Found problems: 6530
2019 Romania EGMO TST, P3
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
2002 Iran MO (3rd Round), 4
$a_{n}$ ($n$ is integer) is a sequence from positive reals that \[a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4\] Prove $a_{n}$ is constant.
1995 IMO Shortlist, 1
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that
\[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}.
\]
2014 Contests, 2
Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.
2010 Saint Petersburg Mathematical Olympiad, 6
For positive is true $$\frac{3}{abc} \geq a+b+c$$
Prove $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq a+b+c$$
2018 International Zhautykov Olympiad, 6
In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$,$BE$ and $CF$,$CF$ and $AD$ of the hexagon intersect at the points $M$,$N$ and$K$, respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that.$$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$ .
2025 Israel National Olympiad (Gillis), P6
Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc+abc=4.$ . Prove that:
$$\sqrt{\frac{ab+ac+1}{a+2}}+\sqrt{\frac{ab+bc+1}{b+2}}+\sqrt{\frac{ac+bc+1}{c+2}}\leq3.$$
[hide="PS"]Dedicated to dear KhuongTrang :-D [/hide]
2019 Romanian Master of Mathematics Shortlist, A2
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
$$
[i](B. Serankou, M. Karpuk)[/i]
2005 IMO Shortlist, 5
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
[i]Hojoo Lee, Korea[/i]
2005 Irish Math Olympiad, 1
Let $ X$ be a point on the side $ AB$ of a triangle $ ABC$, different from $ A$ and $ B$. Let $ P$ and $ Q$ be the incenters of the triangles $ ACX$ and $ BCX$ respectively, and let $ M$ be the midpoint of $ PQ$. Prove that: $ MC>MX$.
1993 IMO Shortlist, 8
The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC.$ If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC,$ prove that
\[ DE \geq \frac{2 \cdot \sqrt{2} \cdot S}{\sqrt{a^2 + b^2 + c^2 + 4 \cdot \sqrt{3} \cdot S}}. \]
the 12th XMO, Problem 2
Let $a_1,a_2,\cdots,a_{22}\in [1,2],$ find the maximum value of
$$\dfrac{\sum\limits_{i=1}^{22}a_ia_{i+1}}{\left( \sum\limits_{i=1}^{22}a_i\right) ^2}$$where $a_{23}=a_1.$
2000 Irish Math Olympiad, 1
Prove that if $ x,y$ are nonnegative real numbers with $ x\plus{}y\equal{}2$, then: $ x^2 y^2 (x^2\plus{}y^2) \le 2$.
2011 National Olympiad First Round, 3
How many positive integer $n$ are there satisfying the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ ?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$
2017 Middle European Mathematical Olympiad, 2
Determine the smallest possible real constant $C$ such that the inequality
$$|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1|$$
holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$.
1987 Nordic, 4
Let $a, b$, and $c$ be positive real numbers. Prove: $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$ .
2007 JBMO Shortlist, 5
The real numbers $x,y,z, m, n$ are positive, such that $m + n \ge 2$. Prove that
$x\sqrt{yz(x + my)(x + nz)} + y\sqrt{xz(y + mx)(y + nz)} + z\sqrt{xy(z + mx)(x + ny) }\le \frac{3(m + n)}{8}
(x + y)(y + z)(z + x)$
2006 AMC 12/AHSME, 24
The expression
\[ (x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006}
\]is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
$ \textbf{(A) } 6018 \qquad \textbf{(B) } 671,676 \qquad \textbf{(C) } 1,007,514 \qquad \textbf{(D) } 1,008,016 \qquad \textbf{(E) } 2,015,028$
2017 Miklós Schweitzer, 8
Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$. (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}$, then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by
$$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and
$$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?$$
2004 Romania Team Selection Test, 9
Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements.
Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.
2016 India PRMO, 12
Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$.
You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$.
2015 China Girls Math Olympiad, 2
Let $a\in(0,1)$ ,$f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3 $ , $g(x)=(1-a)x^3-x^2+(2-a)x-3a-1 $.
Prove that:For any real number $x$ ,at least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$.
1991 Czech And Slovak Olympiad IIIA, 3
For any permutation $p$ of the set $\{1,2,...,n\}$, let us denote $d(p) = |p(1)-1|+|p(2)-2|+...+|p(n)-n|$. Let $i(p)$ be the number of inversions of $p$, i.e. the number of pairs $1 \le i < j \le n$ with $p(i) > p(j)$. Prove that $d(p)\le 2i(p)$$.
2011 Morocco TST, 3
The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
1998 Tournament Of Towns, 4
Among all sets of real numbers $\{ x_1 , x_2 , ... , x_{20} \}$ from the open interval $(0, 1 )$ such that $$x_1x_2...x_{20}= ( 1 - x_1 ) ( 1 -x_2 ) ... (1 - x_{20} )$$ find the one for which $x_1 x_2... x_{20}$ is maximal.
(A Cherniatiev)