Find all real numbers \(\lambda\) and the corresponding functions \(f\) such that the equation

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Find all real numbers \(\lambda\) and the corresponding functions \(f\) such that the equation

Let a, b, c, d be positive rational numbers. Prove that if \(\sqrt a+\sqrt b+\sqrt c+\sqrt d\) is rational, then each of \(\sqrt a\), \(\sqrt b\), \(\sqrt c\), and \(\sqrt d\) is rational.

The best solution was submitted by Yang, Hae Hun (양해훈), 2008학번. Congratulations!

Here is his Solution of Problem 2008-11.

Let a, b, c, d be positive rational numbers. Prove that if \(\sqrt a+\sqrt b+\sqrt c+\sqrt d\) is rational, then each of \(\sqrt a\), \(\sqrt b\), \(\sqrt c\), and \(\sqrt d\) is rational.

Let \(x_1,x_2,\ldots,x_n\) be nonnegative real numbers. Show that

\(\displaystyle \left(\sum_{i=1}^n x_i\right) \left(\sum_{i=1}^n x_i^{n-1}\right) \le (n-1) \sum_{i=1}^n x_i^n + n \prod_{i=1}^n x_i \).

The best solution was submitted by Sang Hoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2008-10.

Let \(x_1,x_2,\ldots,x_n\) be nonnegative real numbers. Show that

\(\displaystyle \left(\sum_{i=1}^n x_i\right) \left(\sum_{i=1}^n x_i^{n-1}\right) \le (n-1) \sum_{i=1}^n x_i^n + n \prod_{i=1}^n x_i \).

Let \(\mathbb{R}\) be the set of real numbers and let \(\mathbb{N}\) be the set of positive integers. Does there exist a function \(f:\mathbb{R}^3\to \mathbb{N}\) such that f(x,y,z)=f(y,z,w) implies x=y=z=w?

The best solution was submitted by Yang, Hae Hun (양해훈), 2008학번. Congratulations!

Here is his Solution of Problem 2008-9.

Let \(\mathbb{R}\) be the set of real numbers and let \(\mathbb{N}\) be the set of positive integers. Does there exist a function \(f:\mathbb{R}^3\to \mathbb{N}\) such that f(x,y,z)=f(y,z,w) implies x=y=z=w?

Let A be a 0-1 square matrix. If all eigenvalues of A are real positive, then those eigenvalues are all equal to 1.

The best solution was submitted by Sang Hoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2008-8.