Olympiad problems

2023 Canadian Mathematical Olympiad Qualification, 6

Given triangle $ABC$ with circumcircle $\Gamma$, let $D$, $E$, and $F$ be the midpoints of sides $BC$, $CA$, and $AB$, respectively, and let the lines $AD$, $BE$, and $CF$ intersect $\Gamma$ again at points $J$, $K$, and $L$, respectively. Show that the area of triangle $JKL$ is at least that of triangle $ABC$.

2022 Kosovo National Mathematical Olympiad, 4

Tags: inequalities , geometric inequality

Assume that in the $\triangle ABC$ there exists a point $D$ on $BC$ and a line $l$ passing through $A$ such that $l$ is tangent to $(ADC)$ and $l$ bisects $BD.$ Prove that $a\sqrt{2}\geq b+c.$

2007 Grigore Moisil Intercounty, 1

Tags: inequalities , geometric inequality , geometry

For a point $ P $ situated in the plane determined by a triangle $ ABC, $ prove the following inequality: $$ BC\cdot PB\cdot PC+AC\cdot PC\cdot PA +AB\cdot PA\cdot PB\ge AB\cdot BC\cdot CA $$

2018 Spain Mathematical Olympiad, 3

Tags: geometry , inequalities , geometric inequality

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$. Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.

2014 Contests, 2

Tags: function , geometric inequality , geometry

Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality: \begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}

1976 Chisinau City MO, 133

Tags: geometry , geometric inequality , square , parallelogram , area

A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.

Champions Tournament Seniors - geometry, 2000.4

Tags: geometry , centroid , geometric inequality , circumcircle

Let $G$ be the point of intersection of the medians in the triangle $ABC$. Let us denote $A_1, B_1, C_1$ the second points of intersection of lines $AG, BG, CG$ with the circle circumscribed around the triangle. Prove that $AG + BG + CG \le A_1C + B_1C + C_1C$. (Yasinsky V.A.)

1973 Dutch Mathematical Olympiad, 1

Tags: geometry , perimeter , geometric inequality

Given is a triangle $ABC$, $\angle C = 60^o$, $R$ the midpoint of side $AB$. There exist a point $P$ on the line $BC$ and a point $Q$ on the line $AC$ such that the perimeter of the triangle $PQR$ is minimal. a) Prove that and also indicate how the points $P$ and $Q$ can be constructed. b) If $AB = c$, $AC = b$, $BC = a$, then prove that the perimeter of the triangle $PQR$ equals $\frac12\sqrt{3c^2+6ab}$ .

1989 IMO Shortlist, 20

Tags: combinatorics , geometric inequality , combinatorial inequality , point set

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that [b]i.)[/b] no three points of $ S$ are collinear, and [b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

1973 IMO, 1

Tags: vector , geometric inequality , inequalities , geometry

Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.

1982 IMO Longlists, 57

Tags: geometry , analytic geometry , convex polygon , area , geometric inequality

Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that \[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\] where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$

Kyiv City MO 1984-93 - geometry, 1990.10.5

Tags: geometry , max , geometric inequality , analytic geometry , parabola , area

A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of the quadrilateral $ABCD$.

2023 Macedonian Team Selection Test, Problem 2

Tags: geometric inequality , geometry , geometric transformation , reflection , inequalities

Let $ABC$ be an acute triangle such that $AB<AC$ and $AB<BC$. Let $P$ be a point on the segment $BC$ such that $\angle APB = \angle BAC$. The tangent to the circumcircle of triangle $ABC$ at $A$ meets the circumcircle of triangle $APB$ at $Q \neq A$. Let $Q'$ be the reflection of $Q$ with respect to the midpoint of $AB$. The line $PQ$ meets the segment $AQ'$ at $S$. Prove that $$\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.$$ [i]Authored by Nikola Velov[/i]

1955 Moscow Mathematical Olympiad, 300

Tags: geometry , geometric inequality

Inside $\vartriangle ABC$, there is fixed a point $D$ such that $AC - DA > 1$ and $BC - BD > 1$. Prove that $EC - ED > 1$ for any point $E$ on segment $AB$.

1984 Tournament Of Towns, (070) T4

Tags: geometry , rectangle , perimeter , geometric inequality , diagonal , inscribed

Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle. (V. V . Proizvolov , Moscow)

2017 Swedish Mathematical Competition, 3

Tags: midpoint , geometric inequality , geometry , 3d geometry

Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that $2 | XY | \le | AD | + | BC |$. When is equality achieved?

2001 Kazakhstan National Olympiad, 6

Tags: geometric inequality , geometry , equilateral

Each interior point of an equilateral triangle with sides equal to $1$ lies in one of six circles of the same radius $ r $. Prove that $ r \geq \frac {{\sqrt 3}} {{10}} $.

2016 Latvia Baltic Way TST, 13

Tags: geometric inequality , geometry , angle

Suppose that $A, B, C$, and $X$ are any four distinct points in the plane with $$\max \,(BX,CX) \le AX \le BC.$$ Prove that $\angle BAC \le 150^o$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.6

Tags: geometry , exradius , geometric inequality

The radius of the circle inscribed in triangle $ABC$ is equal to $r$, and the radius of the circle tangent to the segment $BC$ and the extensions of sides $AB$ and $AC$ (the exscribed circle corresponding to angle $A$) is equal to $R$. A circle with radius $x < r$ is inscribed in angle $\angle BAC$. Tangents to this circles passing through points $B$ and $C$ and different from $BA$ and $AC$ intersect at point $A'$. Let $y$ be the radius of the circle inscribed in triangle $BCK$. Find the greatest value of the sum $x + y$ as x changes from $0$ to $r$. (In this case, it is necessary to prove that this largest value is the same in any triangle with given $r$ and $R$).

2006 Sharygin Geometry Olympiad, 21

Tags: cevian , area of a triangle , geometric inequality , geometry , area , concurrency

On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken. Prove that for the areas of the corresponding triangles, the inequality holds: $$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$ and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.

2008 Balkan MO Shortlist, G1

Tags: geometric inequality , geometry , median , exradius , excircle

In acute angled triangle $ABC$ we denote by $a,b,c$ the side lengths, by $m_a,m_b,m_c$ the median lengths and by $r_{b}c,r_{ca},r_{ab}$ the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that $$\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}$$

1981 Romania Team Selection Tests, 5.

Tags: inequalities , geometry , geometric inequality

Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$. [i]Laurențiu Panaitopol[/i]