Found problems: 6530
1984 Austrian-Polish Competition, 8
The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $ f_0(x) = 2x$ and$ f_1(x) =\frac{x}{x-1}$. Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k \in \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$.
2006 Baltic Way, 2
Suppose that the real numbers $a_i\in [-2,17],\ i=1,2,\ldots,59,$ satisfy $a_1+a_2+\ldots+a_{59}=0.$
Prove that
\[a_1^2+a_2^2+\ldots+a_{59}^2\le 2006\]
1984 IMO Longlists, 35
Prove that there exist distinct natural numbers $m_1,m_2, \cdots , m_k$ satisfying the conditions
\[\pi^{-1984}<25-\left(\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}\right)<\pi^{-1960}\]
where $\pi$ is the ratio between a circle and its diameter.
2020 Ecuador NMO (OMEC), 5
In triangle $ABC$, $D$ is the middle point of side $BC$ and $M$ is a point on segment $AD$ such that $AM=3MD$.
The barycenter of $ABC$ and $M$ are on the inscribed circumference of $ABC$.
Prove that $AB+AC>3BC$.
2010 Postal Coaching, 5
Prove that there exist a set of $2010$ natural numbers such that product of any $1006 $ numbers is divisible by product of remaining $1004$ numbers.
1967 Swedish Mathematical Competition, 5
$a_1, a_2, a_3, ...$ are positive reals such that $a_n^2 \ge a_1 + a_2 +... + a_{n-1}$.
Show that for some $C > 0$ we have $a_n \ge C n$ for all $n$.
1974 IMO Longlists, 5
A straight cone is given inside a rectangular parallelepiped $B$, with the apex at one of the vertices, say $T$ , of the parallelepiped, and the base touching the three faces opposite to $T .$ Its axis lies at the long diagonal through $T.$ If $V_1$ and $V_2$ are the volumes of the cone and the parallelepiped respectively, prove that
\[V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.\]
2022 Thailand Mathematical Olympiad, 2
Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that
$$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$
for every positive integers $m,n$. Determine the minimum possible value of
$$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$
across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.
Estonia Open Senior - geometry, 2004.1.5
Find the smallest real number $x$ for which there exist two non-congruent triangles with integral side lengths having area $x$.
2019 China Team Selection Test, 5
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.
2010 Puerto Rico Team Selection Test, 4
Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$.
1988 Romania Team Selection Test, 12
The four vertices of a square are the centers of four circles such that the sum of theirs areas equals the square's area. Take an arbitrary point in the interior of each circle. Prove that the four arbitrary points are the vertices of a convex quadrilateral.
[i]Laurentiu Panaitopol[/i]
1997 Croatia National Olympiad, Problem 2
Given are real numbers $a<b<c<d$. Determine all permutations $p,q,r,s$ of the numbers $a,b,c,d$ for which the value of the sum
$$(p-q)^2+(q-r)^2+(r-s)^2+(s-p)^2$$is minimal.
2005 China Team Selection Test, 1
Find all positive integers $m$ and $n$ such that the inequality:
\[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \]
is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.
2024 Brazil Cono Sur TST, 4
An infinite sequence of positive real numbers $x_0,x_1,x_2,...$ is called $vasco$ if it satisfies the following properties:
(a) $x_0=1,x_1=3$; and
(b) $x_0+x_1+...+x_{n-1}\ge3x_{n}-x_{n+1}$, for every $n\ge1$.
Find the greatest real number $M$ such that, for every $vasco$ sequence, the inequality $\frac{x_{n+1}}{x_{n}}>M$ is true for every $n\ge0$.
1987 Greece National Olympiad, 3
Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$
1988 Swedish Mathematical Competition, 5
Show that there exists a constant $a > 1$ such that, for any positive integers $m$ and $n$, $\frac{m}{n} < \sqrt7$ implies that $$7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .$$
2006 Moldova Team Selection Test, 3
Let $a,b,c$ be sides of a triangle and $p$ its semiperimeter. Show that
$a\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p$
2013 Regional Competition For Advanced Students, 3
For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$
(a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$).
(b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$
2012 Romania Team Selection Test, 1
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]
2009 Bosnia Herzegovina Team Selection Test, 3
$a_{1},a_{2},\dots,a_{100}$ are real numbers such that:\[
a_{1}\geq a_{2}\geq\dots\geq a_{100}\geq0\]
\[
a_{1}^{2}+a_{2}^{2}\geq100\]
\[
a_{3}^{2}+a_{4}^{2}+\dots+a_{100}^{2}\geq100\]
What is the minimum value of sum $a_{1}+a_{2}+\dots+a_{100}.$
2012 India Regional Mathematical Olympiad, 4
Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have
\[(a+b)(b+c)(c+a)\geq 8.\]
Also determine the case of equality.
1982 IMO Longlists, 3
Given $n$ points $X_1,X_2,\ldots, X_n$ in the interval $0 \leq X_i \leq 1, i = 1, 2,\ldots, n$, show that there is a point $y, 0 \leq y \leq 1$, such that
\[\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.\]
2008 China Team Selection Test, 2
Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$
1972 IMO Longlists, 9
Given natural numbers $k$ and $n, k \le n, n \ge 3,$ find the set of all values in the interval $(0, \pi)$ that the $k^{th}-$largest among the interior angles of a convex $n$-gon can take.