Found problems: 2529
Discriminants of square trinomials $f(x),g(x),h(x),f(x)+g(x),f(x)+h(x),g(x)+h(x)$ equals $1$.
Prove that $f(x)+h(x)+g(x) \equiv 0$
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
Let $a,b,c \in\mathbb{R^*_+}$.Prove the inequality $\frac{a^2+4}{b+c}+\frac{b^2+9}{c+a}+\frac{c^2+16}{a+b}\ge9$.
Given: (i) $a$, $b > 0$; (ii) $a$, $A_1$, $A_2$, $b$ is an arithmetic progression; (iii) $a$, $G_1$, $G_2$, $b$ is a geometric progression. Show that
\[A_1 A_2 \ge G_1 G_2.\]
Let $x,y{}$ and $z{}$ be positive real numbers satisfying $x+y+z=1$. Prove that
[list=a]
[*]\[1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x};\]
[*]\[\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.\]
[/list]
Find the maximum value of \[abcd(a+b)(b+c)(c+d)(d+a)\] such that $a,b,c$ and $d$ are positive real numbers satisfying $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+\sqrt[3]{d}=4$
Find the maximum of $x+y$ given that $x$ and $y$ are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\]
Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.
Given $a$, $b$, and $c$ be positive real numbers with $a+b+c=1$, prove that
\[\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2\]
Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that
$$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$
If $a_1$, $a_2$ and $a_3$ are nonnegative real numbers for which $a_1+a_2+a_3=1$, then prove the inequality $a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}$
Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.
Given is a $K_{2024}$ in which every edge has weight $1$ or $2$. If every cycle has even total weight, find the minimal value of the sum of all weights in the graph.
Let $a,b,c$ be the lengths of the sides of a triangle, let $p=(a+b+c)/2$, and $r$ be the radius of the inscribed circle. Show that $$\frac{1}{(p-a)^2}+ \frac{1}{(p-b)^2}+\frac{1}{(p-c)^2} \geq \frac{1}{r^2}.$$
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]
Prove that for any positive real numbers $x,y$ and $z,$
$xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$
Let $x$ and $y$ be positive reals such that \[ x^3 + y^3 + (x + y)^3 + 30xy = 2000. \] Show that $x + y = 10$.
Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}$.
$M$ is an interior point of a rectangle $ABCD$ and $S$ is its area.
Prove that $S \le AM \cdot CM + BM \cdot DM$.
(I.J . Goldsheyd)
Let $ a,b,c$ be positive real numbers such that $ abc\equal{}\frac{9}{4}$. Prove the inequality:
$ a^3 \plus{} b^3 \plus{} c^3 > a\sqrt {b \plus{} c} \plus{} b\sqrt {c \plus{} a} \plus{} c\sqrt {a \plus{} b}$
Jury's variant:
Prove the same, but with $ abc\equal{}2$
$\boxed{\text{A4}}$ Let $x,y$ be positive reals satisfying the condition $x^3+y^3\leq x^2+y^2$.Find the maximum value of $xy$.
Show that for any non-negative real numbers $a,b,c,d$ such that $a^2+b^2+c^2+d^2=1$ the following inequality hold:
$$a+b+c+d-1\geq 16abcd$$
When does equality hold?
[b](a)[/b] Prove that
\[\frac {x^{2}}{\left(x \minus{} 1\right)^{2}} \plus{} \frac {y^{2}}{\left(y \minus{} 1\right)^{2}} \plus{} \frac {z^{2}}{\left(z \minus{} 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.
[b](b)[/b] Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.
[i]Author: Walther Janous, Austria[/i]
Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression
$$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$
Find all real numbers $x,y,z$ so that
\begin{align*}
x^2 y + y^2 z + z^2 &= 0 \\
z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4).
\end{align*}