Found problems: 6530
2003 Baltic Way, 3
Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Prove that
$$\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).$$
1999 China National Olympiad, 2
Determine the maximum value of $\lambda$ such that if $f(x) = x^3 +ax^2 +bx+c$ is a cubic polynomial with all its roots nonnegative, then \[f(x)\geq\lambda(x -a)^3\] for all $x\geq0$. Find the equality condition.
2009 Indonesia TST, 4
Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality
\[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3.
\]
2014 Taiwan TST Round 2, 1
Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$,
\[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]
2016 Czech-Polish-Slovak Junior Match, 2
Let $x$ and $y$ be real numbers such that $x^2 + y^2 - 1 < xy$. Prove that $x + y - |x - y| < 2$.
Slovakia
2016 Estonia Team Selection Test, 4
Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$
2007 Mediterranean Mathematics Olympiad, 1
Let $x \geq y \geq z$ be real numbers such that $xy + yz + zx = 1$. Prove that $xz < \frac 12.$ Is it possible to improve the value of constant $\frac 12 \ ?$
2019 Brazil National Olympiad, 2
Let $a, b$ and $k$ be positive integers with $k> 1$ such that
$lcm (a, b) + gcd (a, b) = k (a + b)$.
Prove that $a + b \geq 4k$
2009 District Olympiad, 1
Let $ f:[0,\infty )\longrightarrow [0,\infty ) $ a nonincreasing function that satisfies the inequality:
$$ \int_0^x f(t)dt <1,\quad\forall x\ge 0. $$ Prove the following affirmations:
[b]a)[/b] $ \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} . $
[b]b)[/b] $ \lim_{x\to\infty} xf(x) =0. $
2007 German National Olympiad, 6
For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$
2002 Hungary-Israel Binational, 2
Let $A', B' , C'$ be the projections of a point $M$ inside a triangle $ABC$ onto the sides $BC, CA, AB$, respectively. Define $p(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC}$ . Find the position of point $M$ that maximizes $p(M )$.
1974 Polish MO Finals, 4
Prove that, so have $k$ for $\forall a_1,a_2,...,a_n$ satisfying
$$|\sum_{i=1}^k a_i -\sum_{j=k+1}^n a_j |\leq \max_{1\leq m\leq n} |a_m|$$
2019 Ramnicean Hope, 3
For this exercise, $ \{\} $ denotes the fractional part.
[b]a)[/b] Let be a natural number $ n. $ Compare $ \left\{ \sqrt{n+1} -\sqrt{n} \right\} $ with $ \left\{ \sqrt{n} -\sqrt{n-1} \right\} . $
[b]b)[/b] Show that there are two distinct natural numbers $ a,b, $ such that $ \left\{ \sqrt{a} -\sqrt{b} \right\} =\left\{ \sqrt{b} -\sqrt{a} \right\} . $
[i]Traian Preda[/i]
2006 AMC 12/AHSME, 10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$
2000 Moldova National Olympiad, Problem 2
Show that if real numbers $x<1<y$ satisfy the inequality
$$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.
2008 Germany Team Selection Test, 1
Show that there is a digit unequal to 2 in the decimal represesentation of $ \sqrt [3]{3}$ between the $ 1000000$-th und $ 3141592$-th position after decimal point.
1986 China Team Selection Test, 3
Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that:
i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$
ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.
2023 UMD Math Competition Part II, 5
Let $0 \le a_1 \le a_2 \le \dots \le a_n \le 1$ be $n$ real numbers with $n \ge 2$. Assume $a_1 + a_2 + \dots + a_n \ge n-1$. Prove that
\[ a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1} \]
1986 IMO Longlists, 1
Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$. Prove the inequality
\[1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k\]
for all real $b \geq 0.$
2019 Switzerland Team Selection Test, 11
Let $n $ be a positive integer. Determine whether there exists a positive real number $\epsilon >0$ (depending on $n$) such that for all positive real numbers $x_1,x_2,\dots ,x_n$, the inequality $$\sqrt[n]{x_1x_2\dots x_n}\leq (1-\epsilon)\frac{x_1+x_2+\dots+x_n}{n}+\epsilon \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}},$$ holds.
2019 JBMO Shortlist, A4
Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that
$a^4 - 2019a = b^4 - 2019b = c$.
Prove that $- \sqrt{c} < ab < 0$.
2011 Kurschak Competition, 2
Let $n$ be a positive integer. Denote by $a(n)$ the ways of expression $n=x_1+x_2+\dots$ where $x_1\leqslant x_2 \leqslant\dots$ are positive integers and $x_i+1$ is a power of $2$ for each $i$. Denote by $b(n)$ the ways of expression $n=y_1+y_2+\dots$ where $y_i$ is a positive integer and $2y_i\leqslant y_{i+1}$ for each $i$.
Prove that $a(n)=b(n)$.
2009 Indonesia TST, 3
Let $ n \ge 2009$ be an integer and define the set:
\[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}.
\]
Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that
\[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}.
\]
2000 Korea - Final Round, 3
The real numbers $a,b,c,x,y,$ and $z$ are such that $a>b>c>0$ and $x>y>z>0$. Prove that
\[\frac {a^2x^2}{(by+cz)(bz+cy)}+\frac{b^2y^2}{(cz+ax)(cx+az)}+\frac{c^2z^2}{(ax+by)(ay+bx)}\ge \frac{3}{4}\]
2004 Baltic Way, 4
Let $x_1$, $x_2$, ..., $x_n$ be real numbers with arithmetic mean $X$. Prove that there is a positive integer $K$ such that for any integer $i$ satisfying $0\leq i<K$, we have $\frac{1}{K-i}\sum_{j=i+1}^{K} x_j \leq X$. (In other words, prove that there is a positive integer $K$ such that the arithmetic mean of each of the lists $\left\{x_1,x_2,...,x_K\right\}$, $\left\{x_2,x_3,...,x_K\right\}$, $\left\{x_3,...,x_K\right\}$, ..., $\left\{x_{K-1},x_K\right\}$, $\left\{x_K\right\}$ is not greater than $X$.)