Found problems: 6530
1990 Bundeswettbewerb Mathematik, 4
In the plane there is a worm of length 1. Prove that it can be always covered by means of half of a circular disk of diameter 1.
[i]Note.[/i] Under a "worm", we understand a continuous curve. The "half of a circular disk" is a semicircle including its boundary.
2014 Denmark MO - Mohr Contest, 5
Let $x_0, x_1, . . . , x_{2014}$ be a sequence of real numbers, which for all $i < j$ satisfy $x_i + x_j \le 2j$. Determine the largest possible value of the sum $x_0 + x_1 + · · · + x_{2014}$.
1992 IberoAmerican, 3
Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle.
a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$, $B$, $C$ is 5.
b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$, $BP$, $CP$. Also, the area of such triangle is $\frac{\sqrt{3}}{4}$.
2002 Balkan MO, 4
Determine all functions $f: \mathbb N\to \mathbb N$ such that for every positive integer $n$ we have: \[ 2n+2001\leq f(f(n))+f(n)\leq 2n+2002. \]
1986 AMC 12/AHSME, 29
Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $
2006 MOP Homework, 4
Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.
2021 Saudi Arabia BMO TST, 3
Let $a$, $b$, and $c$ be positive real numbers. Prove that $$(a^5 - a^2 +3)(b^5 - b^2 +3)(c^5 - c^2 +3)\ge (a+b+c)^3$$
2001 Slovenia National Olympiad, Problem 1
Let $a,b,c,d,e,f$ be positive numbers such that $a,b,c,d$ is an arithmetic progression, and $a,e,f,d$ is a geometric progression. Prove that $bc\ge ef$.
1996 Romania National Olympiad, 1
Let $a, b, c \in R,$ $a \ne 0$, such that $a$ and $4a+3b+2c$ have the same sign. Show that the equation $ax^2+bx+c=0$ cannot have both roots in the interval $(1,2)$.
1994 Putnam, 1
Suppose that a sequence $\{a_n\}_{n\ge 1}$ satisfies $0 < a_n \le a_{2n} + a_{2n+1}$ for all $n\in \mathbb{N}$. Prove that the series$\sum_{n=1}^{\infty} a_n$ diverges.
1999 Romania Team Selection Test, 9
Let $O,A,B,C$ be variable points in the plane such that $OA=4$, $OB=2\sqrt3$ and $OC=\sqrt {22}$. Find the maximum value of the area $ABC$.
[i]Mihai Baluna[/i]
2019 Peru EGMO TST, 5
Define the sequence sequence $a_0,a_1, a_2,....,a_{2018}, a_{2019}$ of real numbers as follows:
$\bullet$ $a_0 = 1$.
$\bullet$ $a_{n + 1} = a_n - \frac{a_n^2}{2019}$ for $n = 0, 1, ...,2018$.
Prove that $a_{2019} < \frac12 <a_{2018}$.
1990 IMO Longlists, 8
Let $a, b, c$ be the side lengths and $P$ be area of a triangle, respectively. Prove that
\[(a^2+b^2+c^2-4\sqrt 3 P) (a^2+b^2+c^2) \geq 2 \left(a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2\right).\]
2014 Contests, 1
Prove that for $n\ge 2$ the following inequality holds:
$$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
1998 Junior Balkan Team Selection Tests - Romania, 2
Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that:
[b]a)[/b] $ p_{MNPQ}\ge AC+BD. $
[b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $
[b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $
[i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]
2015 Taiwan TST Round 3, 1
Let $x,y$ be the positive real numbers with $x+y=1$, and $n$ be the positive integer with $n\ge2$. Prove that
\[\frac{x^n}{x+y^3}+\frac{y^n}{x^3+y}\ge\frac{2^{4-n}}{5}\]
2013 Sharygin Geometry Olympiad, 11
a) Let $ABCD$ be a convex quadrilateral and $r_1 \le r_2 \le r_3 \le r_4$ be the radii of the incircles of triangles $ABC, BCD, CDA, DAB$. Can the inequality $r_4 > 2r_3$ hold?
b) The diagonals of a convex quadrilateral $ABCD$ meet in point $E$. Let $r_1 \le r_2 \le r_3 \le r_4$ be the radii of the incircles of triangles $ABE, BCE, CDE, DAE$. Can the inequality $r_2 > 2r_1$ hold?
1965 IMO, 1
Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]
2000 Saint Petersburg Mathematical Olympiad, 11.2
Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality:
$$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$
[I]Proposed by A. Khrabrov[/i]
2013 China Team Selection Test, 3
Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]
2014 Macedonia National Olympiad, 4
Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that:
\[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]
2000 IMO, 2
Let $ a, b, c$ be positive real numbers so that $ abc \equal{} 1$. Prove that
\[ \left( a \minus{} 1 \plus{} \frac 1b \right) \left( b \minus{} 1 \plus{} \frac 1c \right) \left( c \minus{} 1 \plus{} \frac 1a \right) \leq 1.
\]
2014 India Regional Mathematical Olympiad, 5
Let $a,b,c$ be positive real numbers such that
\[ \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1. \]
Prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$. When does equality hold?
2020 Jozsef Wildt International Math Competition, W58
In all triangles $ABC$ does it hold that:
$$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
2007 Switzerland - Final Round, 1
Determine all positive real solutions of the following system of equations:
$$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$
$$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$