Olympiad problems

1986 French Mathematical Olympiad, Problem 2

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 $M'A'\le MA,M'B'\le MB,M'C'\le MC$?

1937 Eotvos Mathematical Competition, 3

Tags: geometry , Geometric Inequalities

Let $n$ be a positive integer. Let $P,Q,A_1,A_2,...,A_n$ be distinct points such that $A_1,A_2,...,A_n$ are not collinear. Suppose that $PA_1 + PA_2 + ...+PA_n$, and $QA_1 + QA_2 +...+ QA_n$, have a common value $s$ for some real number $s$. Prove that there exists a point $R$ such that $$RA_1 + RA_2 +... + RA_n < s.$$

2020 Jozsef Wildt International Math Competition, W35

Tags: inequalities , geometry , Geometric Inequalities

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]

2019 Durer Math Competition Finals, 3

Tags: geometry , perimeter , Geometric Inequalities

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 $ABC$?

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]

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]

1975 Kurschak Competition, 2

Tags: geometry , rhombus , inscribed , Geometric Inequalities

Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , area , Geometric Inequalities , combinatorial 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 $3$.

1970 Poland - Second Round, 4

Tags: geometry , perimeter , Geometric Inequalities

Prove that if triangle $T_1$ contains triangle $T_2$, then the perimeter of triangle $T_1$ is not less than the perimeter of triangle $T_2$.

2010 Estonia Team Selection Test, 3

Tags: inequalities , Geometric Inequalities , trigonometry , circumradius , geometry

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

2010 Junior Balkan Team Selection Tests - Moldova, 7

Tags: Geometric Inequalities , isosceles

In the triangle $ABC$ with $| AB | = c, | BC | = a, | CA | = b$ the relations hold simultaneously $$a \ge max \{ b, c, \sqrt{bc}\}, \sqrt{(a - b) (a + c)} + \sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc}$$ Prove that the triangle $ABC$ is isosceles.

1989 French Mathematical Olympiad, Problem 3

Tags: geometry , 3D geometry , tetrahedron , inequalities , geometric inequality , Geometric Inequalities

Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$, the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$.

1999 Romania National Olympiad, 2

Tags: geometry , 3D geometry , tetrahedron , geometric inequality , Geometric Inequalities

On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that $$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$

1958 Polish MO Finals, 6

Tags: geometry , perimeter , Geometric Inequalities , tangential

Prove that of all the quadrilaterals circuscribed around a given circle, the square has the smallest perimeter.

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]

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 $$ AK+BL+CM > AB+BC+CA.$$

1929 Eotvos Mathematical Competition, 3

Tags: geometry , concurrent , Geometric Inequalities

Let $p, q$ and $r$ be three concurrent lines in the plane such that the angle between any two of them is $60^o$. Let $a$, $b$ and $c$ be real numbers such that $0 < a \le b \le c$. (a) Prove that the set of points whose distances from $p, q$ and $r$ are respectively less than $a, b$ and $c$ consists of the interior of a hexagon if and only if $a + b > c$. (b) Determine the length of the perimeter of this hexagon when $a + b > c$.

1997 Brazil Team Selection Test, Problem 5

Tags: geometry , geometric inequality , inequalities , Geometric Inequalities , Triangles , Triangle , area

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

2017 Hanoi Open Mathematics Competitions, 13

Tags: inequalities , distance , Geometric Inequalities , geometry

Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?

1972 Kurschak Competition, 1

Tags: geometry , Geometric Inequalities , inequalities

A triangle has side lengths $a, b, c$. Prove that $$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$

2003 Turkey MO (2nd round), 2

Tags: geometry , inequalities , parallelogram , inequalities solved , Geometric 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 $s=\text{Area}(ABCD)$.

1965 German National Olympiad, 6

Tags: trigonometry , inequalities , Geometric Inequalities , geometry

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.

2004 Swedish Mathematical Competition, 6

Tags: geometry , Geometric Inequalities , convex , areas

Prove that every convex $n$-gon of area $1$ contains a quadrilateral of area at least $\frac12 $. .

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 $$BM + CM > AY.$$

1997 Singapore MO Open, 1

Tags: Equilateral , geometry , Geometric Inequalities

$\vartriangle ABC$ is an equilateral triangle. $L, M$ and $N$ are points on $BC, CA$ and $AB$ respectively. Prove that $MA \cdot AN + NB \cdot BL + LC \cdot CM < BC^2$.