Found problems: 6530
2018 Balkan MO Shortlist, A4
Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that:
$$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$
2002 Belarusian National Olympiad, 2
Given rational numbers $a_1,...,a_n$ such that $\sum_{i=1}^n \{ka_i\}<\frac{n}{2}$ for any positive integer $k$.
a) Prove that at least one of $a_1,...,a_n$ is integer.
b) Is the previous statement true, if the number $\frac{n}{2}$ is replaced by the greater number? (Here $\{x\}$ means a fractional part of $x$.)
(N. Selinger)
2004 Federal Competition For Advanced Students, P2, 1
Prove without using advanced (differential) calculus:
(a) For any real numbers a,b,c,d it holds that $a^6+b^6+c^6+d^6-6abcd \ge -2$.
When does equality hold?
(b) For which natural numbers $k$ does some inequality of the form $a^k +b^k +c^k +d^k -kabcd \ge M_k$ hold for all real $a,b,c,d$? For each such $k$,
1997 Korea - Final Round, 2
The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that
\[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\]
and find the conditions for equality.
2011 Middle European Mathematical Olympiad, 2
Let $a, b, c$ be positive real numbers such that
\[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2.\]
Prove that
\[\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.\]
1990 Czech and Slovak Olympiad III A, 4
Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)
2013 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c, d > 0$ satisfying $abcd = 1$. Prove that $$\frac{1}{a + b + 2}+\frac{1}{b + c + 2}+\frac{1}{c + d + 2}+\frac{1}{d + a + 2} \le 1$$
2023 OlimphÃada, 1
Let $n \geq 2023$ be an integer. For each real $x$, we say that $\lfloor x \rceil$ is the closest integer to $x$, and if there are two closest integers then it is the greater of the two. Suppose there is a positive real $a$ such that $$\lfloor an \rceil = n + \bigg\lfloor\frac{n}{a} \bigg\rceil.$$
Show that $|a^2 - a - 1| < \frac{n\varphi+1}{n^2}$.
2011 Mediterranean Mathematics Olympiad, 2
Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$.
Show that $|A|\cdot|B|\le|C|^2$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
1997 Israel National Olympiad, 3
Let $n?$ denote the product of all primes smaller than $n$.
Prove that $n? > n$ holds for any natural number $n > 3$.
2017 Romania National Olympiad, 3
Let $n \in N, n\ge 2$, and $a_1, a_2, ..., a_n, b_1, b_2, ..., b_n$ be real positive numbers such that
$$\frac{a_1}{b_1} \le \frac{a_2}{b_2} \le ... \le\frac{a_n}{b_n}.$$
Find the largest real $c$ so that $$(a_1-b_1c)x_1+(a_2-b_2c)x_2+...+(a_n-b_nc)x_n \ge 0,$$
for every $x_1, x_2,..., x_n > 0$, with $x_1\le x_2\le ...\le x_n$.
1979 IMO Shortlist, 13
Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$
2008 IberoAmerican Olympiad For University Students, 7
Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$, where $pA = \{pa |a \in A\}$, $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$).
Prove that each subgroup of $A$ of finite index is isomorphic to $A$.
1988 China Team Selection Test, 1
Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds:
\[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\]
Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).
I Soros Olympiad 1994-95 (Rus + Ukr), 9.8
Let $f(x) =x^2-2x$. Find all $x$ for which $f(f(x))<3$.
2013 Kazakhstan National Olympiad, 1
Find maximum value of
$|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.
2010 Contests, 2
Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by
\[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \]
is a decreasing function.
[i]Dan Marinescu et al.[/i]
1998 India National Olympiad, 5
Suppose $a,b,c$ are three rela numbers such that the quadratic equation \[ x^2 - (a +b +c )x + (ab +bc +ca) = 0 \] has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that
(i) The numbers $a,b,c$ are all positive.
(ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.
2018 Kazakhstan National Olympiad, 6
Inside of convex quadrilateral $ABCD$ found a point $M$ such that $\angle AMB=\angle ADM+\angle BCM$ and $\angle AMD=\angle ABM+\angle DCM$.Prove that $$AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.$$
2022 Austrian MO National Competition, 1
Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality?
[i](Walther Janous)[/i]
2005 Korea Junior Math Olympiad, 7
If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}}
$$
2024 District Olympiad, P4
Let $f:[0,\infty)\to\mathbb{R}$ be a differentiable function, with a continous derivative. Given that $f(0)=0$ and $0\leqslant f'(x)\leqslant 1$ for every $x>0$ prove that\[\frac{1}{n+1}\int_0^af(t)^{2n+1}\mathrm{d}t\leqslant\left(\int_0^af(t)^n\mathrm{d}t\right)^2,\]for any positive integer $n{}$ and real number $a>0.$
2016 Croatia Team Selection Test, Problem 1
Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that
$$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$
2016 Singapore MO Open, 2
Let $a, b, c$ be real numbers such that $0 < a, b, c < 1/2$ and $a + b + c= 1$. Prove that for all real numbers $x,y,z$,
$$abc(x + y + z)^2 \ge ayz( 1- 2a) + bxz( 1 - 2b) + cxy( 1 - 2c)$$. When does equality hold?
2007 USA Team Selection Test, 2
Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that
\[ a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n \]
and
\[ a_1 + \dots + a_n = b_1 + \dots + b_n. \]
Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.