Olympiad problems

2017 Macedonia JBMO TST, 4

In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$

1987 Swedish Mathematical Competition, 2

Tags: geometric inequalities , equal area , arc , geometry

A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.

1969 Czech and Slovak Olympiad III A, 2

Tags: area , geometric inequalities , quadrilateral , geometrical inequalities , convex , geometry

Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.

2018 Flanders Math Olympiad, 1

Tags: geometric inequalities , angle , geometry

In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .

2000 Swedish Mathematical Competition, 4

Tags: lattice , area , geometric inequalities , 3d geometry , geometry , combinatorial geometry

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

1924 Eotvos Mathematical Competition, 1

Tags: number theory , geometric inequalities , triangle inequality , geometry , inequalities

Let $a, b, c$ be fìxed natural numbers. Suppose that, for every positive integer n, there is a triangle whose sides have lengths $a^n$, $b^n$, and $c^n$ respectively. Prove that these triangles are isosceles.

1990 Swedish Mathematical Competition, 2

Tags: sum , geometric inequalities , collinear , geometry

The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

1989 Greece National Olympiad, 4

Tags: tangential , geometric inequalities , geometric inequality , trapezoid , geometry

A trapezoid with bases $a,b$ and altitude $h$ is circumscribed around a circl.. Prove that $h^2\le ab$.

1914 Eotvos Mathematical Competition, 1

Tags: geometric inequalities , geometry , arc

Let $A$ and $B$ be points on a circle $k$. Suppose that an arc $k'$ of another circle, $\ell$, connects $A$ with $B$ and divides the area inside the circle $k$ into two equal parts. Prove that arc $k'$ is longer than the diameter of $k$.

1927 Eotvos Mathematical Competition, 3

Tags: exradius , geometric inequalities , inradius , geometry

Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.

2011 Indonesia TST, 1

Tags: geometric inequalities , inequalities , algebra

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

1972 Spain Mathematical Olympiad, 2

Tags: inequalities , geometric inequalities , geometry , analytic geometry

A point moves on the sides of the triangle $ABC$, defined by the vertices $A(-1.8, 0)$, $B(3.2, 0)$, $C(0, 2.4)$ . Determine the positions of said point, in which the sum of their distance to the three vertices is absolute maximum or minimum. [img]https://cdn.artofproblemsolving.com/attachments/2/5/9e5bb48cbeefaa5f4c069532bf5605b9c1f5ea.png[/img]

1963 Polish MO Finals, 3

Tags: geometric inequalities , rectangle , geometry

From a given triangle, cut out the rectangle with the largest area.

1991 Swedish Mathematical Competition, 6

Tags: area , geometric inequalities , equilateral , geometry

Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

2020 Jozsef Wildt International Math Competition, W57

Tags: triangle , geometric inequalities , inequalities , geometry

In all triangles $ABC$ does it hold that: $$\sum\sin^2\frac A2\cos^2A\ge\frac{3\left(s^2-(2R+r)^2\right)}{8R^2}$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

2018 Czech-Polish-Slovak Match, 4

Tags: geometric inequalities , geometry , inequalities

Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i]

2009 Postal Coaching, 3

Tags: polygon , cyclic , geometric inequalities , inequalities

Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.

1930 Eotvos Mathematical Competition, 3

Tags: circumradius , geometric inequalities , geometry

Inside an acute triangle $ABC$ is a point $P$ that is not the circumcenter. Prove that among the segments $AP$, $BP$ and $CP$, at least one is longer and at least one is shorter than the circumradius of $ABC$.

2021 Israel National Olympiad, P3

Tags: geometric inequalities , inequalities , geometry

Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle. Prove that \[AX+AY+BC>AB+AC\]

2019 Durer Math Competition Finals, 3

Tags: perimeter , geometric inequalities , geometry

Let $P$ be an interior point of triangle $ABC$. The lines $AP$, $BP$ and $CP$ divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of $ABC$?

2020 Jozsef Wildt International Math Competition, W35

Tags: geometric inequalities , inequalities , geometry

In all triangles $ABC$ does it hold: $$(b^n+c^p)\tan^{n+p}\frac A2+(c^n+a^p)\tan^{n+p}\frac B2+(a^n+b^p)\tan^{n+p}\frac C2\ge6\sqrt{\left(\frac{4r^2}{R\sqrt3}\right)^{n+p}}$$ where $n,p\in(0,\infty)$. [i]Proposed by Nicolae Papacu[/i]

2000 Belarus Team Selection Test, 1.4

Tags: trigonometry , geometric inequalities , 3d geometry , sphere , geometry , inequalities

A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$. Prove that $\ell \le \sin \frac{2\pi}{5}$ .

2009 Estonia Team Selection Test, 4

Tags: geometric inequalities , geometry , ratio , inequalities

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

1965 German National Olympiad, 6

Tags: inequalities , geometric inequalities , geometry , trigonometry

Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.

2007 Estonia Team Selection Test, 2

Tags: altitude , geometric inequalities , circumradius , incircle , circumcircle , inradius , geometry

Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$