Olympiad problems

2016 Sharygin Geometry Olympiad, 7

From the altitudes of an acute-angled triangle, a triangle can be composed. Prove that a triangle can be composed from the bisectors of this triangle.

1972 Spain Mathematical Olympiad, 2

Tags: geometry , geometric inequalities , analytic geometry , inequalities

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]

2009 Postal Coaching, 4

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

Let $ABC$ be a triangle, and let $DEF$ be another triangle inscribed in the incircle of $ABC$. If $s$ and $s_1$ denote the semiperimeters of $ABC$ and $DEF$ respectively, prove that $2s_1 \le s$. When does equality hold?

1985 Polish MO Finals, 4

Tags: geometry , inradius , inequalities , geometric inequalities

$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$

2009 Iran MO (2nd Round), 3

Tags: inequalities , circumcircle , trigonometry , angle bisector , geometric inequalities , geometry

Let $ ABC $ be a triangle and the point $ D $ is on the segment $ BC $ such that $ AD $ is the interior bisector of $ \angle A $. We stretch $ AD $ such that it meets the circumcircle of $ \Delta ABC $ at $ M $. We draw a line from $ D $ such that it meets the lines $ MB,MC $ at $ P,Q $, respectively ($ M $ is not between $ B,P $ and also is not between $ C,Q $). Prove that $ \angle PAQ\geq\angle BAC $.

1998 North Macedonia National Olympiad, 4

Tags: inequalities , geometric inequalities , sidelenghts , triangle area , geometry

If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$

1964 Swedish Mathematical Competition, 1

Tags: geometry , min , geometric inequalities

Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.

1998 Yugoslav Team Selection Test, Problem 2

Tags: inequalities , geometric inequality , geometric inequalities , geometry

In a convex quadrilateral $ABCD$, the diagonal $AC$ intersects the diagonal $BD$ at its midpoint $S$. The radii of incircles of triangles $ABS,BCS,CDS,DAS$ are $r_1,r_2,r_3,r_4$, respectively. Prove that $$|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.$$

1999 Singapore MO Open, 4

Tags: cyclic , geometric inequalities , geometry

Let $ABCD$ be a quadrilateral with each interior angle less than $180^o$. Show that if $A, B, C, D$ do not lie on a circle, then $AB \cdot CD + AD\cdot BC > AC \cdot BD$

1988 Dutch Mathematical Olympiad, 4

Tags: geometry , geometric inequality , geometric inequalities , square

Given is an isosceles triangle $ABC$ with $AB = 2$ and $AC = BC = 3$. We consider squares where $A, B$ and $C$ lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.

1986 French Mathematical Olympiad, Problem 2

Tags: inequalities , geometric inequalities , geometry

Points $A,B,C$, and $M$ are given in the plane. (a) Let $D$ be the point in the plane such that $DA\le CA$ and $DB\le CB$. Prove that there exists point $N$ satisfying $NA\le MA,NB\le MB$, and $ND\le MC$. (b) Let $A',B',C'$ be the points in the plane such that $A'B'\le AB,A'C'\le AC,B'C'\le BC$. Does there exist a point $M'$ which satisfies the inequalities $M'A'\le MA,M'B'\le MB,M'C'\le MC$?

1925 Eotvos Mathematical Competition, 3

Tags: geometry , inradius , geometric inequalities , right triangle

Let $r$ be the radius of the inscribed circle of a right triangle $ABC$. Show that $r$ is less than half of either leg and less than one fourth of the hypotenuse.

2003 Turkey MO (2nd round), 2

Tags: geometry , parallelogram , geometric inequalities , inequalities

Let $ABCD$ be a convex quadrilateral and $K,L,M,N$ be points on $[AB],[BC],[CD],[DA]$, respectively. Show that, \[ \sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s} \] where $s_1=\text{Area}(AKN)$, $s_2=\text{Area}(BKL)$, $s_3=\text{Area}(CLM)$, $s_4=\text{Area}(DMN)$ and $s=\text{Area}(ABCD)$.

2004 Germany Team Selection Test, 2

Tags: geometry , locus problems , geometric inequalities

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

1971 Bulgaria National Olympiad, Problem 6

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

In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that (a) $m^2+n^2+p^2\ge18r^2$; (b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$.

2001 Abels Math Contest (Norwegian MO), 3b

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

The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$

1969 Czech and Slovak Olympiad III A, 2

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

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.

2000 Belarus Team Selection Test, 5.1

Tags: geometric inequalities , inequalities , angle bisector , median

Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$). If $BC = a, AM = m_a, AL = l_a$, prove the inequalities: (a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$ (b) $2l_a \le a\cot \frac{a}{2} $.

2020 Jozsef Wildt International Math Competition, W57

Tags: inequalities , geometric inequalities , geometry , triangle

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]

1970 Bulgaria National Olympiad, Problem 5

Tags: inequalities , geometric inequalities , geometry , polygon

Prove that for $n\ge5$ the side of regular inscribable $n$-gon is bigger than the side of regular $n+1$-gon circumscribed around the same circle and if $n\le4$ the opposite statement is true.

1994 French Mathematical Olympiad, Problem 2

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

Let be given a semi-sphere $\Sigma$ whose base-circle lies on plane $p$. A variable plane $Q$, parallel to a fixed plane non-perpendicular to $P$, cuts $\Sigma$ at a circle $C$. We denote by $C'$ the orthogonal projection of $C$ onto $P$. Find the position of $Q$ for which the cylinder with bases $C$ and $C'$ has the maximum volume.

2022 Durer Math Competition Finals, 9

Tags: geometry , perimeter , geometric inequalities

Every side of a right triangle is an integer when measured in cm, and the difference between the hypotenuse and one of the legs is $75$ cm. What is the smallest possible value of its perimeter?

2020 Jozsef Wildt International Math Competition, W48

Tags: inequalities , geometry , geometric inequalities

Let $ABC$ be a triangle such that $$S^2=2R^2+8Rr+3r^2$$ Then prove that $\frac Rr=2$ or $\frac Rr\ge\sqrt2+1$. [i]Proposed by Marian Cucoanoeş and Marius Drăgan[/i]

Cono Sur Shortlist - geometry, 2018.G5

Tags: ratio , inscribed , square , circumscribed , geometric inequalities , geometry

We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .

2002 Federal Math Competition of S&M, Problem 2

Tags: geometric inequalities , geometric inequality , inequalities , geometry

Let $O$ be a point inside a triangle $ABC$ and let the lines $AO,BO$, and $CO$ meet sides $BC,CA$, and $AB$ at points $A_1,B_1$, and $C_1$, respectively. If $AA_1$ is the longest among the segments $AA_1,BB_1,CC_1$, prove that $$OA_1+OB_1+OC_1\le AA_1.$$