Found problems: 6530
2009 China Team Selection Test, 3
Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that
$ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$
Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.
2018 Middle European Mathematical Olympiad, 1
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$ Prove that$$\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$$
1976 Chisinau City MO, 131
The sum of the real numbers $x_1, x_2, ...,x_n$ belonging to the segment $[a, b]$ is equal to zero.
Prove that $$x_1^2+ x_2^2+ ...+x_n^2 \le - nab.$$
1996 Bosnia and Herzegovina Team Selection Test, 1
$a)$ Let $a$, $b$ and $c$ be positive real numbers. Prove that for all positive integers $m$ holds: $$(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)$$
$b)$ Does inequality $a)$ holds for
$1)$ arbitrary real numbers $a$, $b$ and $c$
$2)$ any integer $m$
2018 Nepal National Olympiad, 2b
[b]Problem Section #2
b) Find the maximal value of $(x^3+1)(y^3+1)$, where $x,y \in \mathbb{R}$, $x+y=1$.
1987 IMO Longlists, 27
Find, with proof, the smallest real number $C$ with the following property:
For every infinite sequence $\{x_i\}$ of positive real numbers such that $x_1 + x_2 +\cdots + x_n \leq x_{n+1}$ for $n = 1, 2, 3, \cdots$, we have
\[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.\]
2010 Turkey MO (2nd round), 3
Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i]
2024 Assara - South Russian Girl's MO, 5
Prove that $(100!)^{99} > (99!)^{100} > (100!)^{98}$.
[i]K.A.Sukhov[/i]
2006 Moldova National Olympiad, 12.5
Let $ a_{1},a_{2},...,a_{n} $ be real positive numbers and $ k>m, k,m $ natural numbers. Prove that
$(n-1)(a_{1}^m +a_{2}^m+...+a_{n}^m)\leq\frac{a_{2}^k+a_{3}^k+...+a_{n}^k}{a_{1}^{k-m}}+\frac{a_{1}^k+a_{3}^k+...+a_{n}^k}{a_2^{k-m}}+...+\frac{a_{1}^k+a_{2}^k+...+a_{n-1}^k}{a_{n}^{k-m}} $
2021 Iran Team Selection Test, 3
Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have:
$$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$
$$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$
Then we have :
$$bP(\frac{a}{c})=dQ(\frac{a}{c})$$
(Two polynomials are relatively prime if they don't have a common root)
Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]
2003 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c$ be positive real numbers with $abc = 1$. Prove that $1 + \frac{3}{a+b+c}\ge \frac{6}{ab+bc+ca}$
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
2022 Canadian Junior Mathematical Olympiad, 3
If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$
2017 District Olympiad, 4
If $ a,b,c>0 $ and $ ab+bc+ca+abc=4, $ then $ \sqrt{ab} +\sqrt{bc} +\sqrt{ca} \le 3\le a+b+c. $
2012 Balkan MO Shortlist, C1
Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$
Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$
2006 Bulgaria National Olympiad, 2
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$
\[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\]
a) Prove that $f(2x)=4f(x)$ for all $x>0$;
b) Find all such functions.
[i]Nikolai Nikolov, Oleg Mushkarov [/i]
1995 Italy TST, 3
A function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the conditions
\[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\]
Prove that $f(x+1)=f(x)+1$ for all real $x$.
2024-IMOC, A2
Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of
\[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\]
[i]Proposed by snap7822[/i]
PEN S Problems, 13
The sum of the digits of a natural number $n$ is denoted by $S(n)$. Prove that $S(8n) \ge \frac{1}{8} S(n)$ for each $n$.
2023 Centroamerican and Caribbean Math Olympiad, 3
Let $a,\ b$ and $c$ be positive real numbers such that $a b+b c+c a=1$. Show that
$$
\frac{a^3}{a^2+3 b^2+3 a b+2 b c}+\frac{b^3}{b^2+3 c^2+3 b c+2 c a}+\frac{c^3}{c^2+3 a^2+3 c a+2 a b}>\frac{1}{6\left(a^2+b^2+c^2\right)^2} .
$$
2007 IMAC Arhimede, 5
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$
2008 Junior Balkan Team Selection Tests - Moldova, 2
[b]BJ2. [/b] Positive real numbers $ a,b,c$ satisfy inequality $ \frac {3}{2}\geq a \plus{} b \plus{} c$. Find the smallest possible value for
$ S \equal{} abc \plus{} \frac {1}{abc}$
2018 Dutch IMO TST, 2
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.
2006 Taiwan National Olympiad, 1
Positive reals $a,b,c$ satisfy $abc=1$. Prove that
$\displaystyle 1+ \frac{3}{a+b+c} \ge \frac{6}{ab+bc+ca}$.
2004 Unirea, 2
Find the maximum value of the expression $ x+y+z, $ where $ x,y,z $ are real numbers satisfying
$$ \left\{ \begin{matrix} x^2+yz\le 2 \\y^2+zx\le 2\\ z^2+xy\le 2 \end{matrix} \right. . $$