# 2018-11 Fallacy

On a math exam, there was a question that asked for the largest angle of the triangle with sidelengths $$21$$, $$41$$, and $$50$$. A student obtained the correct answer as follows:

Let $$x$$ be the largest angle. Then,
$\sin x = \frac{50}{41} = 1 + \frac{9}{41}.$
Since $$\sin 90^{\circ} = 1$$ and $$\sin 12^{\circ} 40′ 49” = 9/41$$, the angle $$x = 90^{\circ} + 12^{\circ} 40′ 49” = 102^{\circ} 40′ 49”$$.

Find the triangle with the smallest area with integer sidelengths and possessing this property (that the wrong argument as above gives the correct answer).

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# 2018-10 Probability of making an acute triangle

Given a stick of length 1, we choose two points at random and break it into three pieces. Compute the probability that these three pieces form an acute triangle.

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# Solution: 2018-08 Large LCM

Let $$a_1$$, $$a_2$$, $$\ldots$$, $$a_m$$ be distinct positive integers. Prove that if $$m>2\sqrt{N}$$, then there exist $$i$$, $$j$$ such that the least common multiple of $$a_i$$ and $$a_j$$ is greater than $$N$$.

The best solution was submitted by Bonwoo Lee (이본우, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-08.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 김태균 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3), 이재우 (함양고등학교 3학년, +3).

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# Solution: 2018-07 A tridiagonal matrix

Let $$S$$ be an $$(n+1) \times (n+1)$$ matrix defined by
$S_{ij} = \begin{cases} (n+1)-i & \text{ if } j=i+1, \\ i-1 & \text{ if } j=i-1, \\ 0 & \text{ otherwise. } \end{cases}$
Find all eigenvalues of $$S$$.

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2018-07.

Alternative solutions were submitted by 한준호 (수리과학과 2015학번, +3), 채지석 (수리과학과 2016학번, +3), Hitesh Kumar (Imperial College London, +2), 고성훈 (2018학번, +2).

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# 2018-09 Sum of digits

For a positive integer $$n$$, let $$S(n)$$ be the sum of all decimal digits in $$n$$, i.e., if $$n = n_1 n_2 \dots n_m$$ is the decimal expansion of $$n$$, then $$S(n) = n_1 + n_2 + \dots + n_m$$. Find all positive integers $$n$$ and $$r$$ such that $$(S(n))^r = S(n^r)$$.

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Let $$a_1$$, $$a_2$$, $$\ldots$$, $$a_m$$ be distinct positive integers. Prove that if $$m>2\sqrt{N}$$, then there exist $$i$$, $$j$$ such that the least common multiple of $$a_i$$ and $$a_j$$ is greater than $$N$$.