Found problems: 2529
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
Let $a,b,c$ be sides of a triangle, show that
\[\frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca.\]
Positive real numbers $a, b, c$ satisfy $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.$ Prove the inequality \[\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.\]
Let $0\le x\le 1$. Show that if $n$ is any positive integer, then $$(1+x)^n\ge (1-x)^n+2nx(1-x^2)^{\frac{n-1}{2}}$$.
Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that \[a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.\]
[i]Proposed by Daniel Liu[/i]
Prove that for all non-negative real numbers $x,y,z$, not all equal to $0$, the following inequality holds
$\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.$
Determine all the triples $(x,y,z)$ for which the equality holds.
[i]Milan Mitreski, Serbia[/i]
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.
Is it possible to partition $3$-dimensional Euclidean space into $1979$ mutually isometric subsets?
Let $x,y$ and $z$ be real numbers such that $0 \leq x,y,z \leq 1.$ Prove that
\[xyz+(1-x)(1-y)(1-z) \leq 1.\]
When does equality hold?
Given a quartet of positive numbers $a,b,c,d$, and is known, that $abcd=1$. Prove that $$a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+dc \ge 10$$
If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that:
$\textbf{(A)}\ |r_1+r_2|>4\sqrt{2}\qquad
\textbf{(B)}\ |r_1|>3 \; \text{or} \; |r_2| >3 \\
\textbf{(C)}\ |r_1|>2 \; \text{and} \; |r_2|>2\qquad
\textbf{(D)}\ r_1<0 \; \text{and} \; r_2<0\qquad
\textbf{(E)}\ |r_1+r_2|<4\sqrt{2}$
Let $a$,$b$,$c$ be positive real numbers satisfying $abc=1$. Prove that inequality $\dfrac{a^k}{a+b}+ \dfrac{b^k}{b+c}+\dfrac{c^k}{c+a}\ge \dfrac{3}{2}$ holds for all integer $k$ ($k \ge 2$).
Let’s consider the real numbers $a, b, c$ satisfying the condition
\[21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.\]
Find the minimal value of the expression
\[P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\]
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
$$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}$$
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.
Let $n > 1$ be an integer. An $n \times n$-square is divided into $n^2$ unit squares. Of these unit squares, $n$ are coloured green and $n$ are coloured blue, and all remaining ones are coloured white. Are there more such colourings for which there is exactly one green square in each row and exactly one blue square in each column; or colourings for which there is exactly one green square and exactly one blue square in each row?
Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality?
[i](Walther Janous)[/i]
Let $a,b,c\ge0$ and $a+b+c=3.$ Prove that $(3a-bc)(3b-ac)(3c-ab)\le8.$ (O. Rudenko)
Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$. Let $C$ be the set of all circles whose center lies in $S$, and which are tangent to $X$-axis. Find the minimum value of $k$ such that any two circles in $C$ have at least one point of intersection.
Find the smallest real number $M$ with the following property: Given nine nonnegative real numbers with sum $1$, it is possible to arrange them in the cells of a $3 \times 3$ square so that the product of each row or column is at most $M$.
[i]Evan O' Dorney.[/i]
Let $x,y,z$ be positive real numbers satisfying $x+y+z=1$. Find the smallest $k$ such that $\frac{x^2y^2}{1-z}+\frac{y^2z^2}{1-x}+\frac{z^2x^2}{1-y}\leq k-3xyz$.
Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$?
$
\textbf{(A) }909\qquad
\textbf{(B) }910\qquad
\textbf{(C) }911\qquad
\textbf{(D) }912\qquad
\textbf{(E) }913\qquad
$
Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?
[i]Second Test, May 16[/i]
Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Prove that
$\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca}\le 1$ .
When does equality hold?
Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Prove that
$$\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).$$