Found problems: 6530
1979 AMC 12/AHSME, 13
The inequality $y-x<\sqrt{x^2}$ is satisfied if and only if
$\textbf{(A) }y<0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad$
$\textbf{(B) }y>0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad$
$\textbf{(C) }y^2<2xy\qquad\textbf{(D) }y<0\qquad\textbf{(E) }x>0\text{ and }y<2x$
2016 Estonia Team Selection Test, 8
Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ . Prove that $xy + yz + zx \ge 3$.
2010 Belarus Team Selection Test, 5.2
Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that
a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$
(I. Bliznets)
1998 Greece National Olympiad, 3
Prove that for any non-zero real numbers $a, b, c,$
\[\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.\]
2015 Saudi Arabia GMO TST, 1
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Prove that $$2 \left( \frac{ab}{a + b} +\frac{bc}{b + c} +\frac{ca}{c+ a}\right)+ 1 \ge 6(ab + bc + ca)$$
Trần Nam Dũng
2013 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c, d > 0$ satisfying $abcd = 1$. Prove that $$\frac{1}{a + b + 2}+\frac{1}{b + c + 2}+\frac{1}{c + d + 2}+\frac{1}{d + a + 2} \le 1$$
2007 Estonia National Olympiad, 2
Let $ x, y, z$ be positive real numbers such that $ x^n, y^n$ and $ z^n$ are side lengths of some triangle for all positive integers $ n$. Prove that at least two of x, y and z are equal.
2019 Belarusian National Olympiad, 11.6
The diagonals of the inscribed quadrilateral $ABCD$ intersect at the point $O$. The points $P$, $Q$, $R$, and $S$ are the feet of the perpendiculars from $O$ to the sides $AB$, $BC$, $CD$, and $DA$, respectively.
Prove the inequality $BD\ge SP+QR$.
[i](A. Naradzetski)[/i]
2013 Finnish National High School Mathematics Competition, 2
In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?
2016 Latvia Baltic Way TST, 5
Given real positive numbers $a, b, c$ and $d$, for which the equalities $a^2 + ab + b^2 = 3c^2$ and $a^3 + a^2b + ab^2 + b^3 = 4d^3$ are fulfilled. Prove that $$a + b + d \le 3c.$$
2021 Bundeswettbewerb Mathematik, 1
A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut.
What is the smallest possible volume of the largest of the three cuboids?
2009 South africa National Olympiad, 4
Let $x_1,x_2,\dots,x_n$ be a finite sequence of real numbersm mwhere $0<x_i<1$ for all $i=1,2,\dots,n$. Put $P=x_1x_2\cdots x_n$, $S=x_1+x_2+\cdots+x_n$ and $T=\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}$. Prove that
\[\frac{T-S}{1-P}>2.\]
PEN A Problems, 80
Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.
2012 Iran MO (3rd Round), 2
Suppose $S$ is a convex figure in plane with area $10$. Consider a chord of length $3$ in $S$ and let $A$ and $B$ be two points on this chord which divide it into three equal parts. For a variable point $X$ in $S-\{A,B\}$, let $A'$ and $B'$ be the intersection points of rays $AX$ and $BX$ with the boundary of $S$. Let $S'$ be those points $X$ for which $AA'>\frac{1}{3} BB'$. Prove that the area of $S'$ is at least $6$.
[i]Proposed by Ali Khezeli[/i]
2017 Thailand TSTST, 5
Let $a, b, c \in \mathbb{R}^+$ such that $a + b + c = 3$. Prove that $$\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.$$
2009 China Team Selection Test, 1
Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$
2008 Estonia Team Selection Test, 3
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
2003 Estonia Team Selection Test, 5
Let $a, b, c$ be positive real numbers satisfying the condition $\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}=1$ . Prove the inequality $$\frac{a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}} \le \frac{3\sqrt3}{2}$$
When does the equality hold?
(L. Parts)
2019 Jozsef Wildt International Math Competition, W. 50
Let $x$, $y$, $z > 0$, $\lambda \in (-\infty, 0) \cup (1,+\infty)$ such that $x + y + z = 1$. Then$$\sum \limits_{cyc} x^{\lambda}y^{\lambda}\sum \limits_{cyc}\frac{1}{(x+y)^{2\lambda}}\geq 9\left(\frac{1}{4}-\frac{1}{9}\sum \limits_{cyc}\frac{1}{(x+1)^2} \right)^{\lambda}$$
2002 Regional Competition For Advanced Students, 3
In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively.
a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$
b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.
1987 IMO Longlists, 20
Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. [i](IMO Problem 3)[/i]
[i]Proposed by Germany, FR[/i]
2006 Switzerland - Final Round, 9
Let $a, b, c, d$ be real numbers. Prove that is
$$(a^2 + b^2 + 1)(c^2 + d^2 + 1) \ge 2(a + c)(b + d).$$
2019 Irish Math Olympiad, 9
Suppose $x, y, z$ are real numbers such that $x^2 + y^2 + z^2 + 2xyz = 1$. Prove that $8xyz \le 1$, with equality if and only if $(x, y,z)$ is one of the following:
$$\left( \frac12, \frac12, \frac12 \right) , \left( -\frac12, -\frac12, \frac12 \right), \left(- \frac12, \frac12, -\frac12 \right), \left( \frac12,- \frac12, - \frac12 \right)$$
2010 Iran MO (3rd Round), 1
[b]two variable ploynomial[/b]
$P(x,y)$ is a two variable polynomial with real coefficients. degree of a monomial means sum of the powers of $x$ and $y$ in it. we denote by $Q(x,y)$ sum of monomials with the most degree in $P(x,y)$.
(for example if $P(x,y)=3x^4y-2x^2y^3+5xy^2+x-5$ then $Q(x,y)=3x^4y-2x^2y^3$.)
suppose that there are real numbers $x_1$,$y_1$,$x_2$ and $y_2$ such that
$Q(x_1,y_1)>0$ , $Q(x_2,y_2)<0$
prove that the set $\{(x,y)|P(x,y)=0\}$ is not bounded.
(we call a set $S$ of plane bounded if there exist positive number $M$ such that the distance of elements of $S$ from the origin is less than $M$.)
time allowed for this question was 1 hour.
2011 National Olympiad First Round, 15
For which pair $(a,b)$, there is no positive real pair $(x,y)$ satisfying $x+2y < a$ and $xy > b$ ?
$\textbf{(A)}\ \left (\frac{15}{7}, \frac{4}{7}\right ) \qquad\textbf{(B)}\ \left (\frac{18}{11}, \frac{1}{3}\right ) \qquad\textbf{(C)}\ \left (\frac{5}{7}, \frac{1}{16}\right ) \qquad\textbf{(D)}\ \left (\frac{6}{7}, \frac{1}{11}\right ) \qquad\textbf{(E)}\ \text{None}$