Found problems: 6530
2018 Mexico National Olympiad, 1
Let $A$ and $B$ be two points on a line $\ell$, $M$ the midpoint of $AB$, and $X$ a point on segment $AB$ other than $M$. Let $\Omega$ be a semicircle with diameter $AB$. Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$. Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$. The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$. Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$. Show that $MX > XY$
2010 Contests, 3
Let $x_1, x_2, \ldots ,x_n(n\ge 2)$ be real numbers greater than $1$. Suppose that $|x_i-x_{i+1}|<1$ for $i=1, 2,\ldots ,n-1$. Prove that
\[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1\]
2018 Peru MO (ONEM), 2
2) Let $a, b, c$ be real numbers such that
$$a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1$$a) Prove that $ab+bc+ca=0$ and $a+b+c=3$.
b) Prove that $|a|+|b|+|c|< 5$
2015 China Second Round Olympiad, 1
Let $a_1, a_2, \ldots, a_n$ be real numbers.Prove that you can select $\varepsilon _1, \varepsilon _2, \ldots, \varepsilon _n\in\{-1,1\}$ such that$$\left( \sum_{i=1}^{n}a_{i}\right)^2 +\left( \sum_{i=1}^{n}\varepsilon _ia_{i}\right)^2 \leq(n+1)\left( \sum_{i=1}^{n}a^2_{i}\right).$$
1993 Poland - First Round, 3
Prove that if $a,b,c$ are the lengths of the sides of a triangle, then
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \leq \frac{1}{a+b-c}+\frac{1}{c+a-b}+\frac{1}{b+c-a}$.
2001 Baltic Way, 15
Let $a_0,a_1,a_2,\ldots $ be a sequence of positive real numbers satisfying $i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}$ for $i=1, 2, \ldots $ Furthermore, let $x$ and $y$ be positive reals, and let $b_i=xa_i+ya_{i-1}$ for $i=1, 2, \ldots $
Prove that the inequality $i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}$ holds for all integers $i\ge 2$.
2010 Sharygin Geometry Olympiad, 14
We have a convex quadrilateral $ABCD$ and a point $M$ on its side $AD$ such that $CM$ and $BM$ are parallel to $AB$ and $CD$ respectively. Prove that $S_{ABCD} \geq 3 S_{BCM}.$
[i]Remark.[/i] $S$ denotes the area function.
2006 Germany Team Selection Test, 2
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]
2013 South africa National Olympiad, 6
Let $ABC$ be an acute-angled triangle with $AC \neq BC$, and let $O$ be the circumcentre and $F$ the foot of the altitude through $C$. Furthermore, let $X$ and $Y$ be the feet of the perpendiculars dropped from $A$ and $B$ respectively to (the extension of) $CO$. The line $FO$ intersects the circumcircle of $FXY$ a second time at $P$. Prove that $OP<OF$.
2011 China Team Selection Test, 2
Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.
2019 Nigeria Senior MO Round 2, 5
Let $a$, $b$, and $c$ be real numbers such that $abc=1$. prove that
$\frac{1+a+ab}{1+b+ab}$ +$\frac{1+b+bc}{1+c+bc}$ + $\frac{1+c+ac}{1+a+ac}$ $>=3$
2024 China Western Mathematical Olympiad, 8
Given a positive integer $n \geq 2$. Let $a_{ij}$ $(1 \leq i,j \leq n)$ be $n^2$ non-negative reals and their sum is $1$. For $1\leq i \leq n$, define $R_i=max_{1\leq k \leq n}(a_{ik})$. For $1\leq j \leq n$, define $C_j=min_{1\leq k \leq n}(a_{kj})$
Find the maximum value of $C_1C_2 \cdots C_n(R_1+R_2+ \cdots +R_n)$
2011 Benelux, 3
If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded:
$a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.
1981 Poland - Second Round, 1
Prove that for any real numbers $ x_1, x_2, \ldots, x_{1981} $, $ y_1, y_2, \ldots, y_{1981} $ such that $ \sum_{j=1}^{1981} x_j = 0 $, $ \sum_{j=1}^{1981} y_j = 0 $ the inequality occurs
$$
\sqrt{\sum_{j=1}^{1981} (x_j^2+y_j^2)} \leq \frac{1}{\sqrt{2}} \sum_{j=1}^{1981} \sqrt{x_j^2+y_j^2}.$$
2014 Contests, 3
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal,
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of
\[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]
2005 Junior Balkan Team Selection Tests - Moldova, 8
The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter.
Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.
2019 Jozsef Wildt International Math Competition, W. 63
If $b_k \geq a_k \geq 0$ $(k = 1, 2, 3)$ and $\alpha \geq 1$ then$$(\alpha+3)\sum \limits_{cyc}(b_1-a_1)\left((b_2+b_3)^{\alpha+2}+(a_2+a_3)^{\alpha+2}-(a_2+b_3)^{\alpha+1}-(b_2+a_3)^{\alpha+1}\right)$$ $$\leq (\alpha+2)(\alpha+3)\sum \limits_{cyc}(b_1-a_1)(b_2-a_2)(b_3^{\alpha+1}-a_3^{\alpha+1})$$ $$+ (b_3 + b_2 + a_1)^{\alpha+3}+(b_3 + a_2 + a_1)^{\alpha+3}+(a_3 + b_2 + a_1)^{\alpha+3}+(a_3 + a_2 + b_1)^{\alpha+3}$$ $$-(b_3 + b_2 + b_1)^{\alpha+3}-(b_3 + a_2 + a_1)^{\alpha+3}-(a_3 + b_2 + b_1)^{\alpha+3}-(a_3 + a_2 + a_1)^{\alpha+3}$$
1982 All Soviet Union Mathematical Olympiad, 348
The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that perimeter of $KLMN$ is less than $4/3$ perimeter of $ABCD$.
1961 AMC 12/AHSME, 20
The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants:
${{ \textbf{(A)}\ \text{I and II} \qquad\textbf{(B)}\ \text{II and III} \qquad\textbf{(C)}\ \text{I and III} \qquad\textbf{(D)}\ \text{III and IV} }\qquad\textbf{(E)}\ \text{I and IV} } $
2020 Saint Petersburg Mathematical Olympiad, 3.
On the side $AD$ of the convex quadrilateral $ABCD$ with an acute angle at $B$, a point $E$ is marked.
It is known that $\angle CAD = \angle ADC=\angle ABE =\angle DBE$.
(Grade 9 version) Prove that $BE+CE<AD$.
(Grade 10 version) Prove that $\triangle BCE$ is isosceles.(Here the condition that $\angle B$ is acute is not necessary.)
1989 Romania Team Selection Test, 3
(a) Find the point $M$ in the plane of triangle $ABC$ for which the sum $MA + MB+ MC$ is minimal.
(b) Given a parallelogram $ABCD$ whose angles do not exceed $120^o$, determine $min \{MA+ MB+NC+ND+ MN | M,N$ are in the plane $ABCD\}$ in terms of the sides and angles of the parallelogram.
1968 Czech and Slovak Olympiad III A, 1
Let $a_1,\ldots,a_n\ (n>2)$ be real numbers with at most one zero. Solve the system
\begin{align*}
x_1x_2 &= a_1, \\
x_2x_3 &= a_2, \\
&\ \vdots \\
x_{n-1}x_n &= a_{n-1}, \\
x_nx_1 &\ge a_n.
\end{align*}
2004 All-Russian Olympiad Regional Round, 10.1
The sum of positive numbers $a, b, c$ is equal to $\pi/2$. Prove that
$$\cos a + \cos b + \cos c > \sin a + \sin b + \sin c.$$
2006 Germany Team Selection Test, 2
Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.
1995 China National Olympiad, 2
Let $a_1,a_2,\cdots ,a_{10}$ be pairwise distinct natural numbers with their sum equal to 1995. Find the minimal value of $a_1a_2+a_2a_3+\cdots +a_9a_{10}+a_{10}a_1$.