2002 Nordic

Part of Nordic

Subcontests

(4)

1

equal products of real numbers

{a_1, a_2, . . . , a_n,}

{b_1, b_2, . . . , b_n}

be real numbers with

{a_1, a_2, . . . , a_n}

distinct. Show that if the product

{(a_i + b_1)(a_i + b_2) \cdot \cdot \cdot (a_i + b_n)}

takes the same value for every

{ i = 1, 2, . . . , n, }

, then the product

{(a_1 + b_j)(a_2 + b_j) \cdot \cdot \cdot (a_n + b_j)}

also takes the same value for every

{j = 1, 2, . . . , n, }

1

probability that 9-digit numbers are mutliples of 11

Eva, Per and Anna play with their pocket calculators. They choose different integers and check, whether or not they are divisible by

{11}

. They only look at nine-digit numbers consisting of all the digits

{1, 2, . . . , 9}

. Anna claims that the probability of such a number to be a multiple of

{11}

{1/11}

. Eva has a different opinion: she thinks the probability is less than

{1/11}

. Per thinks the probability is more than

{1/11}

. Who is correct?

1

largest possible value of balls between two balls

In two bowls there are in total

{N}

balls, numbered from

{1}

{N}

. One ball is moved from one of the bowls into the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount,

{x}

. Determine the largest possible value of

{x}

1

equal segments in an isosceles inscribed trapezoid

{ABCD}

{AB}

{CD}

are parallel and

{AD < CD}

, is inscribed in the circle

{c}

{DP}

be a chord of the circle, parallel to

{AC}

. Assume that the tangent to

{c}

{D}

{AB}

{E}

{PB}

{DC}

{Q}

{EQ = AC}