Found problems: 6530
2015 Romania National Olympiad, 4
Let be a finite set $ A $ of real numbers, and define the sets $ S_{\pm }=\{ x\pm y| x,y\in A \} . $
Show that $ \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 . $
2016 Estonia Team Selection Test, 4
Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$
2004 IMC, 5
Let $S$ be a set of $\displaystyle { 2n \choose n } + 1$ real numbers, where $n$ is an positive integer. Prove that there exists a monotone sequence $\{a_i\}_{1\leq i \leq n+2} \subset S$ such that
\[ |x_{i+1} - x_1 | \geq 2 | x_i - x_1 | , \]
for all $i=2,3,\ldots, n+1$.
2007 Stars of Mathematics, 3
Let $ n\ge 3 $ be a natural number and $ A_0A_1...A_{n-1} $ a regular polygon. Consider $ B_0 $ on the segment $ A_0A_1 $ such that $ A_0B_0<\frac{1}{2}A_0A_1; B_1 $ on $ A_1A_2 $ so that $ A_1B_1<\frac{1}{2} A_1A_2; $ etc.; $ B_{n-2} $ on $ A_{n-2}A_{n-1} $ so that $ A_{n-2}B_{n-2} <\frac{1}{2} A_{n-2}A_{n-1} , $ and $ B_{n-1} $ on $ A_{n-1}A_0 $ with $ A_{n-1}B_{n-1} <\frac{1}{2} A_{n-1}A_{0} . $
Show that the perimeter of any ploygon that has its vertices on the segments $ A_1B_1,A_2B_2,...,A_{n-1}B_{n-1}, $ is equal or greater than the perimeter of $ B_0B_1...B_{n-1} . $
2017 Thailand TSTST, 3
Let $a, b, c \in\mathbb{R}^+$. Prove that $$\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}.$$
2021 Estonia Team Selection Test, 3
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
MathLinks Contest 7th, 3.3
Find the greatest positive real number $ k$ such that the inequality below holds for any positive real numbers $ a,b,c$:
\[ \frac ab \plus{} \frac bc \plus{} \frac ca \minus{} 3 \geq k \left( \frac a{b \plus{} c} \plus{} \frac b{c \plus{} a} \plus{} \frac c{a \plus{} b} \minus{} \frac 32 \right).
\]
1999 Singapore MO Open, 3
For each positive integer $n$, let $f(n)$ be a positive integer. Show that if $f(n + 1) > f(f(n))$ for every positive integer n, then $f(x) = x$ for all positive integers $x$.
2012 Stars of Mathematics, 3
For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality
$$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | \geq 2$$
and determine all cases of equality.
Prove that if we also impose $a,b,c \geq 0$, then
$$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | > 3,$$
with the value $3$ being the best constant possible.
([i]Dan Schwarz[/i])
1968 Miklós Schweitzer, 2
Let $ a_1,a_2,...,a_n$ be nonnegative real numbers. Prove that \[ ( \sum_{i=1}^na_i)( \sum_{i=1}^na_i^{n-1}) \leq n \prod_{i=1}^na_i+ (n-1) ( \sum_{i=1}^na_i^n).\]
[i]J. Suranyi[/i]
PEN A Problems, 82
Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?
2021 Thailand TST, 1
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
2011 AMC 12/AHSME, 25
Triangle $ABC$ has $\angle BAC=60^\circ$, $\angle CBA \le 90^\circ$, $BC=1$, and $AC \ge AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of the pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$?
$\textbf{(A)}\ 60 ^\circ \qquad
\textbf{(B)}\ 72 ^\circ\qquad
\textbf{(C)}\ 75 ^\circ \qquad
\textbf{(D)}\ 80 ^\circ\qquad
\textbf{(E)}\ 90 ^\circ$
2013 ELMO Shortlist, 8
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that
\[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]
2004 Croatia National Olympiad, Problem 2
If $a,b,c$ are positive numbers, prove the inequality
$$\frac{a^2}{(a+b)(a+c)}+\frac{b^2}{(b+c)(b+a)}+\frac{c^2}{(c+a)(c+b)}\ge\frac34.$$
2005 IMO Shortlist, 7
In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively.
Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ .
[i]Proposed by Hojoo Lee, Korea[/i]
JBMO Geometry Collection, 2011
Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that
\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]
If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$
2008 Harvard-MIT Mathematics Tournament, 3
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.
1963 IMO, 3
In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation
\[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \]
Prove that $a_{1}=a_{2}= \ldots= a_{n}$.
1992 AMC 12/AHSME, 18
The increasing sequence of positive integers $a_{1},a_{2},a_{3},\ldots$ has the property that $a_{n+2} = a_{n} + a_{n+1}$ for all $n \ge 1$. If $a_{7} = 120$, then $a_{8}$ is
$ \textbf{(A)}\ 128\qquad\textbf{(B)}\ 168\qquad\textbf{(C)}\ 193\qquad\textbf{(D)}\ 194\qquad\textbf{(E)}\ 210 $
1997 China Team Selection Test, 1
Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions:
[b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2}
x^2 + a_{2n}, a_0 > 0$;
[b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left(
\begin{array}{c}
2n\\
n\end{array} \right) a_0 a_{2n}$;
[b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.
2023 China Team Selection Test, P11
Let $n\in\mathbb N_+.$ For $1\leq i,j,k\leq n,a_{ijk}\in\{ -1,1\} .$ Prove that: $\exists x_1,x_2,\cdots ,x_n,y_1,y_2,\cdots ,y_n,z_1,z_2,\cdots ,z_n\in \{-1,1\} ,$ satisfy
$$\left| \sum\limits_{i=1}^n\sum\limits_{j=1}^n\sum\limits_{k=1}^na_{ijk}x_iy_jz_k\right| >\frac {n^2}3.$$
[i]Created by Yu Deng[/i]
2016 CCA Math Bonanza, L4.3
Let $ABC$ be a non-degenerate triangle with perimeter $4$ such that $a=bc\sin^2A$. If $M$ is the maximum possible area of $ABC$ and $m$ is the minimum possible area of $ABC$, then $M^2+m^2$ can be expressed in the form $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $a+b$.
[i]2016 CCA Math Bonanza Lightning #4.3[/i]
2019 China Team Selection Test, 5
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.
2021 Israel Olympic Revenge, 4
Prove that the inequality
$$\frac{4}{a+bc+4}+\frac{4}{b+ca+4}+\frac{4}{c+ab+4}\le 1+\frac{1}{2a+1}+\frac{1}{2b+1}+\frac{1}{2c+1}$$
holds for all positive reals $a,b,c$ such that $a^2+b^2+c^2+abc=4$.