2020-19 Continuous functions

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 $$x_1 + \dots + x_n = 1$$.

GD Star Rating

Notice: Mid-term break

POW will resume on Oct. 30.

GD Star Rating

Solution: 2020-18 A way of shuffling cards

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).

GD Star Rating

2020-18 A way of shuffling cards

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.

GD Star Rating

Solution: 2020-17 Endomorphisms of abelian groups

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).

GD Star Rating

Solution: 2020-16 A convex function of matrices

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).

GD Star Rating