Olympiad problems

2001 Mongolian Mathematical Olympiad, Problem 2

In an acute-angled triangle $ABC$, $a,b,c$ are sides, $m_a,m_b,m_c$ the corresponding medians, $R$ the circumradius and $r$ the inradius. Prove the inequality $$\frac{a^2+b^2}{a+b}\cdot\frac{b^2+c^2}{b+c}\cdot\frac{a^2+c^2}{a+c}\ge16R^2r\frac{m_a}a\cdot\frac{m_b}b\cdot\frac{m_c}c.$$

2000 Junior Balkan Team Selection Tests - Moldova, 3

Tags: angle bisector , geometric inequality , isosceles , geometry

Let $ABC$ be a triangle with $AB = AC$ ¸ $\angle BAC = 100^o$ and $AD, BE$ angle bisectors. Prove that $2AD <BE + EA$

1969 Spain Mathematical Olympiad, 7

Tags: geometry , regular polygon , geometric inequality

A convex polygon $A_1A_2 . . .A_n$ of $n$ sides and inscribed in a circle, has its sides that satisfy the inequalities $$A_nA_1 > A_1A_2 > A_2A_3 >...> A_{n-1}A_n$$ Show that its interior angles satisfy the inequalities $$\angle A_1 < \angle A_2 < \angle A_3 < ... < \angle A_{n-1}, \angle A_{n-1} > \angle A_n> \angle A_1.$$

Indonesia MO Shortlist - geometry, g8

Tags: geometry , angle bisector , geometric inequality

Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that $$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$

Kyiv City MO Juniors 2003+ geometry, 2014.7.41

Tags: triangle inequality , geometric inequality , geometry

The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?

VI Soros Olympiad 1999 - 2000 (Russia), 10.4

Tags: geometry , geometric inequalities , exradius , geometric inequality , inequalities

Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.

Durer Math Competition CD 1st Round - geometry, 2008.D1

Tags: geometry , geometric inequality , algebra , inequalities

Prove the following inequality if we know that $a$ and $b$ are the legs of a right triangle , and $c$ is the length of the hypotenuse of this triangle: $$3a + 4b \le 5c.$$ When does equality holds?

2015 Indonesia MO Shortlist, G1

Tags: rectangle , angle , geometric inequality , geometry

Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

1986 IMO Shortlist, 21

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

Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that \[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\] where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.

1992 IMO Longlists, 73

Tags: geometry , point set , geometric inequality

Let $\{A_n | n = 1, 2, \cdots \} $ be a set of points in the plane such that for each $n$, the disk with center $A_n$ and radius $2^n$ contains no other point $A_j$ . For any given positive real numbers $a < b$ and $R$, show that there is a subset $G$ of the plane satisfying: [b](i)[/b] the area of $G$ is greater than or equal to $R$; [b](ii) [/b]for each point $P$ in $G$, $a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.$

Croatia MO (HMO) - geometry, 2017.7

Tags: geometry , geometric inequality , circumcircle

The point $M$ is located inside the triangle $ABC$. The ray $AM$ intersects the circumcircle of the triangle $MBC$ once more at point $D$, the ray $BM$ intersects the circumcircle of the triangle $MCA$ once more at point $E$, and the ray $CM$ intersects the circumcircle of the triangle $MAB$ once more at point $F$. Prove that holds $$\frac{AD}{MD}+\frac{BE}{ME} +\frac{CF}{MF}\ge \frac92 $$

1999 Spain Mathematical Olympiad, 5

Tags: inequalities , centroid , geometric inequality , distance , geometry

The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that: a) $g_a,g_b,g_c \ge \frac{2}{3}r$ b) $g_a+g_b+g_c \ge 3r$

2021 Taiwan TST Round 1, G

Tags: geometry , inequalities , disks , geometric inequality

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.)

2009 Switzerland - Final Round, 1

Tags: geometric inequality , geometry , locus , hexagon

Let $P$ be a regular hexagon. For a point $A$, let $d_1\le d_2\le ...\le d_6$ the distances from $A$ to the six vertices of $P$, ordered by magnitude. Find the locus of all points $A$ in the interior or on the boundary of $P$ such that: (a) $d_3$ takes the smallest possible value. (b) $d_4$ takes the smallest possible value.

1981 IMO Shortlist, 15

Tags: geometry , trigonometry , minimization , geometric inequality , incenter

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]

2010 Balkan MO Shortlist, G2

Tags: midpoint , cyclic , geometric inequality , circles , area , cyclic quadrilateral , geometry

Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle

1967 IMO, 2

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

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

1950 Polish MO Finals, 4

Tags: geometry , geometric inequality , geometric inequalities , square , hexagon

Someone wants to unscrew a square nut with side $a$, with a wrench whose hole has the form of a regular hexagon with side $b$. What condition should the lengths $a$ and $b$ meet to make this possible?

2009 Junior Balkan Team Selection Tests - Romania, 1

Tags: right triangle , geometric inequality , inequalities , geometry

Show that in any triangle $ABC$ with $A = 90^0$ the following inequality holds: $$(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 \le 2BC^6$$

2001 IMO Shortlist, 2

Tags: geometric inequality , circumcenter , triangle , geometry

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

1991 IMO, 1

Tags: incenter , geometry , triangle inequality , geometric inequality , angle bisector

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

2021 Kosovo National Mathematical Olympiad, 4

Tags: geometric inequality , inequalities , geometry

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: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$

1989 IMO, 3

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

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: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

2005 Portugal MO, 2

Tags: right triangle , geometry , geometric inequality

Consider the triangles $[ABC]$ and $[EDC]$, right at $A$ and $D$, respectively. Show that, if $E$ is the midpoint of $[AC]$, then $AB <BD$. [img]https://cdn.artofproblemsolving.com/attachments/c/3/75bc1bda1a22bcf00d4fe7680c80a81a9aaa4c.png[/img]

Estonia Open Junior - geometry, 2018.2.5

Tags: centroid , geometric inequality , area , geometry

Medians $AD, BE$, and $CF$ of triangle $ABC$ intersect at point $M$. Is it possible that the circles with radii $MD, ME$, and $MF$ a) all have areas smaller than the area of triangle $ABC$, b) all have areas greater than the area of triangle $ABC$, c) all have areas equal to the area of triangle $ABC$?