Category Archives: problem

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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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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2008-6 Many primes

Let f(x) be a polynomial with integer coefficients. Prove that if f(x) is not constant, then there are infinitely many primes p such that \(f(x)\equiv 0\pmod p\) has a solution x.

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2008-5 Monochromatic lines (10/2)

Suppose that P is a finite set of points in the plane colored by red or blue. Show that if no straight line contains all points of P, then there exists a straight line L with at least two points of P on L such that all points on \(P\cap L\) have the same color.

P는 평면 상에 점들의 집합으로, 각 점은 파랑색 혹은 빨간색으로 칠해져있다. 모든 P의 점을 포함하는 직선이 없다면, P의 두 점이상을 포함하는 직선 중에 \(P\cap L\)의 모든 점이 같은 색이 되도록 하는 직선이 있음을 보여라.

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2008-4 Limit (9/25)

Let \(a_1=\sqrt{1+2}\),
\(a_2=\sqrt{1+2\sqrt{1+3}}\),
\(a_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4}}}\), …,
\(a_n=\sqrt{1+2\sqrt{1+3\sqrt {\cdots \sqrt{\sqrt{\sqrt{\cdots\sqrt{1+n\sqrt{1+(n+1)}}}}}}}}\), … .

Prove that \(\displaystyle\lim_{n\to \infty} \frac{a_{n+1}-a_{n}}{a_n-a_{n-1}}=\frac12\).

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2008-3 Integer Matrices (9/18)

Let A, B be \(3\times 3\) integer matrices such that A, A+B, A+2B, A+3B, A-B, A-2B, A-3B are invertible and their inverse matrices are all integer matrices. 

Prove that A+4B also has an inverse, and its inverse is again an integer matrix.

A, B가 \(3\times 3\) 정수 행렬이면서, A, A+B, A+2B, A+3B, A-B, A-2B, A-3B가 모두 역행렬을 가지고 그 역행렬이 모두 정수행렬이라고 하자. 이때 A+4B 역시 역행렬을 가지고 그 역행렬은 정수행렬임을 보여라.

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2008-2 Strange representation (9/11)

Prove that if x is a real number such that \(0<x\le \frac12\), then x can be represented as an infinite sum

\(\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}\),

where each \(n_k\) is an integer such that \(\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}\).

x가 \(0<x\le \frac12\)을 만족하는 실수일때, x는 아래와 같은 무한급수로 표현할 수 있음을 보여라.

\(\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}\).

여기서 각 \(n_k\)는 정수이며 \(\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}\)을 만족한다.

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