Found problems: 6530
2010 USA Team Selection Test, 2
Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]
1966 Polish MO Finals, 4
ff nonnegative real numbers$ x_1,x_2,...,x_n$ satisfy $x_1 +...+x_n\le \frac12$, prove that
$$(1-x_1)(1-x_2)...(1-x_n) \ge \frac12$$
1977 Bulgaria National Olympiad, Problem 1
For natural number $n$ and real numbers $\alpha$ and $x$ satisfy the inequalities $\alpha^{n+1}\le x\le1$ and $0<\alpha<1$. Prove that
$$\prod_{k=1}^n\left|\frac{x-\alpha^k}{x+\alpha^k}\right|\le\prod_{k=1}^n\left|\frac{1-\alpha^k}{1+\alpha^k}\right|.$$
[i]Borislav Boyanov[/i]
2014 Balkan MO Shortlist, A4
$\boxed{A4}$Let $m_1,m_2,m_3,n_1,n_2$ and $n_3$ be positive real numbers such that
\[(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3\]
Prove that
\[(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3\]
1997 Argentina National Olympiad, 3
Let $x_1,x_2,x_3,\ldots ,x_{100}$ be one hundred real numbers greater than or equal to $0$ and less than or equal to $1$. Find the maximum possible value of the sum$$S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).$$
2015 China Girls Math Olympiad, 2
Let $a\in(0,1)$ ,$f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3 $ , $g(x)=(1-a)x^3-x^2+(2-a)x-3a-1 $.
Prove that:For any real number $x$ ,at least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$.
2017 Kosovo Team Selection Test, 3
If $a$ and $b$ are positive real numbers with sum $3$, and $x, y, z$ positive real numbers with product $1$, prove that :
$(ax+b)(ay+b)(az+b)\geq 27$
2007 Turkey MO (2nd round), 3
If $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that
$ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $
MathLinks Contest 5th, 3.3
Let $x_1, x_2,... x_n$ be positive numbers such that $S = x_1+x_2+...+x_n =\frac{1}{x_1}+...+\frac{1}{x_n}$
Prove that $$\sum_{i=1}^{n} \frac{1}{n - 1 + x_i} \ge \sum_{i=1}^{n} \frac{1}{1+S - x_i}$$
2004 239 Open Mathematical Olympiad, 4
Let the sum of positive reals $a,b,c$ be equal to 1. Prove an inequality \[
\sqrt{{ab}\over {c+ab}}+\sqrt{{bc}\over {a+bc}}+\sqrt{{ac}\over {b+ac}}\le 3/2
\].
[b]proposed by Fedor Petrov[/b]
2015 Costa Rica - Final Round, 5
Let $f: N^+ \to N^+$ be a function that satisfies that
$$kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+$$
Prove that
$$f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+$$
1997 Canada National Olympiad, 3
Prove that $\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}$.
2013 Baltic Way, 18
Find all pairs $(x,y)$ of integers such that $y^3-1=x^4+x^2$.
2018 Switzerland - Final Round, 8
Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression
$$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$
2018 Balkan MO Shortlist, A4
Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that:
$$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$
2015 Irish Math Olympiad, 10
Prove that, for all pairs of nonnegative integers, $j,n$, $$\sum_{K=0}^{n}k^j\binom n k \ge 2^{n-j} n^j$$
2017 Silk Road, 3
Prove that among any $42$ numbers from the interval $[1,10^6]$, you can choose four numbers so that for any permutation $(a, b, c, d)$ of these numbers, the inequality
$$25 (ab + cd) (ad + bc) \ge 16 (ac + bd)^ 2$$
holds.
2020 Jozsef Wildt International Math Competition, W45
Let $a_1,a_2,a_3,a_4$ be strictly positive numbers. Then is the following inequality true:
$$4\left(a_1a_2^n+a_2a_3^n+a_3a_4^n+a_4a_1^n\right)^n\le\left(a_1^n+a_2^n+a_3^n+a_4^n\right)^{n+1}$$
for each $n\in\mathbb N$?
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
2007 Greece National Olympiad, 2
Let $a,b,c$ be sides of a triangle, show that
\[\frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca.\]
2023 Austrian MO Regional Competition, 1
Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that
$$(a - b)(b - c)(a- c) \le 2.$$
When does equality hold?
[i](Karl Czakler)[/i]
2002 India IMO Training Camp, 20
Let $a,b,c$ be positive real numbers. Prove that
\[\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}\]
2023 IMC, 8
Let $T$ be a tree with $n$ vertices; that is, a connected simple graph on $n$ vertices that contains no cycle. For every pair $u$, $v$ of vertices, let $d(u,v)$ denote the distance between $u$ and $v$, that is, the number of edges in the shortest path in $T$ that connects $u$ with $v$.
Consider the sums
\[W(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}d(u,v) \quad \text{and} \quad H(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}\frac{1}{d(u,v)}\]
Prove that
\[W(T)\cdot H(T)\geq \frac{(n-1)^3(n+2)}{4}.\]
1990 India Regional Mathematical Olympiad, 5
$P$ is any point inside a triangle $ABC$. The perimeter of the triangle $AB + BC + Ca = 2s$. Prove that $s < AP +BP +CP < 2s$.
2013 Bosnia Herzegovina Team Selection Test, 5
Let $x_1,x_2,\ldots,x_n$ be nonnegative real numbers of sum equal to $1$.
Let $F_n=x_1^{2}+x_2^{2}+\cdots +x_n^{2}-2(x_1x_2+x_2x_3+\cdots +x_nx_1)$.
Find:
a) $\min F_3$;
b) $\min F_4$;
c) $\min F_5$.
2009 AMC 10, 12
In quadrilateral $ ABCD$, $ AB \equal{} 5$, $ BC \equal{} 17$, $ CD \equal{} 5$, $ DA \equal{} 9$, and $ BD$ is an integer. What is $ BD$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair C=(0,0), B=(17,0);
pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0];
pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0];
pair[] dotted={A,B,C,D};
draw(D--A--B--C--D--B);
dot(dotted);
label("$D$",D,NW);
label("$C$",C,W);
label("$B$",B,E);
label("$A$",A,NE);[/asy]$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$