Prove that \(\displaystyle \dfrac{x^2}{1+x^4}\leq \dfrac12\).

Difficulty: 2.

Prove that \(\displaystyle x^2+\dfrac1{x^2}\geq 2\) if \(\displaystyle x\neq 0\).

Difficulty: 2.

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.

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.

Prove that the following inequality holds for all \(\displaystyle a,b,c>0\)!

\(\displaystyle \frac ab+\frac bc+\frac ca \ge 3.\)

Difficulty: 3.

Let \(\displaystyle a,b>0\). For which \(\displaystyle x\) the expression \(\displaystyle \dfrac{a+bx^4}{x^2}\) is minimal?

Difficulty: 4.

Suppose that the product of three positive numbers is \(\displaystyle 1\).

 (1)  What is the maximum of their sum?
 (2)  What is the minimum of their sum?
 (3)  What is the maximum of the sum of their inverses?
 (4)  What is the minimum of the sum of their inverses?

Difficulty: 4.

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.

Which rectangular box has the greatest volume among the ones with given surface area?

Difficulty: 4. Solution is available for this problem.

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.

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.

Prove that \(\displaystyle n!<\left(\dfrac{n+1}2\right)^n\).

Difficulty: 5. Solution is available for this problem.

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.

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.

What is the maximum of the function \(\displaystyle x^3-x^5\) on the interval \(\displaystyle [0,1]\)?

Difficulty: 6.

What is the greatest volume of a cylinder inscribed into a right circular cone?

Difficulty: 6.

What is the greatest volume of a cylinder inscribed into the unit sphere?

Difficulty: 6.

Which one is the greater? \(\displaystyle 1000001^{1000000}\) or \(\displaystyle 1000000^{1000001}\).

Difficulty: 8.

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.