Let \( n \) be a positive integer. Determine all continuous functions \(f: [0, 1] \to \mathbb{R}\) such that

\[

f(x_1) + \dots + f(x_n) =1

\]

for all \( x_1, \dots, x_n \in [0, 1] \) satisfying \( x_1 + \dots + x_n = 1\).

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Let \( n \) be a positive integer. Determine all continuous functions \(f: [0, 1] \to \mathbb{R}\) such that

\[

f(x_1) + \dots + f(x_n) =1

\]

for all \( x_1, \dots, x_n \in [0, 1] \) satisfying \( x_1 + \dots + x_n = 1\).

POW will resume on Oct. 30.

Consider the cards with labels \( 1,\dots, n \) in some order. If the top card has label \(m \), we reverse the order of the top \( m \) cards. The process stops only when the card with label \( 1\) is on the top. Prove that the process must stop in at most \( (1.7)^n \) steps.

The best solution was submitted by 길현준 (수리과학과 2018학번). Congratulations!

Here is his solution of problem 2020-18.

Other solutions was submitted by 김유일 (2020학번, +3), 이준호 (수리과학과 2016학번, +3).

Consider the cards with labels \( 1,\dots, n \) in some order. If the top card has label \(m \), we reverse the order of the top \( m \) cards. The process stops only when the card with label \( 1\) is on the top. Prove that the process must stop in at most \( (1.7)^n \) steps.

Prove or disprove that a surjective homomorphism from a finitely generated abelian group to itself is an isomorphism.

The best solution was submitted by 김유일 (2020학번). Congratulations!

Here is his solution of problem 2020-17.

A different solution was submitted by 박은아 (수리과학과 2015학번, +3).

Let \( A \) be an \( n \times n \) Hermitian matrix and \( \lambda_1 (A) \geq \lambda_2 (A) \geq \dots \geq \lambda_n (A) \) the eigenvalues of \( A \). Prove that for any \( 1 \leq k \leq n \)

\[

A \mapsto \lambda_1 (A) + \lambda_2 (A) + \dots + \lambda_k (A)

\]

is a convex function.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-16.

Other solutions were submitted by 길현준 (수리과학과 2018학번, +3), 이준호 (수리과학과 2016학번, +3).

Prove or disprove that a surjective homomorphism from a finitely generated abelian group to itself is an isomorphism.