# 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.

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# Solution: 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.

The best solution was submitted by 고성훈 (수리과학과 2018학번, +4). Congratulations!

Here is his solution of problem 2020-24.

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

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# 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.

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# 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?

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# Notice on POW 2020-21

POW 2020-21 is still open and anyone who first submits a correct solution will get the full credit.

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# Solution: 2020-22 Regular simplex

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 $$n$$).

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

Here is his solution of problem 2020-22.

Other solutions was submitted by 고성훈 (수리과학과 2018학번, +3), 채지석 (수리과학과 2016학번, +3).

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# 2020-22 Regular simplex

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 $$n$$).

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# Notice on POW 2020-20

POW 2020-20 is still open and anyone who first submits a correct solution will get the full credit.

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# 2020-21 이적이 부릅니다. 거짓말 거짓말 거짓말(A lie, a lie, a lie)

Alice and Bob play the following game with $$S=\{1,\dots, 777\}$$.

Alice picks a number $$x \in S$$ without telling anyone and Bob will guess what the number is at the end of the game. Alice is malicious so that she can always change her number $$x$$ at any time until the end of the game.

In each round, Bob picks a subset $$T\subseteq S$$ and asks a following question to Alice: “is your $$x$$ belong to $$T$$?” Alice must say either Yes or No. At the end of the game, Bob guesses her $$x$$ first and then Alice reveals her number $$x$$ (Alice can still change her number after she listen to Bob’s guess and before revealing her number). According to her final number $$x$$, each of her previous answers are determined to be either a truth or a lie.

Bob wins if Alice end up lying more than three times or his answer is correct. Alice wins if Bob’s answer is wrong and at most three of her answers are lies. Prove that if a game consists of twenty rounds, then no matter what Bob does Alice can always win.

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Let $$S_g$$ denote the closed orientable connected surface of genus $$g$$. Suppose we glue triangles along the edges so that the resulting space is $$S_g$$ and the intersection of any two triangles are either empty or a single edge. Let $$n(g)$$ be the minimum number of triangles one needs to make $$S_g$$ while satisfying the above rule. What are $$n(1), n(2), n(3)$$? Does the limit $$\lim_{g \to \infty} n(g)/g$$ exist?