2017 Japan TST

Part of Japan TST

Subcontests

(3)

1

Prove sequence is bounded

A sequence of positive integers

a_1,a_2,\ldots

a_n > \frac{a_{n+1}+a_{n+2}+\ldots +a_{2n}}{n+2016}

for all positive integers

n

. Show that there exists a positive real

C

a_n<C

for all positive integers

n

1

Sum of quartic and power of 2 is perfect square

Find all pairs of positive integers

(n,k)

n^4+10n^2+2^k

is a perfect square.

1

Sum of exponents for all n

Find all positive integers

k

such that there exist positive integer sequences

a_1,a_2,\ldots

r_1,r_2,\ldots

satisfying the following conditions:
[*]

a_1<a_2<a_3<\ldots

a_1^k+a_2^k+\ldots +a_n^k=(a_1+a_2+\ldots +a_n)^{r_n}

holds for all positive integers

n

.