Found problems: 1671
Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that
\[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]
Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$.
Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations:
\[
\begin{cases}
x-\lambda y=a,\\
y-\lambda z=b,\\
z-\lambda x=c.
\end{cases}
\]
we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$.
[i]Radu Gologan[/i]
Let $x, y, z \in R^+$. Prove that
$$\frac{x}{x +\sqrt{(x + y)(x + z)}}+\frac{y}{y +\sqrt{(y + z)(y + x)}}+\frac{z}{z +\sqrt{(x + z)(z + y)}} \le 1$$
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,
$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.
Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$.
[hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]
Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that
\[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]
Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:
[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]
Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
Let $a$, $b$ and $c$ be real numbers larger than $1$. Prove the inequality $$\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.$$
When does equality hold?
[i](Karl Czakler)[/i]
Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$.
Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that
\[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0.
\]
Find the greatest positive real number $ k$ such that the inequality below holds for any positive real numbers $ a,b,c$:
\[ \frac ab \plus{} \frac bc \plus{} \frac ca \minus{} 3 \geq k \left( \frac a{b \plus{} c} \plus{} \frac b{c \plus{} a} \plus{} \frac c{a \plus{} b} \minus{} \frac 32 \right).
\]
Let $a_1,a_2,\cdots ,a_{100}$ be non-negative integers such that
$(1)$ There are positive integers$ k\leq 100$ such that $a_1\leq a_2\leq \cdots\leq a_{k}$
and $a_i=0$ $(i>k);$
$(2)$ $ a_1+a_2+a_3+\cdots +a_{100}=100;$
$(3)$ $ a_1+2a_2+3a_3+\cdots +100a_{100}=2022.$
Find the minimum of $ a_1+2^2a_2+3^2a_3+\cdots +100^2a_{100}.$
For $k=1,2 \dots, n$ let $z_k=x_k+iy_k,$ where the $x_k$ and $y_k$ are real and $i=\sqrt{-1}$. Let $r$ bet the absolute value of the real part of $$\pm \sqrt{z_1^2+z_2^2+\dots z_n^2}.$$ Prove that $r\leq |x_1|+|x_2|+ \dots +|x_n|.$
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]
For all natural numbers $n$, consider the polynomial $P_n(x) = x^{n+2}-2x+1$.
(a) Show that the equation $P_n(x)=0$ has exactly one root $c_n$ in the open interval $(0,1)$.
(b) Find $lim_{n \to \infty}c_n$.
Let $I \subset \mathbb{R}$ be a nonempty open interval and let $f: I \cap \mathbb{Q} \to \mathbb{R}$ be a function such that for all $x, y \in I \cap \mathbb{Q}$,
\[ 4f\left(\frac{3x + y}{4}\right)+ 4f\left(\frac{x + 3y}{4}\right) \le f(x) + 6f\left(\frac{x + y}{2}\right)+ f(y). \] Show that $f$ can be continuously extended to $I$.
If $a, b, c > 0$ satisfy $a + b + c = 3$, then prove that
$$\frac{a^2(b + 1)}{ ab + a + b} + \frac{b^2(c + 1)}{ bc + b + c} + \frac{c^2(a + 1)}{ ca + c + a} \ge 2$$
Mathematical Excalibur P322/Vol.14, no.2
Let $I$ be the incenter of triangle $ABC$. The excircle with center $I_A$ touches the side $BC$ at point $A'$. The line $l$ passing through $I$ and perpendicular to $BI$ meets $I_AA'$ at point $K$ lying on the medial line parallel to $BC$. Prove that $\angle B \leq 60^\circ$.
Consider the following inequality for real numbers $x,y,z$: $|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}$ .
(a) Prove that the inequality is valid for $a = 2\sqrt2$
(b) Assuming that $x,y,z$ are nonnegative, show that the inequality is also valid for $a = 2$.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)+1=f(1)$ and for any real numbers $x$ and $y$,
$$ f(xy-x)+f(x+f(y))=yf(x)+3 $$
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality
\[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]
For positive integer $n\geq 3$ and real numbers $a_1,...,a_n,b_1,...,b_n$, prove the following.
$$\sum_{i=1}^n a_i(b_i-b_{i+3})\leq\frac{3n}{8}\sum_{i=1}^n((a_i-a_{i+1})^2+(b_i-b_{i+1})^2)$$
($a_{n+1}=a_1$, and for $i=1,2,3$ $b_{n+i}=b_i$.)