Olympiad problems

2015 Stars Of Mathematics, 4

Let $S$ be a finite set of points in the plane,situated in general position(any three points in $S$ are not collinear),and let $$D(S,r)=\{\{x,y\}:x,y\in S,\text{dist}(x,y)=r\},$$ where $R$ is a positive real number,and $\text{dist}(x,y)$ is the euclidean distance between points $x$ and $y$.Prove that $$\sum_{r>0}|D(S,r)|^2\le\frac{3|S|^2(|S|-1)}{4}.$$

2006 MOP Homework, 4

Tags: geometry , geometric inequality , excenter , right triangle

Let $ABC$ be a right triangle with$ \angle A = 90^o$. Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD$. Point $I_a$ is the excenter of the triangle opposite $A$. Prove that $\frac{AD}{DI_a } \le \sqrt{2} -1$

1966 IMO Shortlist, 21

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

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?

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.

2024 Regional Olympiad of Mexico West, 6

Tags: geometry , inequalities , geometric inequality

We say that a triangle of sides $a,b,c$ is [i] virtual[/i] if such measures satisfy $$\begin{cases} a^{2024}+b^{2024}> c^{2024},\\ b^{2024}+c^{2024}> a^{2024},\\ c^{2024}+a^{2024}> b^{2024} \end{cases}$$ Find the number of ordered triples $(a,b,c)$ such that $a,b,c$ are integers between $1$ and $2024$ (inclusive) and $a,b,c$ are the sides of a [i]virtual [/i] triangle.

Kyiv City MO Seniors 2003+ geometry, 2019.11.2

Tags: geometry , midpoint , geometric inequality

In an acute-angled triangle $ABC$, in which $AB<AC$, the point $M$ is the midpoint of the side $BC, K$ is the midpoint of the broken line segment $BAC$ . Prove that $\sqrt2 KM > AB$. (George Naumenko)

2000 Romania National Olympiad, 2b

Tags: geometry , inequalities , geometric inequality

If $a, b, c$ represent the lengths of the sides of a triangle, prove that: $$\frac{a}{b-a+c}+ \frac{b}{b-a+c}+ \frac{c}{b-a+c} \ge 3$$

1984 IMO, 2

Tags: algebra , geometry , convex polygon , perimeter , geometric inequality

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

1996 All-Russian Olympiad Regional Round, 10.2

Tags: geometry , combinatorics , combinatorial geometry , geometric inequality

Is it true that from an arbitrary triangle you can cut three equal figures, the area of each of which is more than a quarter of the area triangle?

2005 Cuba MO, 8

Tags: geometry , integer , geometric inequality , area

Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.

2021 International Zhautykov Olympiad, 4

Tags: geometric inequality , geometry , inequalities

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

2022 Yasinsky Geometry Olympiad, 5

Tags: geometry , geometric inequality , right triangle

Let $ABC$ be a right triangle with leg $CB = 2$ and hypotenuse $AB= 4$. Point $K$ is chosen on the hypotenuse $AB$, and point $L$ is chosen on the leg $AC$. a) Describe and justify how to construct such points $K$ and $ L$ so that the sum of the distances $CK+KL$ is the smallest possible. b) Find the smallest possible value of $CK+KL$. (Olexii Panasenko)

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: $$ BC\cdot PB\cdot PC+AC\cdot PC\cdot PA +AB\cdot PA\cdot PB\ge AB\cdot BC\cdot CA $$

OMMC POTM, 2023 12

Tags: geometry , triangle inequality , geometric inequality

All four angles of quadrilateral are greater than $60^o$. Prove that we can choose three sides to make a triangle.

2021 Kosovo National Mathematical Olympiad, 4

Tags: geometry , geometric inequality , inequalities

Let $M$ be the midpoint of segment $BC$ of $\triangle ABC$. Let $D$ be a point such that $AD=AB$, $AD\perp AB$ and points $C$ and $D$ are on different sides of $AB$. Prove that: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$

1979 IMO Shortlist, 13

Tags: trigonometry , inequality , geometric inequality , trigonometric identities , inequalities

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

2018 Kazakhstan National Olympiad, 6

Tags: inequalities , geometric inequality , geometry

Inside of convex quadrilateral $ABCD$ found a point $M$ such that $\angle AMB=\angle ADM+\angle BCM$ and $\angle AMD=\angle ABM+\angle DCM$.Prove that $$AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.$$

2014 Contests, 3

Tags: inequalities , geometric inequality , geometry

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.

1967 IMO, 2

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}.$

1998 Singapore Team Selection Test, 1

Tags: geometry , geometric inequality , hexagon

The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that: \[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}. \]

2000 Romania Team Selection Test, 2

Tags: geometry , circumcircle , perimeter , geometric inequality , triangle

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

2010 Sharygin Geometry Olympiad, 3

Tags: geometry , convex polygon , inequality , geometric inequality , diagonal

All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?

1967 IMO Shortlist, 5

Tags: algebra , vector , inequality , geometric inequality , 3d geometry

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

2013 Turkmenistan National Math Olympiad, 4

Tags: geometry , geometric inequality , convex quadrilateral

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

2003 Romania National Olympiad, 1

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

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]