Olympiad problems

1960 Polish MO Finals, 6

On the perimeter of a rectangle, point $ M $ is chosen. Find the shortest path whose beginning and end are point $ M $ and which has a point in common with each side of the rectangle.

1968 Swedish Mathematical Competition, 3

Tags: geometry , geometric inequalities

Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?

2000 Switzerland Team Selection Test, 15

Tags: point , geometry , combinatorial geometry , geometric inequalities , combinatorics

Let $S = \{P_1,P_2,...,P_{2000}\}$ be a set of $2000$ points in the interior of a circle of radius $1$, one of which at its center. For $i = 1,2,...,2000$ denote by $x_i$ the distance from $P_i$ to the closest point $P_j \ne P_i$. Prove that $x_1^2 +x_2^2 +...+x_{2000}^2<9$ .

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: area , geometric inequalities , combinatorial geometry , geometry , combinatorics

Five points are given in the plane that each of $10$ triangles they define has area greater than $2$. Prove that there exists a triangle of area greater than $3$.

1972 Polish MO Finals, 4

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

Points $A$ and $B$ are given on a line having no common points with a sphere $K$. The feet $P$ of the perpendicular from the center of $K$ to the line $AB$ is positioned between $A$ and $B$, and the lengths of segments $AP$ and $BP$ both exceed the radius of $K$. Consider the set $Z$ of all triangles $ABC$ whose sides $AC$ and $BC$ are tangent to $K$. Prove that among all triangles in $Z$, a triangle $T$ with a maximum perimeter also has a maximum area.

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 $\frac{ab+bc+ca}{4P} \ge \sqrt3$

1955 Kurschak Competition, 1

Tags: geometry , trapezoid , geometric inequalities

Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting from the smaller angle is longer than the other diagonal. [img]https://cdn.artofproblemsolving.com/attachments/7/1/77cf4958931df1c852c347158ff1e2bbcf45fd.png[/img]

2017 Macedonia JBMO TST, 4

Tags: circumcircle , geometry , geometric inequalities

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 $$BM + CM > AY.$$

1994 Swedish Mathematical Competition, 2

Tags: median , geometric inequalities , geometry

In the triangle $ABC$, the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge \frac23$.

1965 Kurschak Competition, 3

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

A pyramid has square base and equal sides. It is cut into two parts by a plane parallel to the base. The lower part (which has square top and square base) is such that the circumcircle of the base is smaller than the circumcircles of the lateral faces. Show that the shortest path on the surface joining the two endpoints of a spatial diagonal lies entirely on the lateral faces. [img]https://cdn.artofproblemsolving.com/attachments/c/8/170bec826d5e40308cfd7360725d2aba250bf6.png[/img]

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 \[AX+AY+BC>AB+AC\]

1992 Czech And Slovak Olympiad IIIA, 2

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

Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

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 $$OA_1+OB_1+OC_1\le AA_1.$$

2019 LIMIT Category A, Problem 3

Tags: inequalities , geometric inequalities , geometry

In $\triangle ABC$, $\left|\overline{AB}\right|=\left|\overline{AC}\right|$, $D$ is the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from $B$ to $AC$, then $\textbf{(A)}~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(B)}~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(C)}~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(D)}~\text{None of the above}$

2000 Swedish Mathematical Competition, 4

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

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

2008 Postal Coaching, 2

Tags: geometry , perpendicular , distance , geometric inequalities

Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.

1962 Poland - Second Round, 4

Tags: inequalities , geometry , geometric inequalities , angle bisector

Prove that if the sides $ a $, $ b $, $ c $ of a triangle satisfy the inequality $$a < b < c$$then the angle bisectors $ d_a $, $ d_b $, $ d_c $ of opposite angles satisfy the inequality $$ d_a > d_b > d_c.$$

1983 Poland - Second Round, 5

Tags: geometry , geometric inequalities

The bisectors of the angles $ CAB, ABC, BCA $ of the triangle $ ABC $ intersect the circle circumcribed around this triangle at points $ K, L, M $, respectively. Prove that $$ AK+BL+CM > AB+BC+CA.$$

2007 Thailand Mathematical Olympiad, 6

Tags: geometry , geometric inequalities

A triangle has perimeter $2s$, inradius $r$, and incenter $I$. If $s_a, s_b$ and $s_c$ are the distances from $I$ to the three vertices, then show that $$\frac34 +\frac{r}{s_a}+\frac{r}{s_b}+\frac{r}{s_c} \le \frac{s^2}{12r^2}$$

1974 Czech and Slovak Olympiad III A, 5

Tags: geometry , geometric inequalities , geometrical inequalities , inequalities , cyclic , hexagon , area

Let $ABCDEF$ be a cyclic hexagon such that \[AB=BC,\quad CD=DE,\quad EF=FA.\] Show that \[[ACE]\le[BDF]\] and determine when the equality holds. ($[XYZ]$ denotes the area of the triangle $XYZ.$)

1924 Eotvos Mathematical Competition, 1

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

1971 Bulgaria National Olympiad, Problem 4

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

It is given a triangle $ABC$. Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$. Prove that $q\le6R\sqrt3$. When does equality hold?

1916 Eotvos Mathematical Competition, 2

Tags: geometry , geometric inequalities

Let the bisector of the angle at $C$ of triangle $ABC$ intersect side $AB$ in point $D$. Show that the segment $CD$ is shorter than the geometric mean of the sides $CA$ and $CB$. (The geometric mean of two positive numbers is the square root of their product; the geometric mean of $n$ numbers is the $n$-th root of their product.

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 $N$).

1972 Bulgaria National Olympiad, Problem 5

Tags: geometry , cyclic quadrilateral , perpendicular , inequalities , geometric inequalities

In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral. (a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed. (b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold? [i]H. Lesov[/i]