MathDB

1997 MMO Grade 8 #4

Source:

10/3/2016

Prove that there exists a positive non-prime integer such that if any three of its neighboring digits are replaced with any given triple of the digits, the number remains non-prime. Does there exist a

1997

-digit such number?

Grade 81997

1997 MMO Grade 9 #4

Source:

10/3/2016

Along a circular railroad,

n

trains circulate in the same direction at equal distances between them. Stations

A, B

and

C

on this railroad (denoted as the trains pass them) form an equilateral triangle. Ira enters station

A

at the same time as Alex enters station

B

in order to take the nearest train. It is knows that if they enter the stations at the same time as the driver Roma passes a forest, then Ira takes her train earlier than Alex; otherwise Alex takes the train earlier than or simultaneously with Ira. What part of the railroad goes through the forest (between which stations)?

Grade 91997

1997 MMO Grade 10 #4

Source:

10/3/2016

Given real numbers

a_1\leq{a_2}\leq{a_3}

and

b_1\leq{b_2}\leq{b_3}

such that

a_1+a_2+a_3=b_1+b_2+b_3,

a_1a_2+a_2a_3+a_1a_3=b_1b_2+b_2b_3+b_1b_3.

Prove that if

a_1\leq{b_1},

then

a_3\leq{b_3}

Grade 101997

4

Problems(3)