Found problems: 2529
Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality
\[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]
$ABC$ is acute-angled triangle, $AA_1,CC_1$ are altitudes. $M$ is centroid. $M$ lies on circumcircle of $A_1BC_1$. Find all values of $\angle B$
Given positive numbers ${a_1},{a_2},...,{a_n}$ with ${a_1} + {a_2} + ... + {a_n} = 1$. Prove that $\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}$.
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
[i]Proposed by Tigran Margaryan, Armenia[/i]
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 $x, y, z$ be positive real numbers such that $xyz = 1$. Prove that:
$$\frac{x^3 + y^3}{x^2 + xy + y^2} +\frac{ y^3 + z^3}{y^2 + yz + z^2} + \frac{z^3 + x^3}{z^2 + zx + x^2}
\ge 2.$$
Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality
\[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\]
holds. Determine all cases of equality.
[i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]
For an acute triangle $ ABC$ prove the inequality:
$ \sum_{cyclic} \frac{m_a^2}{\minus{}a^2\plus{}b^2\plus{}c^2}\ge \frac{9}{4}$ where $ m_a,m_b,m_c$ are lengths of corresponding medians.
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$
What is the largest real number $C$ that satisfies the inequality $x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)$ for every real $x$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 9
\qquad\textbf{(E)}\ 25
$
Let $a_{ij}, i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ be positive real numbers. Prove that
\[ \sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1} \]
Let $t_{k} = a_{1}^k + a_{2}^k +...+a_{n}^k$, where $a_{1}$, $a_{2}$, ... $a_{n}$ are positive real numbers and $k \in \mathbb{N}$. Prove that $$\frac{t_{5}^2 t_1^{6}}{15} - \frac{t_{4}^4 t_{2}^2 t_{1}^2}{6} + \frac{t_{2}^3 t_{4}^5}{10} \geq 0 $$
[i]Proposed by Daniel Velinov[/i]
Solve in positive reals the system:
$x+y+z+w=4$
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$
Two right triangles are given, of which the incircle of the first triangle is the circumcircle of the second triangle. Let the areas of the triangles be $S$ and $S'$ respectively. Prove that $\frac{S}{S'} \ge 3 +2\sqrt2$
Let $ a,b$ and $ c $ be positive real numbers with $ abc = 1 $. Prove that
\[ \sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)\]
Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that:
a/(1-x)+b/(1-y)=1
Prove that:
∛ay+∛bx≤1.
If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that
\[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]
Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.
Let $a, b, c$ be positive real numbers. Prove the inequality
$(a^2+ac+c^2) \left( \frac{1}{a+b+c}+\frac{1}{a+c} \right)+b^2 \left( \frac{1}{b+c}+\frac{1}{a+b} \right)>a+b+c$.
[i]Proposed by Tajikistan[/i]
Let $a,b,c,d$ be positive integers such that $a+b+c+d=99$. Find the maximum and minimum of product $abcd$
Let $ x,y$ be reals s.t. $ x^2\plus{}y^2\leq1$ and $ n$ a natural number.Prove that:
$ (x^n\plus{}y)^2\plus{}y^2\geq\dfrac{1}{n\plus{}2}(x^2\plus{}y^2)^n$
Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$.
[i]Alternative version.[/i] Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with
[b]a)[/b] vertices on the sides of the polygon (or)
[b]b)[/b] vertices among the vertices of the polygon
such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon.
[i]Proposed by Ben Green and Edward Crane, United Kingdom[/i]
On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $
Let $a$, $b$, $c$, and $d$ be positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \[a^2+b^2+c^2+a(b+c+d) + b(c+d) + cd.\] Compute $\lfloor 17m\rfloor$.
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality
\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]
holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.