Found problems: 6530
2016 Tuymaada Olympiad, 7
For every $x$, $y$, $z>{3\over 2}$ prove the inequality
$$
x^{24} + \root 5\of {y^{60}+z^{40}} \geq
\left(x^4 y^3 + {1\over 3} y^2 z^2 + {1\over 9} x^3 z^3 \right)^2.
$$
1998 IMO Shortlist, 1
Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that
\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]
1997 Turkey Team Selection Test, 3
If $x_{1}, x_{2},\ldots ,x_{n}$ are positive real numbers with $x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1$, find the minimum value of $\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}$.
2006 Bulgaria Team Selection Test, 3
[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$?
[i]Alexandar Ivanov[/i]
1990 National High School Mathematics League, 9
Let $n$ be a natural number. For all real numbers $x,y,z$, $(x^2+y^2+z^2)^2\geq n(x^4+y^4+z^4)$, then the minumum value of $n$ is________.
2010 BMO TST, 4
Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number.
[b]a)[/b] Prove the inequality for $ k\equal{}1$.
[b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.
2001 Moldova National Olympiad, Problem 1
Prove that $\frac1{2002}<\frac12\cdot\frac34\cdot\frac56\cdots\frac{2001}{2002}<\frac1{44}$.
1996 Iran MO (3rd Round), 1
Let $a,b,c,d$ be positive real numbers. Prove that
\[\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.\]
2018 Pan-African Shortlist, A3
Akello divides a square up into finitely many white and red rectangles, each (rectangle) with sides parallel to the sides of the parent square. Within each white rectangle, she writes down the value of its width divided by its height, while within each red rectangle, she writes down the value of its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of the white rectangles equals the total area of the red rectangles, what is the least possible value of $x$ she can get?
2006 Moldova National Olympiad, 9.1
Let $a,b,c$ be positive real numbers such that $a+b+c=2005$. Find the minimum value of the expression:
$$E=a^{2006}+b^{2006}+c^{2006}+\frac{(ab)^{2004}+(bc)^{2004}+(ca)^{2004}}{(abc)^{2004}}$$
2007 India IMO Training Camp, 2
Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that
\[a^2+b^2+c^2\leq 2abc+1.\]
1993 Polish MO Finals, 2
A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.
1962 All-Soviet Union Olympiad, 7
Let $a;b;c;d>0$ such that $abcd=1$. Prove that $a^2+b^2+c^2+d^2+a(b+c)+b(c+d)+c(d+a)\ge 10$
2004 Singapore MO Open, 1
Let $m,n$ be integers so that $m \ge n > 1$. Let $F_1,...,F_k$ be a collection of $n$-element subsets of $\{1,...,m\}$ so that $F_i\cap F_j$ contains at most $1$ element, $1 \le i < j \le k$. Show that $k\le \frac{m(m-1)}{n(n-1)} $
1990 Irish Math Olympiad, 4
The real number $x$ satisfies all the inequalities $$2^k<x^k+x^{k+1}<2^{k+1}$$ for $k=1,2,\dots ,n$. What is the greatest possible value of $n$?
1994 All-Russian Olympiad Regional Round, 11.1
Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.
2022 Kosovo National Mathematical Olympiad, 4
Let $a,b$ and $c$ be positive real numbers such that $a+b+c+3abc\geq (ab)^2+(bc)^2+(ca)^2+3$. Show that the following inequality hold,
$$\frac{a^3+b^3+c^3}{3}\geq\frac{abc+2021}{2022}.$$
2019 LIMIT Category B, Problem 5
The set of values of $m$ for which $mx^2-6mx+5m+1>0$ for all real $x$ is
$\textbf{(A)}~m<\frac14$
$\textbf{(B)}~m\ge0$
$\textbf{(C)}~0\le m\le\frac14$
$\textbf{(D)}~0\le m<\frac14$
2005 Germany Team Selection Test, 3
Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that
[b](a)[/b] $\triangle ABC$ is acute.
[b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.
2017 Taiwan TST Round 1, 2
Given $a,b,c,d>0$, prove that:
\[\sum_{cyc}\frac{c}{a+2b}+\sum_{cyc}\frac{a+2b}{c}\geq 8(\frac{(a+b+c+d)^2}{ab+ac+ad+bc+bd+cd}-1),\]
where $\sum_{cyc}f(a,b,c,d)=f(a,b,c,d)+f(d,a,b,c)+f(c,d,a,b)+f(b,c,d,a)$.
2002 AMC 12/AHSME, 25
Let $ f(x)\equal{}x^2\plus{}6x\plus{}1$, and let $ R$ denote the set of points $ (x,y)$ in the coordinate plane such that
\[ f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0
\]The area of $ R$ is closest to
$ \textbf{(A)}\ 21 \qquad
\textbf{(B)}\ 22 \qquad
\textbf{(C)}\ 23 \qquad
\textbf{(D)}\ 24 \qquad
\textbf{(E)}\ 25$
2021 Latvia TST, 2.5
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
2023 Macedonian Balkan MO TST, Problem 1
Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers defined by $a_{1}=1$, $a_{2}=2$ and
$$\frac{a_{n+1}^{4}}{a_{n}^3} = 2a_{n+2}-a_{n+1}.$$
Prove that the following inequality holds for every positive integer $N>1$:
$$\sum_{k=1}^{N}\frac{a_{k}^{2}}{a_{k+1}}<3.$$
[i]Note: The bound is not sharp.[/i]
[i]Authored by Nikola Velov[/i]
2013 Kazakhstan National Olympiad, 3
Consider the following sequence : $a_1=1 ; a_n=\frac{a_[{\frac{n}{2}]}}{2}+\frac{a_[{\frac{n}{3}]}}{3}+\ldots+\frac{a_[{\frac{n}{n}]}}{n}$. Prove that $ a_{2n}< 2*a_{n } (\forall n\in\mathbb{N})$
2011 Turkey Team Selection Test, 2
Let $a,b,c$ be positive real numbers satisfying $a^2+b^2+c^2 \geq 3.$ Prove that
\[ \frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2} \]