# Solution: 2022-16 Identity for continuous functions

For a positive integer $$n$$, find all continuous functions $$f: \mathbb{R} \to \mathbb{R}$$ such that
$\sum_{k=0}^n \binom{n}{k} f(x^{2^k}) = 0$
for all $$x \in \mathbb{R}$$.

The best solution was submitted by Kawano Ren (Kaisei Senior High School, +4). Congratulations!

Other solutions were submitted by 김찬우 (연세대학교 수학과, +3), 기영인 (KAIST 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3). Late solutions were not graded.

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# 2022-17 The smallest number of subsets

Let $$n, i$$ be integers such that $$1 \leq i \leq n$$. Each subset of $$\{ 1, 2, \ldots, n \}$$ with $$i$$ elements has the smallest number. We define $$\phi(n,i)$$ to be the sum of these smallest numbers. Compute $\sum_{i=1}^n \phi(n,i).$

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# Solution: 2022-15 A determinant of Stirling numbers of second kind

Let $$S(n,k)$$ be the Stirling number of the second kind that is the number of ways to partition a set of $$n$$ objects into $$k$$ non-empty subsets. Prove the following equality $\det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\ S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\ \cdots & \cdots & \cdots & \cdots \\ S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m$

The best solution was submitted by 기영인 (KAIST 22학번, +4). Congratulations!

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과, +3). Late solutions were not graded.

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# 2022-16 Identity for continuous functions

For a positive integer $$n$$, find all continuous functions $$f: \mathbb{R} \to \mathbb{R}$$ such that
$\sum_{k=0}^n \binom{n}{k} f(x^{2^k}) = 0$
for all $$x \in \mathbb{R}$$.

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# Solution: 2022-14 The number of eigenvalues of a symmetric matrix

For a positive integer $$n$$, let $$B$$ and $$C$$ be real-valued $$n$$ by $$n$$ matrices and $$O$$ be the $$n$$ by $$n$$ zero matrix. Assume further that $$B$$ is invertible and $$C$$ is symmetric. Define $A := \begin{pmatrix} O & B \\ B^T & C \end{pmatrix}.$ What is the possible number of positive eigenvalues for $$A$$?

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

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# 2022-15 A determinant of Stirling numbers of second kind

Let $$S(n,k)$$ be the Stirling number of the second kind that is the number of ways to partition a set of $$n$$ objects into $$k$$ non-empty subsets. Prove the following equality $\det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\ S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\ \cdots & \cdots & \cdots & \cdots \\ S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m$

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# Solution: 2022-13 Inequality involving sums with different powers

Prove for any $$x \geq 1$$ that

$\left( \sum_{n=0}^{\infty} (n+x)^{-2} \right)^2 \geq 2 \sum_{n=0}^{\infty} (n+x)^{-3}.$

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Another solution was submitted by 김찬우 (연세대학교 수학과, +3).

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# 2022-14 The number of eigenvalues of a symmetric matrix

For a positive integer $$n$$, let $$B$$ and $$C$$ be real-valued $$n$$ by $$n$$ matrices and $$O$$ be the $$n$$ by $$n$$ zero matrix. Assume further that $$B$$ is invertible and $$C$$ is symmetric. Define $A := \begin{pmatrix} O & B \\ B^T & C \end{pmatrix}.$ What is the possible number of positive eigenvalues for $$A$$?

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# 2022-13 Inequality involving sums with different powers

Prove for any $$x \geq 1$$ that

$\left( \sum_{n=0}^{\infty} (n+x)^{-2} \right)^2 \geq 2 \sum_{n=0}^{\infty} (n+x)^{-3}.$

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Consider the power set $$P([n])$$ consisting of $$2^n$$ subsets of $$[n]=\{1,\dots,n\}$$.
Find the smallest $$k$$ such that the following holds: there exists a partition $$Q_1,\dots, Q_k$$ of $$P([n])$$ so that there do not exist two distinct sets $$A,B\in P([n])$$ and $$i\in [k]$$ with $$A,B,A\cup B, A\cap B \in Q_i$$.