Problem 60
Prove that \(\displaystyle \dfrac{x^2}{1+x^4}\leq \dfrac12\). Difficulty: 2. |
Problem 66
Prove that \(\displaystyle x^2+\dfrac1{x^2}\geq 2\) if \(\displaystyle x\neq 0\). Difficulty: 2. |
Problem 59
Prove that if \(\displaystyle a,b,c>0\), then the following inequality holds \(\displaystyle \frac {a^2}{bc}+\frac {b^2}{ac}+\frac {c^2}{ab} \ge 3.\) Difficulty: 3. Solution is available for this problem. |
Problem 62
Let \(\displaystyle a_i>0\). Prove that \(\displaystyle {a_1\over a_2}+{a_2\over a_3}+\ldots+{a_{n-1}\over a_n}+{a_n \over a_1}\geq n\) Difficulty: 3. |
Problem 69
Prove that the following inequality holds for all \(\displaystyle a,b,c>0\)! \(\displaystyle \frac ab+\frac bc+\frac ca \ge 3.\) Difficulty: 3. |
Problem 61
Let \(\displaystyle a,b>0\). For which \(\displaystyle x\) the expression \(\displaystyle \dfrac{a+bx^4}{x^2}\) is minimal? Difficulty: 4. |
Problem 64
Suppose that the product of three positive numbers is \(\displaystyle 1\).
Difficulty: 4. |
Problem 65
What is the maximum value of \(\displaystyle xy\), if \(\displaystyle x,y\ge0\) and (a) \(\displaystyle x+y=10\); (b) \(\displaystyle 2x+3y=10\)? Difficulty: 4. |
Problem 67
Which rectangular box has the greatest volume among the ones with given surface area? Difficulty: 4. Solution is available for this problem. |
Problem 68
What is the maximum value of \(\displaystyle a^3b^2c\), if \(\displaystyle a,b,c\) are nonnegative and \(\displaystyle {a+2b+3c=5}\)? Difficulty: 4. |
Problem 70
Calculate the maximum value of the function \(\displaystyle x^2 \cdot (1-x)\) for \(\displaystyle x\in[0,1].\) Difficulty: 4. Solution is available for this problem. |
Problem 72
Prove that \(\displaystyle n!<\left(\dfrac{n+1}2\right)^n\). Difficulty: 5. Solution is available for this problem. |
Problem 58
Let \(\displaystyle a,b\geq 0\) and \(\displaystyle r,s\) be positive rational numbers with \(\displaystyle r+s=1\). Show that \(\displaystyle a^r\cdot b^s\leq ra+sb.\) Difficulty: 6. |
Problem 71
Prove that the cylinder with the least surface area among the ones with given volume is the cylinder with height equals to the diameter of its base. Difficulty: 6. Solution is available for this problem. |
Problem 73
What is the maximum of the function \(\displaystyle x^3-x^5\) on the interval \(\displaystyle [0,1]\)? Difficulty: 6. |
Problem 74
What is the greatest volume of a cylinder inscribed into a right circular cone? Difficulty: 6. |
Problem 75
What is the greatest volume of a cylinder inscribed into the unit sphere? Difficulty: 6. |
Problem 63
Which one is the greater? \(\displaystyle 1000001^{1000000}\) or \(\displaystyle 1000000^{1000001}\). Difficulty: 8. |
Problem 76
Prove that for any sequence \(\displaystyle a_1,a_2,\ldots,a_n\) of positive real numbers, \(\displaystyle \frac1{\frac1{a_1}} + \frac2{\frac1{a_1}+\frac1{a_2}} + \frac3{\frac1{a_1}+\frac1{a_2}+\frac1{a_3}} + \ldots + \frac{n}{\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}} < 2 (a_1+a_2+\ldots+a_n). \) (KöMaL N. 189., November 1998) Difficulty: 10. Solution is available for this problem. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |