Olympiad problems

1997 All-Russian Olympiad Regional Round, 8.5

Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.

2018 Ramnicean Hope, 3

Tags: geometric inequality , inequalities , geometry

Prove that for any noncollinear points $ A,B,C $ and positive real numbers $ x,y, $ the following inequality is true. $$ xAB^2- \frac{xy}{x+y}BC^2 +yCA^2\ge 0 $$ [i]Constantin Rusu[/i]

2009 Sharygin Geometry Olympiad, 1

Tags: geometry , geometric inequality

Let $a, b, c$ be the lengths of some triangle's sides, $p, r$ be the semiperimeter and the inradius of triangle. Prove an inequality $\sqrt{\frac{ab(p- c)}{p}} +\sqrt{\frac{ca(p- b)}{p}} +\sqrt{\frac{bc(p-a)}{p}} \ge 6r$ (D.Shvetsov)

1989 All Soviet Union Mathematical Olympiad, 496

Tags: geometry , perimeter , inequalities , geometric inequality

A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.

1984 IMO Shortlist, 13

Tags: geometry , 3d geometry , tetrahedron , volume , geometric inequality

Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

2022 IFYM, Sozopol, 3

Tags: geometric inequality , geometry , tangential quadrilateral , tangential

Quadrilateral $ABCD$ is circumscribed around circle $k$. Gind the smallest possible value of $$\frac{AB + BC + CD + DA}{AC + BD}$$, as well as all quadrilaterals with the above property where it is reached.

2005 Cuba MO, 1

Tags: geometric inequality , geometry , circles

Determine the smallest real number $a$ such that there is a square of side $a$ such that contains $5$ unit circles inside it without common interior points in pairs.

1991 IMO Shortlist, 4

Tags: geometry , geometric inequality , angle , triangle , brocard

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

Kyiv City MO 1984-93 - geometry, 1993.11.4

Tags: geometry , geometric inequality

Let $a, b, c$ be the lengths of the sides of a triangle, and let $S$ be it's area. Prove that $$S \le \frac{a^2+b^2+c^2}{4\sqrt3}$$ and the equality is achieved only for an equilateral triangle.

1997 Akdeniz University MO, 5

Tags: geometry , geometric inequality

A $ABC$ triangle divide by a $d$ line such that, new two pieces' areas are equal. $d$ line intersects with $[AB]$ at $D$, $[AC]$ at $E$. Prove that $$\frac{AD+AE}{BD+DE+EC+CB} > \frac{1}{4}$$

2021 Estonia Team Selection Test, 3

Tags: geometric inequality , disks , inequalities , geometry

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Ukrainian TYM Qualifying - geometry, VI.14

Tags: geometry , bicentric quadrilateral , geometric inequality

A quadrilateral whose perimeter is equal to $P$ is inscribed in a circle of radius $R$ and is circumscribed around a circle of radius $r$. Check whether the inequality $P\le \frac{r+\sqrt{r^2+4R^2}}{2}$ holds. Try to find the corresponding inequalities for the $n$-gon ($n \ge 5$) inscribed in a circle of radius $R$ and circumscribed around a circle of radius $r$.

2014 Contests, 3

Tags: geometry , inequalities , geometric inequality

Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation \[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \] holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.

2011 Saudi Arabia Pre-TST, 3.3

Tags: geometry , geometric inequality

Let $P$ be a point in the interior of triangle $ABC$. Lines $AP$, $BP$, $CP$ intersect sides $BC$, $CA$, $AB$ at $L$, $M$, $N$, respectively. Prove that $$AP \cdot BP \cdot CP \ge 8PL \cdot PM \cdot PN.$$

1952 Moscow Mathematical Olympiad, 215

Tags: geometry , incircle , inradius , geometric inequality

$\vartriangle ABC$ is divided by a straight line $BD$ into two triangles. Prove that the sum of the radii of circles inscribed in triangles $ABD$ and $DBC$ is greater than the radius of the circle inscribed in $\vartriangle ABC$.

1996 IMO, 5

Tags: geometry , circumcircle , geometric inequality , hexagon

Let $ ABCDEF$ be a convex hexagon such that $ AB$ is parallel to $ DE$, $ BC$ is parallel to $ EF$, and $ CD$ is parallel to $ FA$. Let $ R_{A},R_{C},R_{E}$ denote the circumradii of triangles $ FAB,BCD,DEF$, respectively, and let $ P$ denote the perimeter of the hexagon. Prove that \[ R_{A} \plus{} R_{C} \plus{} R_{E}\geq \frac {P}{2}. \]

2020 Estonia Team Selection Test, 2

Tags: area , geometric inequality , geometry

The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$. Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.

1987 IMO Shortlist, 19

Tags: trigonometry , geometry , geometric inequality , trigonometric inequality

Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than \[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\] [i]Proposed by Soviet Union[/i]

1966 IMO Longlists, 21

Tags: geometric inequality , geometry , area , 3d geometry , volume

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

1967 IMO Shortlist, 1

Tags: geometry , 3d geometry , tetrahedron , volume , geometric inequality

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

1970 IMO Longlists, 17

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

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

2012 Mathcenter Contest + Longlist, 4

Tags: geometry , inequalities , geometric inequality , algebra

Let $a,b,c$ be the side lengths of any triangle. Prove that $$\frac{a}{\sqrt{2b^2+2c^2-a^2}}+\frac{b}{\sqrt{2c^2+2a^2-b^2 }}+\frac{c}{\sqrt{2a^2+2b^2-c^2}}\ge \sqrt{3}.$$ [i](Zhuge Liang)[/i]

1973 IMO Shortlist, 3

Tags: geometry , inequalities , vector , geometric inequality

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.

2018 Irish Math Olympiad, 7

Tags: inequalities , algebra , geometric inequality

Let $a, b, c$ be the side lengths of a triangle. Prove that $2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)$

2015 Czech and Slovak Olympiad III A, 5

Tags: incenter , geometric inequality , geometry

In given triangle $\triangle ABC$, difference between sizes of each pair of sides is at least $d>0$. Let $G$ and $I$ be the centroid and incenter of $\triangle ABC$ and $r$ be its inradius. Show that $$[AIG]+[BIG]+[CIG]\ge\frac{2}{3}dr,$$ where $[XYZ]$ is (nonnegative) area of triangle $\triangle XYZ$.