2015 Nordic

Part of Nordic

Subcontests

(4)

1

permutations of 2000 volumes of encyclopedia with two rules

An encyclopedia consists of

{2000}

numbered volumes. The volumes are stacked in order with number

{1}

{2000}

in the bottom. One may perform two operations with the stack: (i) For

{n}

even, one may take the top

{n}

volumes and put them in the bottom of the stack without changing the order. (ii) For

{n}

odd, one may take the top

{n}

volumes, turn the order around and put them on top of the stack again. How many different permutations of the volumes can be obtained by using these two operations repeatedly?

1

3 primes with sum 101 times their product or the opposite

Find the primes

{p, q, r}

, given that one of the numbers

{pqr}

{p + q + r}

{101}

times the other.

1

circle with diameter a side of a triangle

{ABC}

be a triangle and

{\Gamma}

the circle with diameter

{AB}

. The bisectors of

{\angle BAC}

{\angle ABC}

{\Gamma}

{D}

{E}

, respectively. The incircle of

{ABC}

{BC}

{AC}

{F}

{G}

, respectively. Prove that

{D, E, F}

{G}

1

Polynomial

n > 1

p(x)=x^n+a_{n-1}x^{n-1} +...+a_0

be a polynomial with

n

real roots (counted with multiplicity). Let the polynomial

q

q(x) = \prod_{j=1}^{2015} p(x + j)

p(2015) = 2015

q

1970

different roots

r_1, ..., r_{1970}

|r_j| < 2015

j = 1, ..., 1970