Tag Archives: integer

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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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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