This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 649

2020 Czech and Slovak Olympiad III A, 2
Tags: geometry , geometric inequality , area of a triangle
The triangle $ABC$ is given. Inside its sides $AB$ and $AC$, the points $X$ and $Y$ are respectively selected Let $Z$ be the intersection of the lines $BY$ and $CX$. Prove the inequality $$[BZX] + [CZY]> 2 [XY Z]$$, where $[DEF]$ denotes the content of the triangle $DEF$. (David Hruska, Josef Tkadlec)

2004 Korea - Final Round, 2
Tags: geometric inequality , geometry , circumcircle , inradius , trigonometry
An acute triangle $ABC$ has circumradius $R$, inradius $r$. $A$ is the biggest angle among $A,B,C$. Let $M$ be the midpoint of $BC$, and $X$ be the intersection of two lines that touches circumcircle of $ABC$ and goes through $B,C$ respectively. Prove the following inequality : $ \frac{r}{R} \geq \frac{AM}{AX}$.

2014 Contests, 2
Tags: geometry , geometric inequality , function
Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality: \begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}

2020 IMO Shortlist, G4
Tags: geometry , disks , geometric inequality , inequalities
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.)

1995 IMO Shortlist, 5
Tags: inequalities , geometry , hexagon , geometric inequality
Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.

Estonia Open Senior - geometry, 2009.1.5
Tags: geometric inequality , geometry , inradius
Let any point $D$ be chosen on the side $BC$ of the triangle $ABC$. Let the radii of the incircles of the triangles $ABC, ABD$ and $ACD$ be $r_1, r_2$ and $r_3$. Prove that $r_1 <r_2 + r_3$.

1978 IMO Shortlist, 4
Tags: geometric inequality , triangle inequality , area of a triangle , geometry
Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

1995 Tuymaada Olympiad, 8
Tags: geometric inequality , geometry , distance , sum , minimum
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.

2022 Spain Mathematical Olympiad, 4
Tags: geometric inequality , geometry
Let $P$ be a point in the plane. Prove that it is possible to draw three rays with origin in $P$ with the following property: for every circle with radius $r$ containing $P$ in its interior, if $P_1$, $P_2$ and $P_3$ are the intersection points of the three rays with the circle, then \[|PP_1|+|PP_2|+|PP_3|\leq 3r.\]

2015 Oral Moscow Geometry Olympiad, 5
Tags: geometric inequality , geometry , rhombus , equilateral
On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.

1979 IMO Longlists, 33
Tags: trigonometry , inequalities , inequality , geometric inequality , trigonometric identities
Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

KoMaL A Problems 2022/2023, A.837
Tags: geometry , inequalities , linear algebra , geometric inequality , 3d geometry , tetrahedron
Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that \[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \] [i]Submitted by Viktor Vígh, Szeged[/i]

2000 Poland - Second Round, 2
Tags: geometric inequality , circumcircle , geometry inequality , geometry
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 $AD \ge BK + CL$.

2003 Junior Macedonian Mathematical Olympiad, Problem 3
Tags: inequalities , geometry , geometric inequality
Let $ABC$ be a given triangle. The circumcircle of the triangle has radius $R$, the incircle has radius $r$, the longest side of the triangle is $a$, while the shortest altitude is $h$. Show that: $\frac{R}{r} > \frac{a}{h}$.

1931 Eotvos Mathematical Competition, 3
Tags: geometric inequality , inequalities , geometry
Let $A$ and $B$ be two given points, distance $1 $ apart. Determine a point $P$ on the line $AB$ such that $$\frac{1}{1 + AP}+\frac{1}{1 + BP}$$ is a maximum.

1967 IMO Shortlist, 5
Tags: geometric inequality , vector , algebra , inequality , 3d geometry
Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

1979 IMO Longlists, 20
Tags: geometric inequality , vector , trigonometry , euclidean geometry , geometry
Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

2011 Balkan MO Shortlist, G4
Tags: geometry , incircle , geometric inequality
Given a triangle $ABC$, the line parallel to the side $BC$ and tangent to the incircle of the triangle meets the sides $AB$ and $AC$ at the points $A_1$ and $A_2$ , the points $B_1, B_2$ and $C_1, C_2$ are de ned similarly. Show that $$AA_1 \cdot AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2 \ge \frac19 (AB^2 + BC^2 + CA^2)$$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.6
Tags: geometric inequality , geometry , exradius
The radius of the circle inscribed in triangle $ABC$ is equal to $r$, and the radius of the circle tangent to the segment $BC$ and the extensions of sides $AB$ and $AC$ (the exscribed circle corresponding to angle $A$) is equal to $R$. A circle with radius $x < r$ is inscribed in angle $\angle BAC$. Tangents to this circles passing through points $B$ and $C$ and different from $BA$ and $AC$ intersect at point $A'$. Let $y$ be the radius of the circle inscribed in triangle $BCK$. Find the greatest value of the sum $x + y$ as x changes from $0$ to $r$. (In this case, it is necessary to prove that this largest value is the same in any triangle with given $r$ and $R$).

VII Soros Olympiad 2000 - 01, 11.8
Tags: geometric inequality , sphere , 3d geometry , geometry
Three spheres are tangent to one plane, to a straight line perpendicular to this plane, and in pairs to each other. The radius of the largest sphere is $1$. Within what limits can the radius of the smallest sphere vary?

1966 IMO Shortlist, 33
Tags: geometric inequality , inequality , tangent circle , geometry
Given two internally tangent circles; in the bigger one we inscribe an equilateral triangle. From each of the vertices of this triangle, we draw a tangent to the smaller circle. Prove that the length of one of these tangents equals the sum of the lengths of the two other tangents.

1989 All Soviet Union Mathematical Olympiad, 497
Tags: geometric inequality , equal ratio , area , geometry
$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

2010 Ukraine Team Selection Test, 7
Tags: geometric inequality , trigonometry , geometry , altitude
Denote in the triangle $ABC$ by $h$ the length of the height drawn from vertex $A$, and by $\alpha = \angle BAC$. Prove that the inequality $AB + AC \ge BC \cdot \cos \alpha + 2h \cdot \sin \alpha$ . Are there triangles for which this inequality turns into equality?

1995 Tournament Of Towns, (479) 3
Tags: geometric inequality , combinatorial geometry , geometry , combinatorics , rectangle
A rectangle with sides of lengths $a$ and $b$ ($a > b$) is cut into rightangled triangles so that any two of these triangles either have a common side, a common vertex or no common points. Moreover, any common side of two triangles is a leg of one of them and the hypotenuse of the other. Prove that $a > 2b$. (A Shapovalov)

2023 Novosibirsk Oral Olympiad in Geometry, 5
Tags: geometry , geometric inequality
One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?