Olympiad problems

2011 IMO Shortlist, 4

Tags: marvio , symmetry , radical axis , circumcircle , homothety , geometry

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

2014 Taiwan TST Round 1, 1

Tags: marvio , inequalities

Prove that for positive reals $a$, $b$, $c$ we have \[ 3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}. \]

2010 Balkan MO, 3

Tags: combinatorics , extremal combinatorics , marvio , induction , trigonometry , geometry

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

2022 Iranian Geometry Olympiad, 1

Tags: marvio , geometry

Four points $A$, $B$, $C$ and $D$ lie on a circle $\omega$ such that $AB=BC=CD$. The tangent line to $\omega$ at point $C$ intersects the tangent line to $\omega$ at $A$ and the line $AD$ at $K$ and $L$. The circle $\omega$ and the circumcircle of triangle $KLA$ intersect again at $M$. Prove that $MA=ML$. [i]Proposed by Mahdi Etesamifard[/i]

1978 USAMO, 1

Tags: inequalities , marvio , vector

Given that $a,b,c,d,e$ are real numbers such that $a+b+c+d+e=8$, $a^2+b^2+c^2+d^2+e^2=16$. Determine the maximum value of $e$.

2012 China Girls Math Olympiad, 1

Tags: marvio , inequalities , induction

Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that $\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$

2012 Morocco TST, 4

Tags: marvio , radical axis , symmetry , homothety , circumcircle , geometry

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

1983 USAMO, 2

Tags: calculus , derivative , algebra , quadratic , marvio

Prove that the roots of\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.

2010 Balkan MO Shortlist, C3

Tags: geometry , extremal combinatorics , marvio , trigonometry , induction , combinatorics

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

2012 Brazil Team Selection Test, 3

Tags: homothety , symmetry , radical axis , marvio , circumcircle , geometry

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

2010 Contests, 3

Tags: trigonometry , induction , combinatorics , extremal combinatorics , marvio , geometry

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

2016 EGMO, 4

Tags: geometry , circles , marvio

Two circles $\omega_1$ and $\omega_2$, of equal radius intersect at different points $X_1$ and $X_2$. Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$. Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$.