# 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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