Olympiad problems

2006 Oral Moscow Geometry Olympiad, 3

On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $C', A'$ and $B'$ are selected, respectively, so that the angle $A'C'B'$ is right. Prove that the segment $A'B'$ is longer than the diameter of the inscribed circle of the triangle $ABC$. (M. Volchkevich)

2018 JBMO Shortlist, G6

Tags: inequalities , geometric inequality , chord , geometry

Let $XY$ be a chord of a circle $\Omega$, with center $O$, which is not a diameter. Let $P, Q$ be two distinct points inside the segment $XY$, where $Q$ lies between $P$ and $X$. Let $\ell$ the perpendicular line drawn from $P$ to the diameter which passes through $Q$. Let $M$ be the intersection point of $\ell$ and $\Omega$, which is closer to $P$. Prove that $$ MP \cdot XY \ge 2 \cdot QX \cdot PY$$

Estonia Open Senior - geometry, 2004.1.5

Tags: inequalities , geometric inequality , area of triangle , geometry

Find the smallest real number $x$ for which there exist two non-congruent triangles with integral side lengths having area $x$.

2012 India Regional Mathematical Olympiad, 4

Tags: geometry , circumcircle , trigonometry , geometric inequality , area of a triangle , heron's formula , inequalities

Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have \[(a+b)(b+c)(c+a)\geq 8.\] Also determine the case of equality.

2021 239 Open Mathematical Olympiad, 5

Tags: geometry , triangle inequality , geometric inequality

The median $AD$ is drawn in triangle $ABC$. Point $E$ is selected on segment $AC$, and on the ray $DE$ there is a point $F$, and $\angle ABC = \angle AED$ and $AF // BC$. Prove that from segments $BD, DF$ and $AF$, you can make a triangle, the area of which is not less half the area of triangle $ABC$.

2022 Sharygin Geometry Olympiad, 6

Tags: geometric inequality , inequalities

The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively. Prove that $$PP' > QQ'$$

2004 Croatia National Olympiad, Problem 2

Tags: inequalities , geometric inequality

If $a,b,c$ are the sides and $\alpha,\beta,\gamma$ the corresponding angles of a triangle, prove the inequality $$\frac{\cos\alpha}{a^3}+\frac{\cos\beta}{b^3}+\frac{\cos\gamma}{c^3}\ge\frac3{2abc}.$$

2014 BMT Spring, 12

Tags: geometry , area of a triangle , geometric inequality , area

Suppose four coplanar points $A, B, C$, and $D$ satisfy $AB = 3$, $BC = 4$, $CA = 5$, and $BD = 6$. Determine the maximal possible area of $\vartriangle ACD$.

1992 Denmark MO - Mohr Contest, 4

Tags: median , geometry , geometric inequality

Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction $$\frac{m_a + m_b + m_c}{a + b + c}$$

1990 All Soviet Union Mathematical Olympiad, 532

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

If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.

2016 Danube Mathematical Olympiad, 1

Tags: geometric inequality , inequalities

1.Let $ABC$ be a triangle, $D$ the foot of the altitude from $A$ and $M$ the midpoint of the side $BC$. Let $S$ be a point on the closed segment $DM$ and let $P, Q$ the projections of $S$ on the lines $AB$ and $AC$ respectively. Prove that the length of the segment $PQ$ does not exceed one quarter the perimeter of the triangle $ABC$.

III Soros Olympiad 1996 - 97 (Russia), 9.5

Tags: geometry , geometric inequality

An ant sits at vertex $A$ of unit square $ABCD$. He needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that an ant must overcome in order to get into the anthill.

VII Soros Olympiad 2000 - 01, 10.8

Tags: geometry , trigonometry , geometric inequality , angle

There is a set of triangles, in each of which the smallest angle does not exceed $36^o$ . A new one is formed from these triangles according to the following rule: the smallest side of the new one is equal to the sum of the smallest sides of these triangles, its middle side is equal to the sum of the middle sides, and the largest is the sum of the largest ones. Prove that the sine of the smallest angle of the resulting triangle is less than $2 \sin 18^o$ .

1995 Tuymaada Olympiad, 8

Tags: geometry , geometric inequality , minimum , sum , distance

Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.

2001 IMO, 1

Tags: geometry , circumcenter , triangle , geometric inequality

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 $\angle A+\angle COP < 90^{\circ}$.

2000 Poland - Second Round, 2

Tags: geometry , circumcircle , geometry inequality , geometric inequality

Bisector of angle $BAC$ of triangle $ABC$ intersects circumcircle of this triangle in point $D \neq A$. Points $K$ and $L$ are orthogonal projections on line $AD$ of points $B$ and $C$, respectively. Prove that $AD \ge BK + CL$.

2002 Federal Math Competition of S&M, Problem 2

Tags: geometry , geometric inequality , area , inequalities

Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.

2017 Hanoi Open Mathematics Competitions, 11

Tags: geometry , inequalities , geometric inequality , equilateral , angle

Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle. Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?

1988 IMO Shortlist, 27

Tags: trigonometry , geometry , function , geometric inequality , area of a triangle

Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.

Champions Tournament Seniors - geometry, 2013.3

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

On the base of the $ABC$ of the triangular pyramid $SABC$ mark the point $M$ and through it were drawn lines parallel to the edges $SA, SB$ and $SC$, which intersect the side faces at the points $A1_, B_1$ and $C_1$, respectively. Prove that $\sqrt{MA_1}+ \sqrt{MB_1}+ \sqrt{MC_1}\le \sqrt{SA+SB+SC}$

2001 All-Russian Olympiad Regional Round, 8.3

Tags: pentagon , geometric inequality , angle , geometry

All sides of a convex pentagon are equal, and all angles are different. Prove that the maximum and minimum angles are adjacent to the same side of the pentagon.

1968 IMO Shortlist, 9

Tags: geometry , inequalities , geometric inequality , area of a triangle , centroid

Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality \[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\] Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

II Soros Olympiad 1995 - 96 (Russia), 10.10

Tags: geometry , geometric inequality , area

The Order "For Faithful Service" of the $7$th degree in shape is a combination of a semicircle with a diameter $AB = 2$ and a triangle $AM B$. The sides$ AM$ and $BM$ intersect the semicircle (the border of the semicircle). The part of the circle outside the triangle and the part of the triangle outside the circle are made of pure copper. What should the side of the triangle be equal to in order for the area of the copper part to be the smallest?

1994 Tournament Of Towns, (432) 2

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

Prove that one can construct two triangles from six edges of an arbitrary tetrahedron. (VV Proizvolov)

1984 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , combinatorial geometry , geometric inequality , geometry

In a square field of side length $12$ there is a source that contains a system of straight irrigation ditches. This is laid out in such a way that for every point of the field the distance to the next ditch is at most $1$. Here, the source is as a point and are the ditches to be regarded as stretches. It must be verified that the total length of the irrigation ditches is greater than $70$ m. The sketch shows an example of a trench system of the type indicated. [img]https://cdn.artofproblemsolving.com/attachments/6/5/5b51511da468cf14b5823c6acda3c4d2fe8280.png[/img]