Found problems: 6530
2011 USA Team Selection Test, 9
Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality:
\[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.\]
2009 Indonesia MO, 2
For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and
\[ \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots\]
Prove that
\[ \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1\]
holds for every positive integer $ n$.
2023 District Olympiad, P1
Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]
2007 Olympic Revenge, 2
Let $a, b, c \in \mathbb{R}$ with $abc = 1$. Prove that
\[a^{2}+b^{2}+c^{2}+{1\over a^{2}}+{1\over b^{2}}+{1\over c^{2}}+2\left(a+b+c+{1\over a}+{1\over b}+{1\over c}\right) \geq 6+2\left({b\over a}+{c\over b}+{a\over c}+{c\over a}+{c\over b}+{b\over c}\right)\]
2006 MOP Homework, 5
Let $a_1, a_2,...,a_{2005}, b_1, b_2,...,b_{2005}$ be real numbers such that $(a_ix - b_i)^2 \ge \sum_{j\ne i,j=1}^{2005} (a_jx - b_j)$ for all real numbers x and every integer $i$ with $1 \le i \le 2005$. What is maximal number of positive $a_i$'s and $b_i$'s?
2019 Jozsef Wildt International Math Competition, W. 4
If $x, y, z, t > 1$ then: $$\left(\log _{zxt}x\right)^2+\left(\log _{xyt}y\right)^2+\left(\log _{xyz}z\right)^2+\left(\log _{yzt}t\right)^2>\frac{1}{4}$$
1992 AMC 8, 17
The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?
[asy]
pair A,B,C;
A=origin; B=(10,0); C=6.5*dir(15);
dot(A); dot(B); dot(C);
draw(B--A--C);
draw(B--C,dashed);
label("$6.5$",3.25*dir(15),NNW);
label("$10$",(5,0),S);
label("$s$",(8,1),NE);
[/asy]
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$
2019 Tournament Of Towns, 1
Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity
a) not greater than the complexity of $n$.
b) less than the complexity of $n$.
(Boris Frenkin)
2022 Mediterranean Mathematics Olympiad, 3
Let $a, b, c, d$ be four positive real numbers. Prove that
$$\frac{(a + b + c)^2}{a^2+b^2+c^2}+\frac{(b + c + d)^3}{b^3+c^3+d^3}+\frac{(c+d+a)^4}{c^4+d^4+a^4}+\frac{(d+a+b)^5}{d^5+a^5+b^5}\le 120$$
2015 Indonesia MO, 7
Let $a,b,c$ be positive real numbers. Prove that
$\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$
2012 Kyiv Mathematical Festival, 2
Positive numbers $x, y, z$ satisfy $x + y + z \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 8$.
1998 Iran MO (3rd Round), 2
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE$ and $EF = FA$. Prove that
\[\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.\]
2021 Korea Winter Program Practice Test, 3
$n\ge2$ is a given positive integer. $i\leq a_i \leq n$ satisfies for all $1\leq i\leq n$, and $S_i$ is defined as $a_1+a_2+...+a_i(S_0=0)$. Show that there exists such $1\leq k\leq n$ that satisfies $a_k^2+S_{n-k}<2S_n-\frac{n(n+1)}{2}$.
2024 Auckland Mathematical Olympiad, 3
Prove that for arbitrary real numbers $a$ and $b$ the following inequality is true $$a^2 +ab+b^2 \geq 3(a+b-1).$$
2022 China Second Round A2, 1
$a_1,a_2,...,a_9$ are nonnegative reals with sum $1$. Define $S$ and $T$ as below:
$$S=\min\{a_1,a_2\}+2\min\{a_2,a_3\}+...+9\min\{a_9,a_1\}$$
$$T=\max\{a_1,a_2\}+2\max\{a_2,a_3\}+...+9\max\{a_9,a_1\}$$
When $S$ reaches its maximum, find all possible values of $T$.
2013 Harvard-MIT Mathematics Tournament, 29
Let $A_1,A_2,\ldots,A_m$ be finite sets of size $2012$ and let $B_1,B_2,\ldots,B_m$ be finite sets of size $2013$ such that $A_i\cap B_j=\emptyset$ if and only if $i=j$. Find the maximum value of $m$.
2014 Contests, 1
Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that\[bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.\]
2006 Petru Moroșan-Trident, 2
Let be an increasing, infinite sequence of natural numbers $ \left( a_n \right)_{n\ge 1} . $
[b]a)[/b] Prove that if $ a_n=n, $ for any natural numbers $ n, $ then
$$ -2+2\sqrt{1+n} <\frac{1}{\sqrt{a_1}} +\frac{1}{\sqrt{a_2}} +\cdots +\frac{1}{\sqrt{a_n}} <2\sqrt n , $$
for any natural numbers $ n. $
[b]b)[/b] Disprove the converse of [b]a).[/b]
[i]Vasile Radu[/i]
2011 N.N. Mihăileanu Individual, 3
Find $ \inf_{z\in\mathbb{C}} \left( |z^2+z+1|+|z^2-z+1| \right) . $
[i]Gheorghe Andrei[/i] and [i]Doru Constantin Caragea[/i]
2015 Romania National Olympiad, 4
Let $a,b,c,d \ge 0$ real numbers so that $a+b+c+d=1$.Prove that
$\sqrt{a+\frac{(b-c)^2}{6}+\frac{(c-d)^2}{6}+\frac{(d-b)^2}{6}} +\sqrt{b}+\sqrt{c}+\sqrt{d} \le 2.$
2019 Polish MO Finals, 5
The sequence $a_1, a_2, \ldots, a_n$ of positive real numbers satisfies the following conditions:
\begin{align*}
\sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1
\end{align*}
for all $i\in \lbrace 1, 2, \ldots, n \rbrace$, where $a_0$ is an integer. Prove that
\begin{align*}
n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i}
\end{align*}
2009 Austria Beginners' Competition, 2
Let $x$ and $y$ be nonnegative real numbers. Prove that $(x +y^3) (x^3 +y) \ge 4x^2y^2$. When does equality holds?
(Task committee)
1976 IMO Longlists, 34
Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas
\[a_{n+1} = a_n + b_n,\]
\[ b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,\]
and the initial values $a_0 = b_0 = 1$. Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegative integers $n$.
1966 IMO Shortlist, 26
Prove the inequality
[b]a.)[/b] $
\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left(
a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $
where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers.
[b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality
\[
a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left(
a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },
\]
then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.
2022 Bulgarian Spring Math Competition, Problem 8.3
Given the inequalities:
$a)$ $\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}$
$b)$ $\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9$
For each of them either prove that it holds for all positive real numbers $a$, $b$, $c$ or present a counterexample $(a,b,c)$ which doesn't satisfy the inequality.