Found problems: 583
2020 Iran MO (2nd Round), P2
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 $a$)
2004 China Western Mathematical Olympiad, 4
Suppose that $ a$, $ b$, $ c$ are positive real numbers, prove that
\[ 1 < \frac {a}{\sqrt {a^{2} \plus{} b^{2}}} \plus{} \frac {b}{\sqrt {b^{2} \plus{} c^{2}}} \plus{} \frac {c}{\sqrt {c^{2} \plus{} a^{2}}}\leq\frac {3\sqrt {2}}{2}
\]
2017 Azerbaijan Junior National Olympiad, P3
Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.
2015 JBMO Shortlist, A1
Let x; y; z be real numbers, satisfying the relations
$x \ge 20$
$y \ge 40$
$z \ge 1675$
x + y + z = 2015
Find the greatest value of the product P = $xy z$
2023 Junior Balkan Mathematical Olympiad, 2
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]
1987 Bundeswettbewerb Mathematik, 4
Let $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$
a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$
b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality holds.
2019 JBMO Shortlist, A1
Real numbers $a$ and $b$ satisfy $a^3+b^3-6ab=-11$. Prove that $-\frac{7}{3}<a+b<-2$.
[i]Proposed by Serbia[/i]
2006 Federal Math Competition of S&M, Problem 1
Let $x,y,z$ be positive numbers with the sum $1$. Prove that
$$\frac x{y^2+z}+\frac y{z^2+x}+\frac z{x^2+y}\ge\frac94.$$
2022 Brazil National Olympiad, 5
Let $n$ be a positive integer number. Define $S(n)$ to be the least positive integer such that $S(n) \equiv n \pmod{2}$, $S(n) \geq n$, and such that there are [b]not[/b] positive integers numbers $k,x_1,x_2,...,x_k$ such that $n=x_1+x_2+...+x_k$ and $S(n)=x_1^2+x_2^2+...+x_k^2$. Prove that there exists a real constant $c>0$ and a positive integer $n_0$ such that, for all $n \geq n_0$, $S(n) \geq cn^{\frac{3}{2}}$.
2022 Korea Winter Program Practice Test, 2
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that
$$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$
and find when the equality holds.
Kvant 2024, M2797
For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\]
[i]Proposed by A. Antropov[/i]
1974 IMO Longlists, 45
The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$
2013 Vietnam Team Selection Test, 4
Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]
1996 IMO Shortlist, 6
Let the sides of two rectangles be $ \{a,b\}$ and $ \{c,d\},$ respectively, with $ a < c \leq d < b$ and $ ab < cd.$ Prove that the first rectangle can be placed within the second one if and only if
\[ \left(b^2 \minus{} a^2\right)^2 \leq \left(bc \minus{} ad \right)^2 \plus{} \left(bd \minus{} ac \right)^2.\]
1969 IMO Shortlist, 14
$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and
$q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$
2023 JBMO Shortlist, A2
For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that
$$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$
1968 IMO Shortlist, 12
If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that
\[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]
2024 Brazil Team Selection Test, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
2023 Brazil Team Selection Test, 3
Show that for all positive real numbers $a, b, c$, we have that $$\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}$$
2013 JBMO Shortlist, 3
Show that
\[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\]
for all positive real numbers $a$ and $b$ such that $ab\geq 1$.
KoMaL A Problems 2017/2018, A. 727
For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1 \le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1 \le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which
\[N(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)})\le \min(N(a_1,a_2,\ldots,a_n),N(b_1,b_2,\ldots,b_n)).\]
2024 Korea National Olympiad, 5
Find the smallest real number $M$ such that
$$\sum_{k = 1}^{99}\frac{a_{k+1}}{a_k+a_{k+1}+a_{k+2}} < M$$
for all positive real numbers $a_1, a_2, \dots, a_{99}$. ($a_{100} = a_1, a_{101} = a_2$)
2017 Bosnia and Herzegovina EGMO TST, 4
Let $a$, $b$, $c$, $d$ and $e$ be distinct positive real numbers such that $a^2+b^2+c^2+d^2+e^2=ab+ac+ad+ae+bc+bd+be+cd+ce+de$
$a)$ Prove that among these $5$ numbers there exists triplet such that they cannot be sides of a triangle
$b)$ Prove that, for $a)$, there exists at least $6$ different triplets
2023 JBMO Shortlist, A1
Prove that for all positive real numbers $a,b,c,d$,
$$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$
and determine when equality occurs.
1992 Irish Math Olympiad, 5
If, for $k=1,2,\dots ,n$, $a_k$ and $b_k$ are positive real numbers, prove that $$\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};$$ and that equality holds if, and only if, $$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.$$