Category Archives: solution

Solution: 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.

The best solution was submitted by 이종원. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김은혜 (+3), 김일희 (+3), 김찬민 (+3), 박훈민 (+3), 안현수 (+3), 어수강 (+3), 윤성철 (+3), 이영민 (+3), 장경석 (+3), 장기정 (+3), 정성진 (+3), 조준영 (+3), 채석주 (+3), 황성호 (+3), 박경호 (+2), 조남경 (+2). An incorrect solutions was submitted by N.J.H. (Some initials here might have been improperly chosen.)

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Solution: 2014-03 Subadditive function

Let \( f: [0, \infty) \to \mathbb{R} \) be a function satisfying the following conditions:

(1) For any \( x, y \geq 0 \), \( f(x+y) \geq f(x)+f(y) \).

(2) For any \( x \in [0, 2] \), \( f(x) \geq x^2 – x \).

Prove that, for any positive integer \( M \) and positive reals \( n_1, n_2, \cdots, n_M \) with \( n_1 + n_2 + \cdots + n_M = M \), we have

\[ f(n_1) + f(n_2) + \cdots + f(n_M) \geq 0. \]

The best solution was submitted by 박훈민. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김일희 (+3), 안현수 (+3), 어수강 (+3), 윤성철 (+3), 이영민 (+3), 이종원 (+3), 장경석 (+3), 장기정 (+3), 정성진 (+3), 정진야 (+3), 조준영 (+3), 채석주 (+3), 황성호 (+3). Incorrect solutions were submitted by K.W.J., K.H.S., N.J.H, M.K.Y., S.W.C., L.H.B., C.W.H. (Some initials here might have been improperly chosen.)

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Solution: 2014-02 Series

Determine all positive integers \(\ell\) such that \[ \sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}\] converges and if it converges, then compute its value.

The best solution was submitted by 황성호 (2013학번). Congratulations!

Alternative solutions were submitted by 박훈민 (+3), 이종원 (+3), 채석주 (+3), 이영민 (+2), 조준영 (+2),정성진 (+3), 장기정 (+3), 오동우 (+3), 이상철 (+3), 어수강 (+3), 엄문용 (+3), 윤성철 (+3), 전한울 (+3), 박경호 (+2), 한대진 (+2), 서진솔 (+2), 이시우 (+2). Four incorrect solutions were submitted (J.K.S., N.J.H., A.H.S., C.J.H.).

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Solution: 2014-01 Uniform convergence

Let \( f \) be a real-valued continuous function on \( [ 0, 1] \). For a positive integer \( n \), define
\[
B_n(f; x) = \sum_{j=0}^n f( \frac{j}{n}) {n \choose j} x^j (1-x)^{n-j}.
\]
Prove that \( B_n (f; x) \) converges to \( f \) uniformly on \( [0, 1 ] \) as \( n \to \infty \).

The best solution was submitted by 김범수. Congratulations!

Similar solutions are submitted by 권현우(+3), 박경호(+3), 오동우(+3), 이시우(+3), 이종원(+3), 이주호(+3), 장경석(+3), 장기정(+3), 정성진(+3), 정진야(+3), 조준영(+3), 채석주(+3), 한대진(+3), 황성호(+3). Thank you for your participation.

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Solution: 2013-23 Polynomials with rational zeros

Find all polynomials \( P(x) = a_n x^n + \cdots + a_1 x + a_0 \) satisfying (i) \( a_n \neq 0 \), (ii) \( (a_0, a_1, \cdots, a_n) \) is a permutation of \( (0, 1, \cdots, n) \), and (iii) all zeros of \( P(x) \) are rational.

The best solution was submitted by 전한솔. Congratulations!

Similar solutions are submitted by 김호진(+3), 박민재(+3), 엄태현(+3), 정성진(+3), 진우영(+3), 한대진(+3). Thank you for your participation.

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Solution: 2013-22 Field automorphisms

Find all field automorphisms of the field of real numbers \( \mathbb{R} \). (A field automorphism of a field \( F \) is a bijective map \( \sigma : F \to F \) that preserves all of \( F \)’s algebraic properties.)

The best solution was submitted by 박지민. Congratulations!

Similar solutions are submitted by 고진용(+3), 김호진(+3), 박경호(+3), 박민재(+3), 박훈민(+3), 어수강(+3), 전한솔(+3), 정성진(+3), 진우영(+3), 한대진(+3). Thank you for your participation.

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Solution: 2013-21 Unique inverse

Let \( f(z) = z + e^{-z} \). Prove that, for any real number \( \lambda > 1 \), there exists a unique \( w \in H = \{ z \in \mathbb{C} : \text{Re } z > 0 \} \) such that \( f(w) = \lambda \).

The best solution was submitted by 박민재. Congratulations!

Similar solutions are submitted by 김동률(+3), 김범수(+3), 김호진(+3), 박지민(+3), 박훈민(+3), 양지훈(+3), 이시우(+3), 전한솔(+3), 정성진(+3), 조정휘(+3), 진우영(+3), Koswara(+3), Harmanto(+3). Thank you for your participation.

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Solution: 2013-20 Eigenvalues of Hermitian matrices

Let \( A, B, C = A+B \) be \( N \times N \) Hermitian matrices. Let \( \alpha_1 \geq \cdots \geq \alpha_N \), \( \beta_1 \geq \cdots \geq \beta_N \), \( \gamma_1 \geq \cdots \geq \gamma_N \) be the eigenvalues of \( A, B, C \), respectively. For any \( 1 \leq i, j \leq N \) with \( i+j -1 \leq N \), prove that
\[ \gamma_{i+j-1} \leq \alpha_i + \beta_j \]

The best solution was submitted by 진우영. Congratulations!

Similar solutions are submitted by 김호진(+3), 박민재(+3), 박훈민(+3), 정성진(+3). Thank you for your participation.

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Solution: 2013-19 Integral inequality

Suppose that a function \( f:[0, 1] \to (0, \infty) \) satisfies that
\[ \int_0^1 f(x) dx = 1. \]
Prove the following inequality.
\[ \left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx. \]

The best solution was submitted by 정성진. Congratulations!

Similar solutions are submitted by 박민재(+3), 진우영(+3). Thank you for your participation.

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Solution: 2013-18 Idempotent elements

Let \( R \) be a ring of characteristic zero. Assume further that \( na \neq 0 \) for a positive integer \( n \) and \( a \in R \) unless \( a = 0 \). Suppose that \( e, f, g \in R \) are idempotent (with respect to the multiplication) and satisfy \( e + f + g = 0 \). Show that \( e = f = g = 0 \). (An element \( a \) is idempotent if \( a^2 = a \). )

The best solution was submitted by 박훈민. Congratulations!

Similar solutions are submitted by 김동현(+3), 김호진(+3), 도수일(+3), 박민재(+3), 정성진(+3), 진우영(+3). Thank you for your participation.

Remark: Special thanks to 김동현, who first reported that the condition `characteristic zero’ is insufficient for the problem.

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