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: 6

2022 Durer Math Competition Finals, 1
Tags: geometry , tangent , Equilateral , Sqaure
To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

1988 Dutch Mathematical Olympiad, 4
Tags: geometry , geometric inequality , Geometric Inequalities , Sqaure
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.

2003 Austria Beginners' Competition, 4
Tags: rectangle , Sqaure , geometry
Prove that every rectangle circumscribed by a square is itself a square. (A rectangle is circumscribed by a square if there is exactly one corner point of the square on each side of the rectangle.)

2014 Czech-Polish-Slovak Junior Match, 5
Tags: Sum , angles , Polygons , Sqaure , combinatorial geometry , combinatorics
A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?

Durer Math Competition CD Finals - geometry, 2022.C3
Tags: geometry , tangent , Equilateral , Sqaure
To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

2008 Flanders Math Olympiad, 4
Tags: geometry , circles , Sqaure , area
A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure). [img]https://cdn.artofproblemsolving.com/attachments/b/6/6e28d94094d69bac13c2702853ac2c906a80a1.png[/img]