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: 467

2014 Contests, 2
Tags: right triangle , rhombus , square , perpendicular , geometry , isosceles
Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

Denmark (Mohr) - geometry, 2001.3
Tags: geometry , square , min
In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

Estonia Open Senior - geometry, 1999.2.5
Tags: area , equal area , square , geometry
Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

1970 Swedish Mathematical Competition, 5
Tags: geometry , square
A $3\times 1$ paper rectangle is folded twice to give a square side $1$. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making $6$ holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?

2016 Junior Balkan Team Selection Tests - Moldova, 7
Tags: geometry , square , equal segments , perpendicular
Let $ABCD$ ba a square and let point $E$ be the midpoint of side $AD$. Points $G$ and $F$ are located on the segment $(BE)$ such that the lines $AG$ and $CF$ are perpendicular on the line $BE$. Prove that $DF= CG$.

2008 Switzerland - Final Round, 5
Tags: geometry , locus , square
Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

Kyiv City MO Juniors 2003+ geometry, 2018.9.51
Tags: geometry , angle , square
Given a circle $\Gamma$ with center at point $O$ and diameter $AB$. $OBDE$ is square, $F$ is the second intersection point of the line $AD$ and the circle $\Gamma$, $C$ is the midpoint of the segment $AF$. Find the value of the angle $OCB$.

2009 Bundeswettbewerb Mathematik, 1
Tags: number theory , square
Determine all possible digits $z$ for which $\underbrace{9...9}_{100}z\underbrace{0...0}_{100}9$ is a square number.

1994 Austrian-Polish Competition, 3
Tags: rectangle , tiling , square , combinatorics , combinatorial geometry
A rectangular building consists of $30$ square rooms situated like the cells of a $2 \times 15$ board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?

1988 Spain Mathematical Olympiad, 5
Tags: combinatorics , square , partition
A well-known puzzle asks for a partition of a cross into four parts which are to be reassembled into a square. One solution is exhibited on the picture. [img]https://cdn.artofproblemsolving.com/attachments/9/1/3b8990baf5e37270c640e280c479b788d989ba.png[/img] Show that there are infinitely many solutions. (Some solutions split the cross into four equal parts!)

1998 Switzerland Team Selection Test, 4
Tags: combinatorial geometry , square , combinatorics
Find all numbers $n$ for which it is possible to cut a square into $n$ smaller squares.

1979 Bundeswettbewerb Mathematik, 2
Tags: concurrency , square , geometry
The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

1979 Austrian-Polish Competition, 7
Tags: combinatorics , combinatorial geometry , square , grid , hexagon , analytic geometry
Let $n$ and $m$ be fixed positive integers. The hexagon $ABCDEF$ with vertices $A = (0,0)$, $B = (n,0)$, $C = (n,m)$, $D = (n-1,m)$, $E = (n-1,1)$, $F = (0,1)$ has been partitioned into $n+m-1$ unit squares. Find the number of paths from $A$ to $C$ along grid lines, passing through every grid node at most once.

2007 Bosnia and Herzegovina Junior BMO TST, 3
Tags: combinatorial geometry , square , circles , geometry
Is it possible to place some circles inside a square side length $1$, such that no two circles intersect and the sum of their radii is $2007$?

2020 AMC 12/AHSME, 10
Tags: geometry , square
In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$ $\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$

2010 NZMOC Camp Selection Problems, 2
Tags: geometry , chord , square
$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$. A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$. Find the area of the square.

Denmark (Mohr) - geometry, 2001.5
Tags: geometry , equilateral , square
Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

Novosibirsk Oral Geo Oly VIII, 2019.7
Tags: geometry , acute , square
The square was cut into acute -angled triangles. Prove that there are at least eight of them.

1950 Polish MO Finals, 2
Tags: concentric , geometry , concentric circles , square , construction
We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.

2024 Kyiv City MO Round 1, Problem 2
Tags: arranging , parity , square , combinatorics
Is it possible to write the numbers from $1$ to $100$ in the cells of a of a $10 \times 10$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $10 \times 10$ square, the numbers in them must have the same parity. The figure below shows examples of such pairs of cells, in which the numbers must have the same parity. [img]https://i.ibb.co/b3P8t36/Kyiv-MO-2024-7-2.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

1998 Tournament Of Towns, 5
Tags: tiling , diagonal , square , combinatorics , combinatorial geometry
A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw? (I Rubanov)

2021 Novosibirsk Oral Olympiad in Geometry, 4
Tags: geometry , square , tangent circle
A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. [img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]

2009 Austria Beginners' Competition, 4
Tags: geometry , midpoint , square
The center $M$ of the square $ABCD$ is reflected wrt $C$. This gives point $E$. The intersection of the circumcircle of the triangle $BDE$ with the line $AM$ is denoted by $S$. Show that $S$ bisects the distance $AM$. (W. Janous, WRG Ursulinen, Innsbruck)

1960 Kurschak Competition, 3
Tags: square , geometry
$E$ is the midpoint of the side $AB$ of the square $ABCD$, and $F, G$ are any points on the sides $BC$, $CD$ such that $EF$ is parallel to $AG$. Show that $FG$ touches the inscribed circle of the square.

2010 Abels Math Contest (Norwegian MO) Final, 1b
Tags: geometry , area of a triangle , area , square
The edges of the square in the figure have length $1$. Find the area of the marked region in terms of $a$, where $0 \le a \le 1$. [img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]