Category Archives: problem

2018-19 Gauss’s theorem

Let
\[
f(x) = 1 + \left( \frac{1}{2} \cdot x \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot x^2 \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot x^3 \right)^2 + \dots
\]
Prove that
\[
(\sin x) f(\sin x) f'(\cos x) + (\cos x) f(\cos x) f'(\sin x) = \frac{2}{\pi \sin x \cos x}.
\]

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2018-18 A random walk on the clock

Suppose that we are given 12 points evenly spaced on a circle. Starting from a point in the 12 o’clock position, a particle P will move to one of the adjacent positions with equal probably, 1/2. P stops if it visits all 12 points. What is the most likely point that P stops for the last?

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2018-15 Diophantine equation

Let \( n \) be a positive integer. Suppose that \( a_1, a_2, \dots, a_n \) are non-zero integers and \( b_1, b_2, \dots, b_n\) are positive integers such that \( (b_i, b_n) = 1 \) for \( i = 1, 2, \dots, n-1 \). Prove that the Diophantine equation
\[
a_1 x_1^{b_1} + a_2 x_2^{b_2} + \dots + a_n x_n^{b_n} = 0
\]
has infinitely many integer solutions \( (x_1, x_2, \dots, x_n) \).

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2018-14 Forests and Planes

Suppose that the edges of a graph \(G\) can be colored by 3 colors so that there is no monochromatic cycle. Prove or disprove that \(G\) has two planar subgraphs \(G_1,G_2\) such that \(E(G)=E(G_1)\cup E(G_2)\).

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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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