Found problems: 6530
2022 Greece JBMO TST, 3
The real numbers $x,y,z$ are such that $x+y+z=4$ and $0 \le x,y,z \le 2$. Find the minimun value of the expression $$A=\sqrt{2+x}+\sqrt{2+y}+\sqrt{2+z}+\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}$$.
2011 Junior Balkan Team Selection Tests - Romania, 3
Let n be a positive integer and let $x_1, x_2,...,x_n$ and $y_1, y_2,...,y_n$ be real numbers. Prove that there exists a number $i, i = 1, 2,...,n$, such that $$\sum_{j=1}^n |x_i - x_j | \le \sum_{j=1}^n |x_i - y_j | $$
1995 Taiwan National Olympiad, 6
Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.
2014 Contests, 4
Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that:
\[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]
1978 Bulgaria National Olympiad, Problem 4
Find the greatest possible real value of $S$ and smallest possible value of $T$ such that for every triangle with sides $a,b,c$ $(a\le b\le c)$ to be true the inequalities:
$$S\le\frac{(a+b+c)^2}{bc}\le T.$$
2012 Romania National Olympiad, 3
[color=darkred]Prove that if $n\ge 2$ is a natural number and $x_1,x_2,\ldots,x_n$ are positive real numbers, then:
\[4\left(\frac {x_1^3-x_2^3}{x_1+x_2}+\frac {x_2^3-x_3^3}{x_2+x_3}+\ldots+\frac {x_{n-1}^3-x_n^3}{x_{n-1}+x_n}+\frac {x_n^3-x_1^3}{x_n+x_1}\right)\le \\ \\
\le(x_1-x_2)^2+(x_2-x_3)^2+\ldots+(x_{n-1}-x_n)^2+(x_n-x_1)^2\, .\][/color]
2002 China Team Selection Test, 3
$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and:
\[ \sum_{i\equal{}1}^n x_i < \frac{1}{k\plus{}1} \left[ k \cdot \frac{n(n\plus{}1)(2n\plus{}1)}{6} \minus{} (k\plus{}1)^2 \cdot \frac{n(n\plus{}1)}{2} \right]\]
and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i \plus{} x_j \equal{} x_t \plus{} x_l$.
2020 Purple Comet Problems, 16
Find the maximum possible value of $$\left( \frac{a^3}{b^2c}+\frac{b^3}{c^2a}+\frac{c^3}{a^2b} \right)^2$$ where $a, b$, and $c$ are nonzero real numbers satisfying $$a \sqrt[3]{\frac{a}{b}}+b\sqrt[3]{\frac{b}{c}}+c\sqrt[3]{\frac{c}{a}}=0$$
1975 Bundeswettbewerb Mathematik, 1
Let $a, b, c, d$ be distinct positive real numbers. Prove that if one of the numbers $c, d$ lies between $a$ and $b$, or one of $a, b$ lies between $c$ and $d$, then
$$\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd}$$
and that otherwise, one can choose $a, b, c, d$ so that this inequality is false.
2011 South africa National Olympiad, 5
Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties:
[list]
[*] $0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$
[*] $x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$[/list]
1980 All Soviet Union Mathematical Olympiad, 299
Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let
$$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$
2005 China Team Selection Test, 1
Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.
2004 Iran MO (3rd Round), 7
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$.
Find the smallest k that:
$S(F) \leq k.P(F)^2$
1972 Bulgaria National Olympiad, Problem 5
In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.
(a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed.
(b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold?
[i]H. Lesov[/i]
2004 National High School Mathematics League, 3
The solution set to the inequality $\sqrt{\log_2 x-1}+\frac{1}{2}\log_{\frac{1}{2}}x^3+2>0$ is
$\text{(A)}[2,3)\qquad\text{(B)}(2,3]\qquad\text{(C)}[2,4)\qquad\text{(D)}(2,4]$
1985 IMO Longlists, 71
For every integer $r > 1$ find the smallest integer $h(r) > 1$ having the following property: For any partition of the set $\{1, 2, . . ., h(r)\}$ into $r$ classes, there exist integers $a \geq 0, 1 \leq x \leq y$ such that the numbers $a + x, a + y, a + x + y$ are contained in the same class of the partition.
2012 NIMO Summer Contest, 2
Compute the number of positive integers $n$ satisfying the inequalities
\[
2^{n-1} < 5^{n-3} < 3^n.
\][i]Proposed by Isabella Grabski[/i]
PEN A Problems, 13
Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.
2012 Online Math Open Problems, 46
If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$.
[i]Author: Alex Zhu[/i]
2018 239 Open Mathematical Olympiad, 8-9.4
In a triangle, each median forms an angle with the side it is drawn to, which is less than $\alpha$. Prove that one of the angles of the triangle is greater than $180^\circ-\frac{3}{2}\alpha$.
[i]Proposed by Sergey Berlov[/i]
2008 IMO Shortlist, 7
Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality
\[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\]
holds. Determine all cases of equality.
[i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]
2015 Moldova Team Selection Test, 1
Let $c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}$. \\Prove that at least one of $a,b$ is bigger than $\sqrt[11]{2015}$.
2019 Yasinsky Geometry Olympiad, p5
In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2 <r_a$ .
(here $r$ is the radius of the circle inscribed in the triangle $ABC$, $r_a$ is the radius of an exscribed circle that touches the sides of $BC$).
(Mykola Moroz)
2007 Postal Coaching, 2
Let $a_1, a_2, a_3$ be three distinct real numbers. Define
$$\begin{cases} b_1=\left(1+\dfrac{a_1a_2}{a_1-a_2}\right)\left(1+\dfrac{a_1a_3}{a_1-a_3}\right) \\ \\
b_2=\left(1+\dfrac{a_2a_3}{a_2-a_3}\right)\left(1+\dfrac{a_2a_1}{a_2-a_1}\right) \\ \\
b_3=\left(1+\dfrac{a_3a_1}{a_3-a_1}\right)\left(1+\dfrac{a_3a_2}{a_3-a_2}\right) \end {cases}$$
Prove that $$1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)$$
When does equality hold?
1998 IMO Shortlist, 2
Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that
\[ \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}. \]