Let \(f(x)\in \mathbb R[x]\) be a polynomial of degree at most \(n\) such that \[ x^2+f(x)^2\le 1\] for all \( -1\le x\le 1 \). Prove that \( \lvert f'(x)\rvert \le 2(n-1)\) for all \( -1\le x\le 1\).
Category Archives: problem
2017-13 Infinite series with recurrence relation
Let \(a_0 = a_1 =1\) and \(a_n = n a_{n-1} + (n-1) a_{n-2}\) for \(n \geq 2\). Find the value of
\[
\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.
\]
2017-12 Invertible matrices
Let \(A\) and \(B\) be \(n\times n\) matrices. Prove that if \(n\) is odd and both \(A+A^T\) and \(B+B^T\) are invertible, then \(AB\neq 0\).
2017-11 Infinite series
Find the value of
\[
\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.
\]
2017-10 An inequality for determinant
Let \(A\), \(B\) be matrices over the reals with \(n\) rows. Let \(M=\begin{pmatrix}A &B\end{pmatrix}\). Prove that \[ \det(M^TM)\le \det(A^TA)\det(B^TB).\]
2017-09 A Diophantine Equation
Find all positive integers \( a, b, c \) satisfying
\[
3^a + 5^b = 2^c.
\]
2017-08 Long arithmetic progression
Does there exist a constant \(\varepsilon>0\) such that for each positive integer \(n\) and each subset \(A\) of \(\{1,2,\ldots,n\}\) with \(\lvert A\rvert<\varepsilon n\), there exists an artihmetic progression \(S\) in \(\{1,2,\ldots,n\}\) such that \( S\cap A=\emptyset\) and \(\lvert S\rvert >\varepsilon n\)?
2017-07 Supremum of a series
For \( \theta>0 \), let
\[
f(\theta) = \sum_{n=1}^{\infty} \left( \frac{1}{n+ \theta} – \frac{1}{n+ 3\theta} \right).
\]
Find \( \sup_{\theta > 0} f(\theta) \).
2017-06 Powers of 2
Does there exist infinitely many positive integers \(n\) such that the first digit of \(2^n\) is \(9\)?
2017-05 Inequality for a continuous function
Suppose that \( f : (2, \infty) \to (-2, 2) \) is a continuous function and there exists a positive constant \( m \) such that \( | 1 + xf(x) + (f(x))^2 | \leq m \) for any \( x > 2 \). Prove that, for any \( x > 2 \),
\[
\left| f(x) – \frac{\sqrt{x^2 -4}-x}{2} \right| \leq 6 \sqrt{m}.
\]