Find all integers \( n \) such that \( \sqrt{1} + \sqrt{2} + \dots + \sqrt{n} \) is an integer.
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Tag Archives: integer
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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2009-12 Colorful sum
Suppose that we color integers 1, 2, 3, …, n with three colors so that each color is given to more than n/4 integers. Prove that there exist x, y, z such that x+y=z and x,y,z have distinct colors.
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2009-1 Integer or not
Let \(a_1\le a_2\le \cdots \le a_n\) be integers. Prove that
\(\displaystyle\prod_{1\le j<i\le n} \frac{a_i-a_j}{i-j}\)
is an integer.
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