MathDB

m be the number of distinct sums a_i +a_j, i\ne j, a_i pairwise distinct

Source: 2002 Estonia National Olympiad Final Round grade 9 p3

3/16/2020

Let

a_1,a_2,...,a_n

be pairwise distinct real numbers and

m

be the number of distinct sums

a_i +a_j

(where

i \ne j

). Find the least possible value of

m

.

Sumcombinatorics

a_i a_j, a_i+a_j , |a_i -a_j |, i \ne j, max no distinct odd integers

Source: 2002 Estonia National Olympiad Final Round grade 10 p3

3/16/2020

John takes seven positive integers

a_1,a_2,...,a_7

and writes the numbers

a_i a_j

,

a_i+a_j

and

|a_i -a_j |

for all

i \ne j

on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.

number theoryoddSumProduct

staring with a 2002-digits number with only 9, digits get replaced

Source: 2002 Estonia National Olympiad Final Round grade 11 p3

3/14/2020

The teacher writes a

2002

-digit number consisting only of digits

9

on the blackboard. The first student factors this number as

ab

with

a > 1

and

b > 1

and replaces it on the blackboard by two numbers

a'

and

b'

with

|a-a'| = |b-b'| = 2

. The second student chooses one of the numbers on the blackboard, factors it as

cd

with

c > 1

and

d > 1

and replaces the chosen number by two numbers

c'

and

d'

with

|c-c'| = |d-d'| = 2

, etc. Is it possible that after a certain number of students have been to the blackboard all numbers written there are equal to

9

?

number theoryDigits

2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2 iff a,b,c>0 are sidelengths

Source: 2002 Estonia National Olympiad Final Round grade 12 p3

3/14/2020

Prove that for positive real numbers

a, b

and

c

the inequality

2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2

holds if and only if

a,b,c

are the sides of a triangle.

inequalitiesGeometric Inequalitiessidelenghts

3

Problems(4)