Found problems: 6530
2002 IMO Shortlist, 2
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
2017 Moldova EGMO TST, 1
Let $a,b,c\geq 0$.
Prove: $$\frac{1+a+a^{2}}{1+b+c^{2}}+\frac{1+b+b^{2}}{1+c+a^{2}}+\frac{1+c+c^{2}}{1+a+b^{2}}\geq 3$$
1975 Bulgaria National Olympiad, Problem 4
In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which
$$\sum_{i=1}^n A_iX^2\ge2nR^2.$$
Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that
$$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$
[i]H. Lesov[/i]
2002 Tournament Of Towns, 2
A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds:
\[ |\ell-1|>\frac{1}{5} \]
1988 IMO Shortlist, 12
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
2012 USAMTS Problems, 1
Several children were playing in the ugly tree when suddenly they all fell.
$\bullet$ Roger hit branches $A$, $B$, and $C$ in that order on the way down.
$\bullet$ Sue hit branches $D$, $E$, and $F$ in that order on the way down.
$\bullet$ Gillian hit branches $G$, $A$, and $C$ in that order on the way down.
$\bullet$ Marcellus hit branches $B$, $D$, and $H$ in that order on the way down.
$\bullet$ Juan-Phillipe hit branches $I$, $C$, and $E$ in that order on the way down.
Poor Mikey hit every branch A through $I$ on the way down. Given only this information, in how many different orders could he have hit these 9 branches on the way down?
2010 Hong kong National Olympiad, 2
Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]
2015 Iran Team Selection Test, 6
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that
$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
2013 AMC 12/AHSME, 20
Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \leq 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$?
$ \textbf{(A)} \ 810 \qquad \textbf{(B)} \ 855 \qquad \textbf{(C)} \ 900 \qquad \textbf{(D)} \ 950 \qquad \textbf{(E)} \ 988$
2008 Croatia Team Selection Test, 1
Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of:
$ (a)\quad \frac{x^2 \plus{} y^2 \plus{} z^2}{xy \plus{} yz}$
$ (b)\quad \frac{x^2 \plus{} y^2 \plus{} 2z^2}{xy \plus{} yz}$
2014 Belarus Team Selection Test, 2
Prove that for all even positive integers $n$ the following inequality holds
a) $\{n\sqrt6\} > \frac{1}{n}$
b)$ \{n\sqrt6\}> \frac{1}{n-1/(5n)} $
(I. Voronovich)
1989 National High School Mathematics League, 7
If $\log_{a}\sqrt2<1$, then the range value of $a$ is________.
2004 USAMO, 6
A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that \[
(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2.
\] Prove that $ABCD$ is an isosceles trapezoid.
1993 All-Russian Olympiad Regional Round, 10.3
Solve in positive numbers the system
$ x_1\plus{}\frac{1}{x_2}\equal{}4, x_2\plus{}\frac{1}{x_3}\equal{}1, x_3\plus{}\frac{1}{x_4}\equal{}4, ..., x_{99}\plus{}\frac{1}{x_{100}}\equal{}4, x_{100}\plus{}\frac{1}{x_1}\equal{}1$
2009 Moldova Team Selection Test, 4
let $ x, y, z$ be real number in the interval $ [\frac12;2]$ and $ a, b, c$ a permutation of them. Prove the inequality:
$ \dfrac{60a^2\minus{}1}{4xy\plus{}5z}\plus{}\dfrac{60b^2\minus{}1}{4yz\plus{}5x}\plus{}\dfrac{60c^2\minus{}1}{4zx\plus{}5y}\geq 12$
2012 Olympic Revenge, 6
Let $ABC$ be an scalene triangle and $I$ and $H$ its incenter, ortocenter respectively.
The incircle touchs $BC$, $CA$ and $AB$ at $D,E$ an $F$. $DF$ and $AC$ intersects at $K$ while $EF$ and $BC$ intersets at $M$.
Shows that $KM$ cannot be paralel to $IH$.
PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect.
PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students.
Sorry if I had some English mistakes here.
2019 India IMO Training Camp, P2
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
2006 Estonia Team Selection Test, 5
Let $a_1, a_2, a_3, ...$ be a sequence of positive real numbers. Prove that for any positive integer $n$ the inequality holds $\sum_{i=1}^n b_i^2 \le 4 \sum_{i=1}^n a_i^2$ where $b_i$ is the arithmetic mean of the numbers $a_1, a_2, ..., a_n$
1989 Federal Competition For Advanced Students, 3
Let $ a$ be a real number. Prove that if the equation $ x^2\minus{}ax\plus{}a\equal{}0$ has two real roots $ x_1$ and $ x_2$, then: $ x_1^2\plus{}x_2^2 \ge 2(x_1\plus{}x_2).$
2021 Macedonian Mathematical Olympiad, Problem 1
Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$
For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.
2011 ISI B.Stat Entrance Exam, 4
Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.
1997 Korea - Final Round, 5
For positive numbers $ a_1,a_2,\dots,a_n$, we define
\[ A\equal{}\frac{a_1\plus{}a_2\plus{}\cdots\plus{}a_n}{n}, \quad G\equal{}\sqrt[n]{a_1\cdots a_n}, \quad H\equal{}\frac{n}{a_1^{\minus{}1}\plus{}\cdots\plus{}a_n^{\minus{}1}}\]
Prove that
(i) $ \frac{A}{H}\leq \minus{}1\plus{}2\left(\frac{A}{G}\right)^n$, for n even
(ii) $ \frac{A}{H}\leq \minus{}\frac{n\minus{}2}{n}\plus{}\frac{2(n\minus{}1)}{n}\left(\frac{A}{G}\right)^n$, for $ n$ odd
2003 Romania National Olympiad, 1
Let be a tetahedron $ OABC $ with $ OA\perp OB\perp OC\perp OA. $ Show that
$$ OH\le r\left( 1+\sqrt 3 \right) , $$
where $ H $ is the orthocenter of $ ABC $ and $ r $ is radius of the inscribed spere of $ OABC. $
[i]Valentin Vornicu[/i]
2008 Saint Petersburg Mathematical Olympiad, 4
The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\frac{1}{2}$ and for every $n$ from $1$ to $99$, $x_{n+1}=1-x_1x_2x_3*...*x_{100}$. Prove that $x_{100}>0.99$.
1996 India National Olympiad, 2
Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocenter of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$. Conversely, show that every point in the interior of $C_2$ is the orthocenter of some triangle inscribed in $C_1$.