Found problems: 2529
Show that for $a,b,c > 0$ the following inequality holds:
$\frac{\sqrt{ab}}{a+b+2c}+\frac{\sqrt{bc}}{b+c+2a}+\frac{\sqrt{ca}}{c+a+2b} \le \frac {3}{4}$.
Let $a,b,c$ be positive real numbers such that
\[ \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1. \]
Prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$. When does equality hold?
Let $a, b, c \in \mathbb{R}^+$ such that $a + b + c = 3$. Prove that $$\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.$$
Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]
Prove that if $0<x<\frac{\pi}{2}$ and $n>m$, where $n$,$m$ are natural numbers, \[ 2 \left| \sin^n x - \cos^n x \right| \le 3 \left| \sin^m x - \cos^m x \right|.\]
Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that
\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]
is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$
Find all the real number triples $(x, y, z)$ satisfying the following system of inequalities under the condition $0 < x, y, z < \sqrt{2}$:
$$y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}$$
$$x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}$$
$$z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}$$.
Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that
\[8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; , \]
with equality if and only if triangle $ABC$ is equilateral.
Let $n$ be a positive integer, and let $1=d_1<d_2<\dots < d_k=n$ denote all positive divisors of $n$, If the following conditions are satisfied:
$$ 2d_2+d_4+d_5=d_7$$
$$ d_3 d_6 d_7=n$$
$$ (d_6+d_7)^2=n+1$$
find all possible values of $n$.
Let $A,B,C,...,Z$ be $26$ nonzero real numbers. Suppose that $T=TNYWR$. Compute the smallest possible value of \[
\left\lceil A^2+B^2+\cdots+Z^2\right\rceil .
\] (The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.)
[i]Proposed by Luke Robitaille[/i]
A block of cheese in the shape of a rectangular solid measures $ 10$ cm by $ 13$ cm by $ 14$ cm. Ten slices are cut from the cheese. Each slice has a width of $ 1$ cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?
If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]
Let $x,y,z \in \mathbb{R}$ such that $xyz = 1$. Prove that:
$$\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).$$
We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]
Given complex numbers $x,y,z$, with $|x|^2+|y|^2+|z|^2=1$. Prove that: $$|x^3+y^3+z^3-3xyz| \le 1$$
Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]
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 $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge ... \ge x_n$ and $y_1 \ge y_2 \ge ... \ge y_n$ be $2n$ real numbers such that $$0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n $$ $$\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2.$$ Prove that $$\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}.$$
[i]Proposed by David Speyer and Kiran Kedlaya[/i]
Find the minimum value of $x^2+2x+ \frac{24}{x}$ for $x > 0$.
$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
Let $ABCD$ be a convex quadrilateral with $AB=a$, $BC=b$, $CD=c$ and $DA=d$. Suppose
\[a^2+b^2+c^2+d^2=ab+bc+cd+da,\]
and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.
Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that
\[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$.