# 2021-06 A nondecreasing subsequence

Let $$\mathcal{A}_n$$ be the collection of all sequences $$\mathbf{a}= (a_1,\dots, a_n)$$ with $$a_i \in [i]$$ for all $$i\in [n]=\{1,2,\dots, n\}$$. A nondecreasing $$k$$-subsequence of $$\mathbf{a}$$ is a subsequence $$(a_{i_1}, a_{i_2},\dots, a_{i_k})$$ such that $$i_1< i_2< \dots < i_k$$ and $$a_{i_1}\leq a_{i_2}\leq \dots \leq a_{i_k}$$. For given $$k$$, determine the smallest $$n$$ such that any sequence $$\mathbf{a}\in \mathcal{A}_n$$ has a nondecreasing $$k$$-subsequence.

GD Star Rating

# 2021-05 Finite generation of a group

Prove or disprove that if all elements of an infinite group G has order less than n for some positive integer n, then G is finitely generated.

GD Star Rating

# 2021-04 Product of matrices

For an $$n \times n$$ matrix $$M$$ with real eigenvalues, let $$\lambda(M)$$ be the largest eigenvalue of $$M$$. Prove that for any positive integer $$r$$ and positive semidefinite matrices $$A, B$$,

$[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.$

GD Star Rating

# Solution: 2021-03 A placement of rooks on a chessboard

Consider an $$n$$ by $$n$$ chessboard with white/black squares alternating on every row and every column. In how many ways can one choose $$k$$ white squares and $$n-k$$ black squares from this chessboard with no two squares in a row or column.

The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!

Here is his solution of problem 2021-03.

Other solutions was submitted by 하석민 (수리과학과 2017학번, +3), 고성훈 (수리과학과 2015학번, +3), 전해구 (기계공학과 졸업생, +3).

GD Star Rating

# 2021-03 A placement of rooks on a chessboard

Consider an $$n$$ by $$n$$ chessboard with white/black squares alternating on every row and every column. In how many ways can one choose $$k$$ white squares and $$n-k$$ black squares from this chessboard with no two squares in a row or column.

GD Star Rating

# 2021-02 Inscribed triangles

Show that for any triangle T and any Jordan curve C in the Euclidean plane, there exists a triangle inscribed in C which is similar to T.

GD Star Rating

# 2021-01 Single-digit number

Prove that for any given positive integer $$n$$, there exists a sequence of the following operations that transforms $$n$$ to a single-digit number (in decimal representation).

1) multiply a given positive integer by any positive integer.

2) remove all zeros in the decimal representation of a given positive integer.

GD Star Rating

# 2020-24 Divisions of Fibonacci numbers and their remainders

For each $$i \in \mathbb{N}$$, let $$F_i$$ be the $$i$$-th Fibonacci number where $$F_0=0, F_1=1$$ and $$F_{i+1}=F_{i}+F_{i-1}$$ for each $$i\geq 1$$.
For $$n>m$$, we divide $$F_n$$ by $$F_m$$ to obtain the remainder $$R$$. Prove that either $$R$$ or $$F_m-R$$ is a Fibonacci number.

GD Star Rating

# 2020-23 The area of a random polygon

Suppose we choose a point on the unit circle in the plane at random with the uniform probability measure on the circle. When we choose n points in that way, what is the probability of the n-gon obtained as the convex hull of the chosen points has the area bigger than $$\pi/2$$ in terms of n?

GD Star Rating
Let $$S$$ be the unit sphere in $$\mathbb{R}^n$$, centered at the origin, and $$P_1 P_2 \dots P_{n+1}$$ a regular simplex inscribed in $$S$$. Prove that for a point $$P$$ inside $$S$$,
$\sum_{i=1}^{n+1} (PP_i)^4$
depends only on the distance $$OP$$ (and $$n$$).