Problem 781
Prove that \(\displaystyle \frac{\sin x + \sin y}{2} \le \sin{\frac{x+y}2} \qquad (x,y\in[0,\pi])\ !\) Difficulty: 4. |
Problem 782
Prove that on the interval \(\displaystyle (0,\pi/2)\) we have \(\displaystyle \tg x>x+{x^3\over 3}\). Difficulty: 4. |
Problem 784
(1) \(\displaystyle 2^x\leq 1+x\leq e^x\), (2) Difficulty: 4. |
Problem 785
Prove that \(\displaystyle |\arctg x -\arctg y|\le |x-y|\) for all \(\displaystyle x,y\). Difficulty: 4. |
Problem 791
Prove that \(\displaystyle \displaystyle\cos x\ge1-\frac{x^2}2\). Difficulty: 4. |
Problem 795
What is the range of the function \(\displaystyle x\mapsto\dfrac{e^x}{x}\) (\(\displaystyle x\in\R\setminus\{0\}\)? Difficulty: 4. |
Problem 786
Let \(\displaystyle x<0\) and \(\displaystyle n\) positive integer. Which one is the greater? \(\displaystyle e^x\) or \(\displaystyle 1+\dfrac{x}{1!}+\dfrac{x^2}{2!}+\ldots+\dfrac{x^n}{n!}\)? Difficulty: 5. |
Problem 792
Prove that \(\displaystyle \cos x < e^{-x^2/2}\) if \(\displaystyle 0<x<\frac{\pi}2\). Difficulty: 5. |
Problem 797
Let \(\displaystyle 0<x,y<\pi\). Which one is greater: \(\displaystyle \sin\sqrt{xy}\), or \(\displaystyle \sqrt{\sin x\cdot\sin y}\)? Difficulty: 5. |
Problem 783
Prove that for all \(\displaystyle x>0\) we have \(\displaystyle \frac{x}{1+x}<\log(1+x)<x.\) Difficulty: 6. |
Problem 794
Let \(\displaystyle |x|<\dfrac\pi2\). Which one is greater, \(\displaystyle \dfrac{\sin x}{x}\) or \(\displaystyle e^{-x^2/2}\)? Difficulty: 7. |
Problem 787
Prove that if \(\displaystyle a>1\) and \(\displaystyle 0<x<\frac{\pi}{a}\), then \(\displaystyle \dfrac{\sin ax}{\sin x}<ae^{-\frac{a^2-1}6x^2}\). Difficulty: 9. |
Problem 788
Prove that for all positive integer \(\displaystyle n\) and\(\displaystyle x>0\) we have \(\displaystyle \dfrac{\displaystyle\binom{n}{0}}{\sqrt{x}}- \dfrac{\displaystyle\binom{n}{1}}{\sqrt{x+1}}+ \dfrac{\displaystyle\binom{n}{2}}{\sqrt{x+2}}- \dfrac{\displaystyle\binom{n}{3}}{\sqrt{x+3}}+-\ldots+(-1)^n \dfrac{\displaystyle\binom{n}{n}}{\sqrt{x+n}} > 0. \) Difficulty: 9. |
Problem 796
Let \(\displaystyle p(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0\) be a polynomial with real coefficients and \(\displaystyle n\ge2\), and suppose that the polynomial \(\displaystyle (x-1)^{k+1}\) divides \(\displaystyle p(x)\) with some positive integer \(\displaystyle k\). Prove that \(\displaystyle \sum_{\ell=0}^{n-1} |a_\ell| > 1+\frac{2k^2}{n}. \) CIIM 4, Guanajuato, Mexico, 2012 Difficulty: 10. Solution is available for this problem. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |