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: 185

1952 Kurschak Competition, 1
Tags: geometry , Geometric Inequalities , tangent circles
A circle $C$ touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.

1961 Polish MO Finals, 3
Tags: geometry , parallelogram , tetrahedron , 3D geometry , Geometric Inequalities
Prove that if a plane section of a tetrahedron is a parallelogram, then half of its perimeter is contained between the length of the smallest and the length of the largest edge of the tetrahedron.

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 $S_{A_1B_1C_1}\ge9S_{ABC}$.

1981 Polish MO Finals, 6
Tags: geometry , 3D geometry , tetrahedron , Geometric Inequalities
In a tetrahedron of volume $V$ the sum of the squares of the lengths of its edges equals $S$. Prove that $$V \le \frac{S\sqrt{S}}{72\sqrt{3}}$$

1927 Eotvos Mathematical Competition, 3
Tags: geometry , inradius , exradius , Geometric Inequalities
Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.

2018 Czech-Polish-Slovak Match, 4
Tags: Geometric Inequalities , geometry , inequalities
Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i]

2020 Jozsef Wildt International Math Competition, W58
Tags: Geometric Inequalities , geometry , inequalities
In all triangles $ABC$ does it hold that: $$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

2021 Israel National Olympiad, P3
Tags: geometry , Geometric Inequalities , inequalities
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 \[AX+AY+BC>AB+AC\]

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?

2004 Germany Team Selection Test, 2
Tags: geometry proposed , geometry , Locus problems , Geometric Inequalities , Germany , TST , Team Selection Test
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 $N$).

1988 Swedish Mathematical Competition, 1
Tags: geometry , altitudes , Geometric Inequalities
Let $a > b > c$ be sides of a triangle and $h_a,h_b,h_c$ be the corresponding altitudes. Prove that $a+h_a > b+h_b > c+h_c$.

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 $AB \cdot CD + AD\cdot BC > AC \cdot BD$

1997 Swedish Mathematical Competition, 1
Tags: Geometric Inequalities , geometry , inequalities
Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$. Show that if $AC = 1$, $AB = a$, and $CD = b$, then $$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$

1965 Swedish Mathematical Competition, 1
Tags: Geometric Inequalities , altitudes , geometry , angles , max
The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?

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 $XYZ.$)

2017 Singapore Junior Math Olympiad, 1
Tags: geometry , rectangle , ratio , Geometric Inequalities , inequalities , square , sidelenghts
A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

1991 Poland - Second Round, 6
Tags: geometry , 3D geometry , sphere , Geometric Inequalities , parallelepiped
The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.

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]

1954 Kurschak Competition, 1
Tags: geometry , Geometric Inequalities
$ABCD$ is a convex quadrilateral with $AB + BD = AC + CD$. Prove that $AB < AC$.

1968 Poland - Second Round, 4
Tags: algebra , inequalities , geometry , Geometric Inequalities
Prove that if the numbers $ a, b, c $, are the lengths of the sides of a triangle and the sum of the numbers $x,y,z$ is zero, then $$a^2yz + b^2zx + c^2xy \leq 0.$$

1991 Swedish Mathematical Competition, 6
Tags: geometry , area , Geometric Inequalities , Equilateral
Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

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 $$ d_a > d_b > d_c.$$

1928 Eotvos Mathematical Competition, 3
Tags: geometry , Geometric Inequalities , construction
Let $\ell$ be a given line, $A$ and $B$ given points of the plane. Choose a point $P$ on $\ell $ so that the longer of the segments $AP$, $BP$ is as short as possible. (If $AP = BP,$ either segment may be taken as the longer one.)

1990 Swedish Mathematical Competition, 2
Tags: Sum , Geometric Inequalities , geometry , collinear
The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

1988 Poland - Second Round, 3
Tags: geometry , Geometric Inequalities
Inside the acute-angled triangle $ ABC $ we consider the point $ P $ and its projections $ L, M, N $ to the sides $ BC, CA, AB $, respectively. Determine the point $ P $ for which the sum $ |BL|^2 + |CM|^2 + |AN|^2 $ is the smallest.