Personal Site of Andrei N. Ciobanu - Olympiad

Problem 00212 Let \(n\ge1\) be an integer and let \(a_1,a_2,\ldots,a_n\ge0\) be real numbers such that: \[ \sum_{i=1}^{n}a_i^2=n. \] Prove that, for every integer \(m\ge1\), \[ \sum_{i=1}^{n}\frac{a_i^{2m+1}}{\sqrt{a_i+(n-1)a_{i+1}}} \ge \frac{n}{\sqrt{\d…

Problem 00210 Let \(a,b,c\) be positive real numbers such that \(a^2+b^2+c^2=1\).

Problem 00209 Prove that for any real number $x$, the following relation holds: \[ \left\lfloor \frac{x+3}{6} \right\rfloor - \left\lfloor \frac{x+4}{6} \right\rfloor + \left\lfloor \frac{x+5}{6} \right\rfloor = \left\lfloor \frac{x+1}{2} \right\rfloor -…

Problem 00208 Consider a function \(f: \mathbb{Z} \to \mathbb{Z}\) with the property: \[ f(m^2 + f(n)) = (f(m))^2 + n \] for any \(m, n \in \mathbb{Z}\).

Problem 00207 In the coordinate system \(xOy\), consider the lines with equations: \[ \begin{aligned} &d_1: 2x - y - 2 = 0 \\ &d_2: x + y - 4 = 0 \\ &d_3: y = 2 \\ &d_4: x - 4y + 3 = 0 \end{aligned} \] Find the vertices of the triangles that have \(d_1, d…

Problem 00206 Consider the equation: $x^2 + (a+b+c)x + \lambda(ab+bc+ca) = 0$, where $a, b, c$ are strictly positive real numbers and $\lambda$ is a real parameter.

Problem 00211 Let \(a,b,c\) be positive real numbers such that \(ab+bc+ca=3\).

Problem 00043 Let \(x,y,z\) be positive real numbers.

Problem 00042 Let \(a,b,c\) be positive real numbers such that \(a^3+b^3+c^3=3\).

Problem 00041 Let \(n\ge3\) be an integer and let \(k,x_1,x_2,\dots,x_n\in(0,1)\) satisfy \(k\gt\max\{x_1,x_2,\dots,x_n\}\).

Problem 00040 Let \(n\ge2\) be an integer, and let \(0 \lt a_1, a_2, \dots, a_n \leq 1\) satisfy \(a_1+a_2+\dots+a_n=n-1\).

Problem 00039 Let \(a,b,c\) be positive real numbers such that \(a+b+c=1\).

Problem 00038 Let \(n \in \mathbb{N}\setminus\{0,1\}\).

Problem 00037 Let \(n>1\) be an integer.

Problem 00036 Let \(n\) be a positive integer.

Problem 00035 Let \(x,y,z\) be positive real numbers such that \(xyz=6\).

Problem 00034 Let \(n\ge2\) be an even integer, and let \(x_1,x_2,\dots,x_n>0\) satisfy \(\prod_{i=1}^nx_i=1\).

Problem 00033 Let \(x,y,z\) be positive real numbers such that \((1+x)(1+y)(1+z)=8\).

Problem 00032 Let \(a,b,c>0\).

Problem 00031 Let \(a,b,c\) be positive real numbers.

Problem 00030 Let \(x, y, z\) be positive real numbers.

Problem 00029 Let \(z \in [0, \infty)\), and \(x, y \in [1, \infty)\).

Problem 00028 Let \(\lambda \ge 3\) be fixed, and let \(x\), \(y\), and \(z\) be positive real numbers such that \(x + y + z = 3\).

Problem 00027 Let \(a\), \(b\), and \(c\) be positive real numbers satisfying \(a^2 + b^2 + c^2 \ge 4\).

Problem 00026 Let \(a, b, c\) be positive real numbers.

Problem 00025 Let \(a, b, c\) be positive real numbers such that \(a^2 + b^2 + c^2 = 1\).

Problem 00024 Let \(x,y,z \ge 1\) such that \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2\).

Problem 00023 Let \(x,y,z\ge1\) and \(x+y+z=6\).

Problem 00022 Let \(a,b,c\) be positive real numbers.

Problem 00021 Let \(a, b\) be positive real numbers satisfying \(a + b = 1\).

Problem 00020 Let \(x,y,z\) be positive real numbers.

Problem 00019 Let \(a, b, c, d, e \in \mathbb{R}\) with \(a^2 + b^2 + c^2 + d^2 + e^2 = 55\).

Problem 00018 Let \(a, b, c\) be positive real numbers such that \(a + b + c = 3\).

Problem 00017 Let \(n\in\mathbb{Z}_{\ge 1}\).

Problem 00016 Let \(x, y, z \in \mathbb{R}\).

Problem 00015 Let \(a,b,c\) be real numbers.

Problem 00014 Let \(x,y,z \in \mathbb{R}\), and assume that \((x+y+z)^3=x^3+y^3+z^3\).

Problem 00013 Find all pairs of distinct non-negative natural numbers \((x,y)\) such that: \[x^3+y^3=(x+y)^2\] Hint 1 Look for a suitable identity that can help simplify the expression.

Problem 00012 Let \(x,y,z \in \mathbb{R}^{*}\), where \(x< y < z\), and \(\frac{x^2}{yz}+\frac{y^2}{xz}+\frac{z^2}{xy}=3\).

Problem 00011 Find \(x,y,z \in [\frac{1}{3}, \infty)\) such that \(x+y+z \leq 4\) and \(\sqrt{3x-1}+\sqrt{3y-1}+\sqrt{3z-1} \geq 3\sqrt{3}\).

Problem 00010 Let \(a,b,c\) be positive real numbers such that \(abc=1\).

Problem 00009 Let \(a_1, a_2, a_3, a_4\) be positive real numbers.

Problem 00008 Let \(x,y,z\) be positive real numbers.

Problem 00007 Let \(x,y,a,b \gt 0\), prove that: \[ \frac{a}{x}+\frac{b}{y} \ge \frac{4(ay+bx)}{(x+y)^2} \] Solution We begin by simplifying the left-hand side: \[ \frac{a}{x}+\frac{b}{y}=\frac{ay+bx}{xy} \] Thus, the inequality becomes: \[ \frac{ay+bx}{x…

Problem 00006 Let \(n\in\mathbb{Z}_{\ge 1}\), and let \(x_1, x_2, \dots, x_n\) be positive real numbers.

Problem 00005 Let \(a,b\) be positive real numbers.

Problem 00004 Let \(n\ge 5\) be an odd integer.

Problem 00003 Let \(n\) be a positive integer greater than \(1\).

Problem 00002 Let \(n\in\mathbb{Z}_{\ge 1}\) and let \(x_1,x_2,\dots,x_n>0\).

Problem 00001 Let \(x>0\).