Olympiad problems

1990 ITAMO, 4

Let $a,b,c$ be side lengths of a triangle with $a+b+c = 1$. Prove that $a^2 +b^2 +c^2 +4abc \le \frac12$ .

2019 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , geometric inequality , convex polygon

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

2010 Chile National Olympiad, 5

Tags: geometry , geometric inequality , distance

Consider a line $ \ell $ in the plane and let $ B_1, B_2, B_3 $ be different points in $ \ell$. Let $ A $ be a point that is not in $ \ell$. Show that there is $ P, Q $ in $ {B_1, B_2, B_3} $ with $ P \ne Q $ so that the distance from $ A $ to $ \ell$ is greater than the distance from $ P $ to the line that passes through $ A $ and $ Q $.

2005 Cuba MO, 8

Tags: geometric inequality , geometry , area , integer

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

2020 Swedish Mathematical Competition, 2

Tags: geometry , median , geometric inequality

The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$.

Ukrainian TYM Qualifying - geometry, IV.8

Tags: geometry , hexagon , convex , diagonal , area , geometric inequality

Prove that in an arbitrary convex hexagon there is a diagonal that cuts off from it a triangle whose area does not exceed $\frac16$ of the area of the hexagon. What are the properties of a convex hexagon, each diagonal of which is cut off from it is a triangle whose area is not less than $\frac16$ the area of the hexagon?

2019 Polish Junior MO Finals, 4.

Tags: geometrical inequalities , geometric inequality , inequalities , geometry

The point $D$ lies on the side $AB$ of the triangle $ABC$. Assume that there exists such a point $E$ on the side $CD$, that $$ \sphericalangle EAD = \sphericalangle AED \quad \text{and} \quad \sphericalangle ECB = \sphericalangle CEB. $$ Show that $AC + BC > AB + CE$.

Indonesia Regional MO OSP SMA - geometry, 2013.5

Tags: geometry , angle , geometric inequality

Given an acute triangle $ABC$. The longest line of altitude is the one from vertex $A$ perpendicular to $BC$, and it's length is equal to the length of the median of vertex $B$. Prove that $\angle ABC \le 60^o$

2014 Hanoi Open Mathematics Competitions, 6

Tags: angle bisector , geometric inequality , inequalities , geometry

Let $a,b,c$ be the length sides of a given triangle and $x,y,z$ be the sides length of bisectrices, respectively. Prove the following inequality $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

1967 IMO Shortlist, 4

Tags: analytic geometry , geometric inequality , circles , geometry

Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that \[PA^2 + PB^2 \geq 2r^2.\]

Estonia Open Senior - geometry, 2015.1.3

Tags: ratio , geometry , inradius , geometric inequality

Let $ABC$ be a triangle. Let $K, L$ and $M$ be points on the sides $BC, AC$ and $AB$, respectively, such that $\frac{|AM|}{|MB|}\cdot \frac{|BK|}{|KC|}\cdot \frac{|CL|}{|LA|} = 1$. Prove that it is possible to choose two triangles out of $ALM, BMK, CKL$ whose inradii sum up to at least the inradius of triangle $ABC$.

1956 Moscow Mathematical Olympiad, 330

Tags: inradius , geometric inequality , inscribed , square , geometry

A square of side $a$ is inscribed in a triangle so that two of the square’s vertices lie on the base, and the other two lie on the sides of the triangle. Prove that if $r$ is the radius of the circle inscribed in the triangle, then $r\sqrt2 < a < 2r$.

2000 Saint Petersburg Mathematical Olympiad, 11.2

Tags: analytic geometry , geometric inequality , vector , vector geometry , 3d geometry , geometry , inequalities

Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality: $$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$ [I]Proposed by A. Khrabrov[/i]

1988 IMO Longlists, 4

Tags: geometry , geometric inequality , circumcircle , inradius , trigonometry

The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$

2000 Kazakhstan National Olympiad, 8

Tags: geometric inequality , distance , geometry

Given a triangle $ ABC $ and a point $ M $ inside it. Prove that $$ \min \{MA, MB, MC\} + MA + MB + MC <AB + BC + AC. $$

1984 All Soviet Union Mathematical Olympiad, 388

Tags: geometric inequality , geometry , collinear

The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $$|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$$

1984 IMO Longlists, 33

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

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$.

1999 Tournament Of Towns, 2

Tags: geometric inequality , area , geometry

Let $ABC$ be an acute-angled triangle, $C'$ and $A'$ be arbitrary points on the sides $AB$ and $BC$ respectively, and $B'$ be the midpoint of the side $AC$. (a) Prove that the area of triangle $A'B'C'$ is at most half the area of triangle $ABC$. (b) Prove that the area of triangle $A'B'C'$ is equal to one fourth of the area of triangle $ABC$ if and only if at least one of the points $A'$, $C'$ is the midpoint of the corresponding side. (E Cherepanov)

2013 BMT Spring, 6

Tags: analytic geometry , geometric inequality , geometry , inequalities

Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Because Bubble Boy loves Bubble Girl, he wants to reach her as quickly as possible, but he needs to bring a gift; luckily, there are plenty of gifts along the $x$-axis. Assuming that Bubble Girl remains stationary, find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then reach Bubble Girl (the bubble is a solid boundary, and anything the bubble can touch, Bubble Boy can touch too)

1972 Spain Mathematical Olympiad, 3

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

Given a regular hexagonal prism. Find a polygonal line that, starting from a vertex of the base, runs through all the lateral faces and ends at the vertex of the face top, located on the same edge as the starting vertex, and has a minimum length.

2009 District Olympiad, 2

Tags: geometric inequality , geometry

Hiven an acute triangle $ABC$, consider the midpoints $M$ and $N$ of the sides $AB$ and $AC$, respectively. If point $S$ is variable on side $BC$, prove that $$(MB - MS)(NC - NS) \le 0$$

Ukrainian TYM Qualifying - geometry, VI.18

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

The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices: 1) $T_3$ and $T_3 (A_2)$ 2) $T_k$ and $T_k (A_1) $ 3) $T_k$ and $T_{k+1}$