For \( a > b > 0 \), find the value of
\[
\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.
\]
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For \( a > b > 0 \), find the value of
\[
\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.
\]
Find the minimum \(m\) (if it exists) such that every convex function \(f:[-1,1]\to[-1,1]\) has a constant \(c\) such that \[ \int_{-1}^1 \lvert f(x)-c\rvert \,dx \le m.\]
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 \( (x_1, x_2, \dots, x_n) \).
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 \(E(G)=E(G_1)\cup E(G_2)\).
Assume that \( x \in \mathbb{R}^n \) with at least \( k \) non-zero entries \( ( k> 0 ) \). Let
\[
A = \{ y \in \{-1, 1\}^n : y \cdot x = 0 \}.
\]
Prove that \( |A| \leq k^{-1/2} 2^n \).
Let \(A\) be a \(2\times 2\) matrix. Prove that if \(Av_1=\lambda_1v_1\) and \(Av_2=\lambda_2v_2\) for distinct reals \(\lambda_1\) and \(\lambda_2\) and nonzero vectors \(v_1\) and \(v_2\), then both columns of \(A-\lambda_1 I\) is a multiple of \(v_2\).
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).
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.
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 \( (S(n))^r = S(n^r) \).
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 \(N\).