Does there exist a nontrivial subgroup \(G\) of \( GL(10, \mathbb{C}) \) such that each element in \(G\) is diagonalizable but the set of all the elements of \(G\) is not simultaneously diagonalizable?
Category Archives: problem
2023-21 A limit
Find the following limit:
\[
\lim_{n \to \infty} \left( \frac{\sum_{k=1}^{n+2} k^k}{\sum_{k=1}^{n+1} k^k} – \frac{\sum_{k=1}^{n+1} k^k}{\sum_{k=1}^{n} k^k} \right)
\]
2023-20 A sequence with small tail
Can we find a sequence \(a_i, i=0,1,2,…\) with the following property: for each given integer \(n\geq 0\), we have \[\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?\]
2023-19 Counting the number of solutions
Let \(N\) be the number of ordered tuples of positive integers \( (a_1,a_2,\ldots, a_{27} )\) such that \( \frac{1}{a_1} + \frac{1}{a_2} + \cdots +\frac{1}{a_{27}} = 1\). Compute the remainder of \(N\) when \(N\) is divided by \(3\).
2023-18 Degrees of a graph
Find all integers \( n \geq 8 \) such that there exists a simple graph with \( n \) vertices whose degrees are as follows:
(i) \( (n-4) \) vertices of the graph are with degrees \( 4, 5, 6, \dots, n-2, n-1 \), respectively.
(ii) The other \( 4 \) vertices are with degrees \( n-2, n-2, n-1, n-1 \), respectively.
2023-17 Comparing area of triangles
Let \(f(x) = x^4 + (2-a)x^3 – (2a+1)x^2 + (a-2)x + 2a\) for some \(a \geq 2\). Draw two tangent lines of its graph at the point \((-1,0)\) and \((1,0)\) and let \(P\) be the intersection point. Denote by \(T\) the area of the triangle whose vertices are \((-1,0), (1,0)\) and \(P\). Let \(A\) be the area of domain enclosed by the interval \([-1,1]\) and the graph of the function on this interval. Show that \(T \leq 3A/2.\)
2023-16 Zeros in a sequence
Define the sequence \( x_n \) by \( x_1 = 0 \) and
\[
x_n = x_{\lfloor n/2 \rfloor} + (-1)^{n(n+1)/2}
\]
for \( n \geq 2\). Find the number of \( n \leq 2023 \) such that \( x_n = 0 \).
2023-15 An inequality for complex polynomials
Let \(p(z), q(z) \)and \(r(z)\) be polynomials with complex coefficients in the complex plane. Suppose that \(|p(z)| + |q(z)| \leq |r(z)|\) for every \(z\). Show that there exist two complex numbers \( a,b \) such that \(|a|^2 +|b|^2 =1\) and \( a p(z) + bq(z) =0 \) for every \(z\).
2023-14 Dividing polynomials
Let \(f(t)=(t^{pq}-1)(t-1) \) and \(g(t)=(t^{p}-1)(t^q-1) \) where \(p\) and \(q\) are relatively prime positive integers. Prove that \(\frac{f(t)}{g(t)}\) can be written as a polynomial where it has just \(1\) or \(-1\) as coefficients. (For example, when \(p=2\) and \(q=3\), we have that \(\frac{f(t)}{g(t)} = t^2-t+1\).)
2023-13 Range of a function on subintervals
Prove or disprove the existence of a function \( f:[0, 1] \to [0, 1] \) with the following property:
for any interval \( (a, b) \subset [0, 1] \) with \( a<b \), \( f((a, b)) = [0, 1] \).