Found problems: 1671
Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]
What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak?
Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.
Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let
$$b_k=\sum_{i=1}^n a_i \omega^{ki}$$
for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.
Find all of the positive real numbers like $ x,y,z,$ such that :
1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$
2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$
Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.
If $x,y,z$ are positive real numbers with $x^2+y^2+z^2=3$, prove that
$\frac32<\frac{1+y^2}{x+2}+\frac{1+z^2}{y+2}+\frac{1+x^2}{z+2}<3$
Let $ a$, $ b$, $ c$ be positive real numbers. Prove that
\[ \dfrac{(2a \plus{} b \plus{} c)^2}{2a^2 \plus{} (b \plus{} c)^2} \plus{} \dfrac{(2b \plus{} c \plus{} a)^2}{2b^2 \plus{} (c \plus{} a)^2} \plus{} \dfrac{(2c \plus{} a \plus{} b)^2}{2c^2 \plus{} (a \plus{} b)^2} \le 8.
\]
The real numbers $a_1,a_2,...,a_{100}$ satisfy the relationship :
$a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101$
Prove that $|a_k| \leq 10$ for all $k \in \{1,2,...,100\}$
Let $ n \geq 3 $ be an integer and let
\begin{align*}
x_1,x_2, \ldots, x_n
\end{align*}
be $ n $ distinct integers. Prove that
\begin{align*}
(x_1 - x_2)^2 + (x_2 - x_3)^2 + \ldots + (x_n - x_1)^2 \geq 4n - 6.
\end{align*}
If $b$, $c$ are the legs, and $a$ is the hypotenuse of a right triangle, prove that$$\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 5+3\sqrt{2}$$
Let $a,b,c$ be positive real numbers satisfying $a^2+b^2+c^2 \geq 3.$ Prove that
\[ \frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2} \]
If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that
\[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]
Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that :
$ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$
Prove that for all positive reals $x, y, z$, the inequality $(x-y)\sqrt{3x^2+y^2}+(y-z)\sqrt{3y^2+z^2}+(z-x)\sqrt{3z^2+x^2} \geq 0$ is satisfied.
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$
Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed into a circle with center $O$. Consider the circular arc with endpoints $A_1,A_6$ not containing $A_2$. For any point $M$ of that arc denote by $h_i$ the distance from $M$ to the line $A_iA_{i+1} \ (1 \leq i \leq 5)$. Construct $M$ such that the sum $h_1 + \cdots + h_5$ is maximal.
Let $E$ be the set of all continuously differentiable real valued functions $f$ on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. Define
$$J(f)=\int^1_0(1+x^2)f'(x)^2\text dx.$$
a) Show that $J$ achieves its minimum value at some element of $E$.
b) Calculate $\min_{f\in E}J(f)$.
Let $a,b,c>0$ such that $a+b+c=1$. Prove: \[\frac{a^{2}}b+\frac{b^{2}}c+\frac{c^{2}}a \ge 3(a^{2}+b^{2}+c^{2}) \]
Find all pairs of real numbers $(x,y)$ which satisfy the system
$$\begin{cases} x-y = 7 \\ \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7\end{cases}$$
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$
Let $x,y,z \in \mathbb{R}^+$ satisfying $xyz=3(x+y+z)$. Prove, that
\begin{align*} \sum \frac{1}{x^2(y+1)} \geq \frac{3}{4(x+y+z)} \end{align*}
What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.
Let $(a_n)_{n=1}^{\infty}$ be a strictly increasing sequence such that inequality
$$a_n(a_n-2a_{n-1})+a_{n-1}(a_{n-1}-2a_{n-2})\geq 0$$
holds for all $n \geq 3$. Prove that for all $n\geq2$ the inequality
$$a_n \geq a_{n-1}+a_{n-2}+\dots+a_1$$
holds as well.
Let $n \in \mathbb{N}, n \ge 2$ and the positive real numbers $a_1,a_2,…,a_n$ and $b_1,b_2,…,b_n$ such that $a_1+a_2+…+a_n=b_1+b_2+…+b_n=S.$
$\textbf{a)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k} \ge \frac{S}{2}.$
$\textbf{b)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k}= \sum\limits_{k=1}^n \frac{b_k^2}{a_k+b_k}.$