# 2023-01 An integral sequence (again)

Suppose $$a_1, a_2, \dots, a_{2023}$$ are real numbers such that
$a_1^3 + a_2^3 + \dots + a_n^3 = (a_1 + a_2 + \dots + a_n)^2$
for any $$n = 1, 2, \dots, 2023$$. Prove or disprove that $$a_n$$ is an integer for any $$n = 1, 2, \dots, 2023$$.

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# 2022-22 An integral sequence

Define a sequence $$a_n$$ by $$a_1 = 1$$ and
$a_{n+1} = \frac{1}{n} \left( 1 + \sum_{k=1}^n a_k^2 \right)$
for any $$n \geq 1$$. Prove or disprove that $$a_n$$ is an integer for all $$n \geq 1$$.

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# 2018-03 Integers from square roots

Find all integers $$n$$ such that $$\sqrt{1} + \sqrt{2} + \dots + \sqrt{n}$$ is an integer.

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# 2014-09 Product of series

For integer $$n \geq 1$$, define
$a_n = \sum_{k=0}^{\infty} \frac{k^n}{k!}, \quad b_n = \sum_{k=0}^{\infty} (-1)^k \frac{k^n}{k!}.$
Prove that $$a_n b_n$$ is an integer.

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# 2014-04 Integer pairs

Prove that there exist infinitely many pairs of positive integers $$(m, n)$$ satisfying the following properties:

(1) gcd$$(m, n) = 1$$.

(2) $$(x+m)^3 = nx$$ has three distinct integer solutions.

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# 2013-02 Functional equation

Let $$\mathbb{Z}^+$$ be the set of positive integers. Suppose that $$f : \mathbb{Z}^+ \to \mathbb{Z}^+$$ satisfies the following conditions.

i) $$f(f(x)) = 5x$$.

ii) If $$m \geq n$$, then $$f(m) \geq f(n)$$.

iii) $$f(1) \neq 2$$.

Find $$f(256)$$.

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# 2012-4 Sum of squares

Find the smallest and the second smallest odd integers n satisfying the following property: $n=x_1^2+y_1^2 \text{ and } n^2=x_2^2+y_2^2$ for some positive integers $$x_1,y_1,x_2,y_2$$ such that $$x_1-y_1=x_2-y_2$$.

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# 2011-6 Equal sums

Let $$a_1\le a_2\le \cdots \le a_k$$ and $$b_1\le b_2\le \cdots \le b_l$$ be sequences of positive integers at most M. Prove that if $\sum_{i=1}^{k} a_i^n = \sum_{j=1}^l b_j^n$ for all $$1\le n\le M$$, then $$k=l$$ and $$a_i=b_i$$ for all $$1\le i\le k$$.

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# 2011-2 Power

Prove that for all positive integers m and n, there is a positive integer k such that $(\sqrt{m}+\sqrt{m-1})^n = \sqrt{k}+\sqrt{k-1}.$

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