Olympiad problems

2001 Cuba MO, 2

Let $ABCD$ be a square. On the sides $BC$ and $CD$ the points $M$ and $K$ respectively, so that $MC = KD$. Let $P$ the intersection point of of segments $MD$ and $BK$. Prove that $AP \perp MK$.

2002 May Olympiad, 3

Tags: geometry , right triangle , isosceles , square

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

1994 Argentina National Olympiad, 3

Tags: geometry , square

Given in the plane the square $ABCD$, the square $A_1B_1C_1D_1$, smaller than the first, and a quadrilateral $PQRS$ that satisfy the following conditions $\bullet$ $ABCD$ and $A_1B_1C_1D_1$ have a common center and respectively parallel sides. $\bullet$$P$, $Q$, $R$, $S$ belong one to each side of the square $ABCD$. $\bullet$ $A_1$, $B_1$, $C_1$, $D_1$ belong one to each side of the quadrilateral $PQRS$. Prove that $PQRS$ is a square.

2020 Novosibirsk Oral Olympiad in Geometry, 4

Tags: equal angles , square

Points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ of square $ABCD$, respectively. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

MathLinks Contest 4th, 6.2

Tags: function , square , number theory , geometry

Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.

2014 Regional Olympiad of Mexico Center Zone, 4

Tags: geometry , ratio , reflection , square

Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $C ^ \prime$ be the reflection of $C$ wrt to $DM$. The parallel to $AB$ passing through $C ^ \prime$ intersects $AD$ at $R$ and $BC$ at $S$. Show that $$\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}$$

Kyiv City MO Juniors 2003+ geometry, 2016.9.51

Tags: equal angles , square , geometry

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected, respectively, so that $PN = NC$, the point $Q$ Is a point on the segment $AN$ for which $\angle NCB = \angle QPN$. Prove that $\angle BCQ = \tfrac {1} {2} \angle PQA$.

1979 Czech And Slovak Olympiad IIIA, 3

Tags: geometric inequality , square , cyclic quadrilateral , geometry

If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it. [hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]

2021 Iranian Geometry Olympiad, 2

Tags: inscribed , square , area , geometry

Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$ is parallel to some side of $ABCD$. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

Novosibirsk Oral Geo Oly IX, 2021.4

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]

2015 JBMO Shortlist, C1

Tags: grid , combinatorics , square , table

A board $ n \times n$ ($n \ge 3$) is divided into $n^2$ unit squares. Integers from $O$ to $n$ included, are written down: one integer in each unit square, in such a way that the sums of integers in each $2\times 2$ square of the board are different. Find all $n$ for which such boards exist.

2007 German National Olympiad, 2

Tags: product , set , number theory , square

Let $A$ be the set of odd integers $\leq 2n-1.$ For a positive integer $m$, let $B=\{a+m\,|\, a\in A \}.$ Determine for which positive integers $n$ there exists a positive integer $m$ such that the product of all elements in $A$ and $B$ is a square.

2006 Sharygin Geometry Olympiad, 4

Tags: geometry , regular , square , pentagon

a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$ F,G$ outside the square $ABCD$. Find the angle between lines $AE$ and $BF$. b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.

Ukrainian TYM Qualifying - geometry, 2012.11

Tags: square , concurrency , geometry

Let $E$ be an arbitrary point on the side $BC$ of the square $ABCD$. Prove that the inscribed circles of triangles $ABE$, $CDE$, $ADE$ have a common tangent.

2017 Czech-Polish-Slovak Match, 2

Tags: board , maximum , square , combinatorics

Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board. (Poland)

1984 Tournament Of Towns, (068) T2

Tags: minimum , distance , combinatorial geometry , square , geometry

A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner? (Muscovite folklore)

2020 Yasinsky Geometry Olympiad, 6

Tags: angle bisector , square , geometry , incenter

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. The point $F$ belongs to the side $AB$, and $DE \perp EF$. The point $G$ lies inside the square, and $GF = FE$ and $GF \perp FE$. Prove that: a) $DE$ is the bisector of the $\angle FDC$ b) $FG$ is the bisector of the $\angle AFD$ c) the point $G$ is the center of the circle inscribed in the triangle $ADF$. (Ercole Suppa, Italy)

2022 Paraguay Mathematical Olympiad, 5

Tags: area , square , geometry

In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$? [img]https://cdn.artofproblemsolving.com/attachments/1/e/794da3bc38096ef5d5daaa01d9c0f8c41a6f84.png[/img]

2007 Estonia Team Selection Test, 4

Tags: angle chasing , geometry , angle , square

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

1999 Estonia National Olympiad, 3

Tags: square , ratio , geometry

Let $E$ and $F$ be the midpoints of the lines $AB$ and $DA$ of a square $ABCD$, respectively and let $G$ be the intersection of $DE$ with $CF$. Find the aspect ratio of sidelengths of the triangle $EGC$, $| EG | : | GC | : | CE |$.

2015 Oral Moscow Geometry Olympiad, 2

Tags: angle , equilateral , geometry , square

The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$. [img]https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png[/img]

2012 Tournament of Towns, 6

Tags: rectangle , square , combinatorics , geometry , combinatorial geometry

We attempt to cover the plane with an infinite sequence of rectangles, overlapping allowed. (a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$? (b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?