllRussian-2014, Grade 9, day2, P3

5/17/2014

In a country, mathematicians chose an

\alpha> 2

and issued coins in denominations of 1 ruble, as well as

\alpha ^k

rubles for each positive integer k.

\alpha

was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?

algorithmalgebrabinomial theoremcombinatorics

Swapping cards with minimum disturbance

Source: All Russian 2014 Grade 10 Day 1 P3

5/17/2014

There are

n

cells with indices from

1

to

n

. Originally, in each cell, there is a card with the corresponding index on it. Vasya shifts the card such that in the

i

-th cell is now a card with the number

a_i

. Petya can swap any two cards with the numbers

x

and

y

, but he must pay

2|x-y|

coins. Show that Petya can return all the cards to their original position, not paying more than

|a_1-1|+|a_2-2|+\ldots +|a_n-n|

coins.

inductioncombinatorics proposedcombinatorics

decimal repres

Source: AllRussian-2014, Grade 11, day1, P3

4/30/2014

Positive rational numbers

a

and

b

are written as decimal fractions and each consists of a minimum period of 30 digits. In the decimal representation of

a-b

, the period is at least

15

. Find the minimum value of

k\in\mathbb{N}

such that, in the decimal representation of

a+kb

, the length of period is at least

15

.
A. Golovanov

number theory proposednumber theory

polynomials on a blackboard

Source: All Russian 2014 Grade 11 Day 2 P3

4/30/2014

If the polynomials

f(x)

and

g(x)

are written on a blackboard then we can also write down the polynomials

f(x)\pm g(x)

,

f(x)g(x)

,

f(g(x))

and

cf(x)

, where

c

is an arbitrary real constant. The polynomials

x^3-3x^2+5

and

x^2-4x

are written on the blackboard. Can we write a nonzero polynomial of form

x^n-1

after a finite number of steps?

algebrapolynomialcalculusderivativefunctionalgebra proposed

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Problems(4)