# 2008-12 Finding eigenvalues and eigenvectors

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

$$\displaystyle \int_0^1 \min(x,y) f(y) \,dy=\lambda f(x)$$
has a non-zero solution $$f$$ that is continuous on the interval [0,1].

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# Solution:2008-11 Sum of square roots

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.

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# 2008-11 Sum of square roots

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.

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# Solution: 2008-10 Inequality with n variables

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.

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# 2008-10 Inequality with n variables

Let $$x_1,x_2,\ldots,x_n$$ be nonnegative real numbers. Show that
$$\displaystyle \left(\sum_{i=1}^n x_i\right) \left(\sum_{i=1}^n x_i^{n-1}\right) \le (n-1) \sum_{i=1}^n x_i^n + n \prod_{i=1}^n x_i$$.

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# Solution: 2008-9 Integer-valued function

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.

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# 2008-9 Integer-valued function

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?

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# Solution: 2008-8 Positive eigenvalues

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.

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