Prove that for all positive reals $a,b,c,d$, we have $\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}$

Let $x, y, z$ be nonnegative real numbers such that $x + y + z = 3$. Prove that $$\frac{x}{4-y}+\frac{y}{4-z}+\frac{z}{4-x}+\frac{1}{16}(1-x)^2(1-y)^2(1-z)^2\leq 1,$$ and determine all such triples $(x, y, z)$ where the equality holds.

let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the $$\max( [x]y + [y]z + [z]x ) $$ ( $[a]$ is the biggest integer not exceeding $a$)

Show that \[ \max\limits_{|x|\leq t} |1 - a \cos x| \geq \tan^2 \frac{t}{2} \] for $a$ positive and $t \in (0, \frac{\pi}{2})$.

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

Let $G$ be the centroid of a triangle $ABC$ and let $d$ be a line that intersects $AB$ and $AC$ at $B_{1}$ and $C_{1}$, respectively, such that the points $A$ and $G$ are not separated by $d$. Prove that: $[BB_{1}GC_{1}]+[CC_{1}GB_{1}] \geq \frac{4}{9}[ABC]$.

Show that \[ abc \geq (a+b-c)(b+c-a)(c+a-b) \] for positive reals $a$, $b$, $c$.

Let $n$ be a positive integer, and $a,b\ge 1$, $c>0$ arbitrary real numbers. Prove that \[\frac{(ab+c)^n-c}{(b+c)^n-c}\le a^n.\]

Positive real $A$ is given. Find maximum value of $M$ for which inequality $ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $ holds for all $x, y>0$

Prove that for positive reals $a,b,c$ satisfying $a+b+c=3$ the following inequality holds: $$ \frac{a}{1+2b^3}+\frac{b}{1+2c^3}+\frac{c}{1+2a^3} \ge 1 $$

Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Because Bubble Boy loves Bubble Girl, he wants to reach her as quickly as possible, but he needs to bring a gift; luckily, there are plenty of gifts along the $x$-axis. Assuming that Bubble Girl remains stationary, find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then reach Bubble Girl (the bubble is a solid boundary, and anything the bubble can touch, Bubble Boy can touch too)

If $f\in C^{(3)}([0,1])$ such that $f(0)=f(1)=f'(0)=0$ and $|f'''(x)|\le1,(\forall)x\in[0,1]$, show that: a) $$|f(x)|\le\frac{x(1-x)}{\sqrt3}\cdot\left(\int^x_0\frac{f(t)}{t(1-t)}dt\right)^{1/2},(\forall)x\in[0,1]$$ b) $$|f'(x)|\le\frac{1-2x}{\sqrt3}\cdot\left(\int^x_0\frac{|f(t)|}{t(1-t)}dt\right)^{1/2},(\forall)x\in\left[0,\frac12\right]$$ c) $$\int^1_0(1-x)^2\cdot\frac{|f(x)|}xdx\ge9\int^1_0\left(\frac{f(x)}x\right)^2dx$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

Prove that $\frac{1}{2} \frac{3}{4} \frac{5}{6} \frac{7}{8} ... \frac{99}{100 } <\frac{1}{10}$.

Let $ I $ be the incentre of $ ABC $ and $ D,E,F $ be the feet of the perpendiculars from $ I $ to $ BC,CA,AB, $ respectively. Show that $$ \frac{AB}{DE} +\frac{BC}{EF} +\frac{CA}{FD}\ge 6. $$

It is given a regular hexagon in the plane. Let P be a point of the plane. Define s(P) as the sum of the distances from P to each side of the hexagon, and v(P) as the sum of the distances from P to each vertex. a) Find the locus of points P that minimize s(P) b) Find the locus of points P that minimize v(P)

Find all real numbers $a$ such that the inequality $3x^2 + y^2 \ge -ax(x + y)$ holds for all real numbers $x$ and $y$.

There is an interval $[a, b]$ that is the solution to the inequality \[|3x-80|\le|2x-105|\] Find $a + b$.

Prove the following inequality. \[ \frac{2}{2\plus{}e^{\frac 12}}<\int_0^1 \frac{dx}{1\plus{}xe^{x}}<\frac{2\plus{}e}{2(1\plus{}e)}\]

Let $x; y$ and $z$ be positive real numbers such that $\sqrt{x}+\sqrt{y}+\sqrt{z}= 1$. Prove that $\frac{x^{2}+yz}{\sqrt{2x^{2}(y+z)}}+\frac{y^{2}+zx}{\sqrt{2y^{2}(z+x)}}+\frac{z^{2}+xy}{\sqrt{2z^{2}(x+y)}}\geq 1.$

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 + c^3 = 5abc$. Show that \[ \left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9. \]

Find all functions $f : R \to R$ such that the inequality $$\sum_{i=1}^{2549} f(x_i + x_{i+1}) + f (\sum_{i=1}^{2550}x_y) \le \sum_{i=1}^{2550}f(2x_i)$$ for all reals $x_1, x_2, . . . , x_{2550}$.

For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!) For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.

Let $j$ and $k$ be integers. Prove that the inequality $$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.

Given three identical $n$- faced dice whose corresponding faces are identically numbered with arbitrary integers. Prove that if they are tossed at random, the probability that the sum of the bottom three face numbers is divisible by three is greater than or equal to $\frac{1}{4}$.

[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$. [url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle. [url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen? [url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$ [url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.