Found problems: 6530
2006 Romania National Olympiad, 4
Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \]
[i]selected by Mircea Lascu[/i]
1995 IberoAmerican, 2
Let $n$ be a positive integer greater than 1. Determine all the collections of real numbers $x_1,\ x_2,\dots,\ x_n\geq1\mbox{ and }x_{n+1}\leq0$ such that the next two conditions hold:
(i) $x_1^{\frac12}+x_2^{\frac32}+\cdots+x_n^{n-\frac12}= nx_{n+1}^\frac12$
(ii) $\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1}$
1957 Poland - Second Round, 4
Prove that if $ a > 0 $, $ b > 0 $, $ c > 0 $, then
$$
\frac{a}{b + c} + \frac{b}{c+ a} + \frac{c}{a+b} \geq \frac{3}{2}.$$
1993 Kurschak Competition, 1
Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.
2011 Mongolia Team Selection Test, 3
Let $G$ be a graph, not containing $K_4$ as a subgraph and $|V(G)|=3k$ (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in $G$?
2020 Jozsef Wildt International Math Competition, W40
If $0\le x_k<k$, for any $k\in\{1,2,\ldots,n\}$, $m\in\mathbb R_{\ge2}$, then prove that
$$\frac1{\sqrt[m]{(1-x_1)(2-x_2)\cdots(n-x_n)}}+\frac1{\sqrt[m]{(1+x_1)(2+x_2)\cdots(n+x_n)}}\ge\frac2{\sqrt[m]{n!}}$$
[i]Proposed by Dorin Mărghidanu[/i]
2018 Belarus Team Selection Test, 1.4
Let $A_1H_1,A_2H_2,A_3H_3$ be altitudes and $A_1L_1,A_2L_2,A_3L_3$ be bisectors of acute-angles triangle $A_1A_2A_3$.
Prove the inequality $S(L_1L_2L_3)\ge S(H_1H_2H_3)$ where $S$ stands for the area of a triangle.
[i](B. Bazylev)[/i]
2010 Contests, 3
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
1985 All Soviet Union Mathematical Olympiad, 403
Find all the pairs $(x,y)$ such that $|\sin x-\sin y| + \sin x \sin y \le 0$.
2008 Romania National Olympiad, 3
Let $ n$ be a positive integer and let $ a_i$ be real numbers, $ i \equal{} 1,2,\ldots,n$ such that $ |a_i|\leq 1$ and $ \sum_{i\equal{}1}^n a_i \equal{} 0$.
Show that $ \sum_{i\equal{}1}^n |x \minus{} a_i|\leq n$, for every $ x\in \mathbb{R}$ with $ |x|\le 1$.
1995 Swedish Mathematical Competition, 4
The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.
2016 Indonesia TST, 2
Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds:
\[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]
2004 IMO Shortlist, 7
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[
n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality.
[i]Proposed by Finbarr Holland, Ireland[/i]
1974 All Soviet Union Mathematical Olympiad, 203
Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $$f(x_1+x_2)\ge f(x_1)+f(x_2).$$
a) Prove that for all $x$ holds $f(x) \le 2x$.
b) Is the inequality $f(x) \le 1.9x$ valid?
1996 Taiwan National Olympiad, 2
Let $0<a\leq 1$ be a real number and let $a\leq a_{i}\leq\frac{1}{a_{i}}\forall i=\overline{1,1996}$ are real numbers. Prove that for any nonnegative real numbers $k_{i}(i=1,2,...,1996)$ such that $\sum_{i=1}^{1996}k_{i}=1$ we have $(\sum_{i=1}^{1996}k_{i}a_{i})(\sum_{i=1}^{1996}\frac{k_{i}}{a_{i}})\leq (a+\frac{1}{a})^{2}$.
2010 Costa Rica - Final Round, 4
Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]
2020 Jozsef Wildt International Math Competition, W46
Let $x_1,x_2,\ldots,x_n\ge0$, $\alpha,\beta>0$, $\beta\ge\alpha$, $t\in\mathbb R$, such that $x_1^{x_2^t}\cdot x_2^{x_3^t}\cdots x_n^{x_1^t}=1$. Then prove that
$$x_1^\beta x_2^t+x_2^\beta x_3^t+\ldots+x_n^\beta x_1^t\ge x_1^\alpha x_2^t+x_2^\alpha x_3^t+\ldots+x_n^\alpha x_1^t.$$
[i]Proposed by Marius Drăgan[/i]
2000 BAMO, 3
Let $x_1, x_2, ..., x_n$ be positive numbers, with $n \ge 2$. Prove that
$$\left(x_1+\frac{1}{x_1}\right)\left(x_2+\frac{1}{x_2}\right)...\left(x_n+\frac{1}{x_n}\right)\ge \left(x_1+\frac{1}{x_2}\right)\left(x_2+\frac{1}{x_3}\right)...\left(x_{n-1}+\frac{1}{x_n}\right)\left(x_n+\frac{1}{x_1}\right)$$
1981 Romania Team Selection Tests, 4.
Consider $x_1,\ldots,x_n>0$. Show that there exists $a_1,a_2,\ldots,a_n\in \{-1,1\}$ such that
\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]
[i]Laurențiu Panaitopol[/i]
2009 Harvard-MIT Mathematics Tournament, 10
Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.
2008 India Regional Mathematical Olympiad, 3
Suppose $ a$ and $ b$ are real numbers such that the roots of the cubic equation $ ax^3\minus{}x^2\plus{}bx\minus{}1$ are positive real numbers. Prove that:
\[ (i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3}
\]
[19 points out of 100 for the 6 problems]
2004 Baltic Way, 8
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.
2004 China Team Selection Test, 2
Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.
2024 New Zealand MO, 5
Determine the least real number $L$ such that $$\dfrac{1}{a}+\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}\leqslant L$$ for all quadruples $(a,b,c,d)$ of integers satisfying $1<a<b<c<d$.
1959 Czech and Slovak Olympiad III A, 2
Let $a, b, c$ be real numbers such that $a+b+c > 0$, $ab+bc+ca > 0$, $abc > 0$. Show that $a, b, c$ are all positive.