Suppose that \( T \) is an \( N \times N \) matrix
\[
T = \begin{pmatrix}
a_1 & b_1 & 0 & \cdots & 0 \\
b_1 & a_2 & b_2 & \ddots & \vdots \\
0 & b_2 & a_3 & \ddots & 0 \\
\vdots & \ddots & \ddots & \ddots & b_{N-1} \\
0 & \cdots & 0 & b_{N-1} & a_N
\end{pmatrix}
\]
with \( b_i > 0 \) for \( i =1, 2, \dots, N-1 \). Prove that \( T \) has \( N \) distinct eigenvalues.
Category Archives: problem
2019-02 Simplification of an expression with factorials
For any positive integers m and n, show that
\[ C_{n,m} = \frac{(mn)!}{(m!)^n n!} \] is an integer.
2019-01 Equilateral polygon
Suppose that \( \Pi \) is a closed polygon in the plane. If \( \Pi \) is equilateral \( k \)-gon, and if \( A \) is the area of \( \Pi \), and \( L \) the length of its boundary, prove that
\[
\frac{A}{L^2} \leq \frac{1}{4k} \cot \frac{\pi}{k} \leq \frac{1}{4\pi}.
\]
2018-23 Game of polynomials
Two players play a game with a polynomial with undetermined coefficients
\[
1 + c_1 x + c_2 x^2 + \dots + c_7 x^7 + x^8.
\]
Players, in turn, assign a real number to an undetermined coefficient until all coefficients are determined. The first player wins if the polynomial has no real zeros, and the second player wins if the polynomial has at least one real zero. Find who has the winning strategy.
2018-22 Two monic quadratic polynomials
Let \(f_1(x)=x^2+a_1x+b_1\) and \(f_2(x)=x^2+a_2x+b_2\) be polynomials with real coefficients. Prove or disprove that the following are equivalent.
(i) There exist two positive reals \(c_1, c_2\) such that \[ c_1f_1(x)+ c_2 f_2(x) > 0\] for all reals \(x\).
(ii) There is no real \(x\) such that \( f_1(x)\le 0\) and \( f_2(x)\le 0\).
2018-21 AM-GM inequality
Does there exist a (possibly \(n\)-dependent) constant \( C \) such that
\[
\frac{C}{a_n} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \leq \frac{a_1+ \dots + a_n}{n} - \sqrt[n]{a_1 \dots a_n} \leq \frac{C}{a_1} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2
\]
for any \( 0 < a_1 \leq a_2 \leq \dots \leq a_n \)?
2018-20 Almost Linear Function
Let \(f:\mathbb R\to\mathbb R\) be a function such that \[ -1\le f(x+y)-f(x)-f(y)\le 1\] for all reals \(x\), \(y\). Does there exist a constant \(c\) such that \( \lvert f(x)-cx\rvert \le 1\) for all reals \(x\)?
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
\[
(\sin x) f(\sin x) f'(\cos x) + (\cos x) f(\cos x) f'(\sin x) = \frac{2}{\pi \sin x \cos x}.
\]
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?
2018-17 Mathematica does not know the answer
For \( a > b > 0 \), find the value of
\[
\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.
\]