Olympiad problems

2016 BMT Spring, 3

Consider an equilateral triangle and square, both with area $1$. What is the product of their perimeters?

2022 Yasinsky Geometry Olympiad, 2

Tags: geometry , square , ratio , area

On the sides $AB$, $BC$, $CD$, $DA$ of the square $ABCD$ points $P, Q, R, T$ are chosen such that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12.$$ The segments $AR$, $BT$, $CP$, $DQ$ in the intersection form the quadrilateral $KLMN$ (see figure). [img]https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.png[/img] a) Prove that $KLMN$ is a square. b) Find the ratio of the areas of the squares $KLMN$ and $ABCD$. (Alexander Shkolny)

2007 Cuba MO, 3

Tags: geometry , cyclic quadrilateral , square

Let $ABCD$ be a quadrilateral that can be inscribed in a circle whose diagonals are perpendicular. Denote by $P$ and $Q$ the feet of the perpendiculars through $D$ and $C$ respectively on the line $AB$, $X$ is the intersection point of the lines $AC$ and $DP$, $Y$ is the intersection point of the lines $BD$ and $CQ$. Show that $XY CD$ is a rhombus.

1963 German National Olympiad, 5

Tags: geometry , locus , midpoint , square

Given is a square with side length $a$. A distance $PQ$ of length $p$, where $p < a$, moves so that its end points are always on the sides of the square. What is the geometric locus of the midpoints of the segments $PQ$?

2010 QEDMO 7th, 4

Tags: square , geometry , area

Let $ABCD$ and $A'B'C'D'$ be two squares, both are oriented clockwise. In addition, it is assumed that all points are arranged as shown in the figure.Then it has to be shown that the sum of the areas of the quadrilaterals $ABB'A'$ and $CDD'C'$ equal to the sum of the areas of the quadrilaterals $BCC'B'$ and $DAA'D'$. [img]https://cdn.artofproblemsolving.com/attachments/0/2/6f7f793ded22fe05a7b0a912ef6c4e132f963e.png[/img]

2013 Estonia Team Selection Test, 4

Tags: right triangle , concyclic , cyclic quadrilateral , square , geometry

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

Estonia Open Junior - geometry, 2007.2.2

Tags: angle bisector , square , right angle , geometry

The center of square $ABCD$ is $K$. The point $P$ is chosen such that $P \ne K$ and the angle $\angle APB$ is right . Prove that the line $PK$ bisects the angle between the lines $AP$ and $BP$.

2014 Romania National Olympiad, 4

Tags: geometry , square , perpendicular , isosceles , right triangle

Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that $MQ \perp CF$.

1962 Poland - Second Round, 5

Tags: equilateral , locus , square , geometry

In the plane there is a square $ Q $ and a point $ P $. The point $ K $ runs through the perimeter of the square $ Q $. Find the locus of the vertex $ M $ of the equilateral triangle $ KPM $.

Denmark (Mohr) - geometry, 1998.3

Tags: geometry , parallel , square , area

The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square. [img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]

1978 Bundeswettbewerb Mathematik, 2

Tags: geometry , square , area , vertices , triangle

Seven distinct points are given inside a square with side length $1.$ Together with the square's vertices, they form a set of $11$ points. Consider all triangles with vertices in $M.$ a) Show that at least one of these triangles has an area not exceeding $1\slash 16.$ b) Give an example in which no four of the seven points are on a line and none of the considered triangles has an area of less than $1\slash 16.$

2005 Slovenia National Olympiad, Problem 3

Tags: square , circumcircle , geometry

Let $T$ be a point inside a square $ABCD$. The lines $TA,TB,TC,TD$ meet the circumcircle of $ABCD$ again at $A',B',C',D'$, respectively. Prove that $A'B'\cdot C'D'=A'D'\cdot B'C'$.

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , parallelogram , square , similar triangles , similar

Let $ABCD$ be a parallelogram of center $O$. Points $M$ and $N$ are the midpoints of $BO$ and $CD$, respectively. Prove that if the triangles $ABC$ and $AMN$ are similar, then $ABCD$ is a square.

1972 Putnam, A4

Tags: ellipse , square

Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square.

2018 Puerto Rico Team Selection Test, 5

Tags: geometry , square

In the square shown in the figure, find the value of $x$. [img]https://cdn.artofproblemsolving.com/attachments/0/1/4659d5afa5b409d9264924735297d1188b0be3.png[/img]

1979 Czech And Slovak Olympiad IIIA, 3

Tags: geometry , geometric inequality , square , cyclic quadrilateral

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]

2013 Czech-Polish-Slovak Junior Match, 6

Tags: geometry , circles , equal circles , square

There is a square $ABCD$ in the plane with $|AB|=a$. Determine the smallest possible radius value of three equal circles to cover a given square.

1998 Estonia National Olympiad, 5

Tags: combinatorics , combinatorial geometry , square

From an $n\times n$ square divided into $n^2$ unit squares, one corner unit square is cut off. Find all positive integers $n$ for which it is possible to tile the remaining part of the square with $L$-trominos. [img]https://cdn.artofproblemsolving.com/attachments/0/4/d13e6e7016d943b867f44375a2205b10ccf552.png[/img]

2011 Saudi Arabia Pre-TST, 4

Tags: geometry , square , fixed

Points $A ,B ,C ,D$ lie on a line in this order. Draw parallel lines $a$ and $b$ through $A$ and $B$, respectively, and parallel lines $c$ and $d$ through $C$ and $D$, respectively, such that their points of intersection are vertices of a square. Prove that the side length of this square does not depend on the length of segment $BC$.

Cono Sur Shortlist - geometry, 2003.G3

Tags: square , geometry , right angle

An interior $P$ point to a square $ABCD$ is such that $PA = a, PB = b$ and $PC = b + c$, where the numbers $a, b$ and $c$ satisfy the relationship $a^2 = b^2 + c^2$. Prove that the angle $BPC$ is right.

2004 May Olympiad, 4

Tags: square , geometry , find angle , area

In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.

1961 Czech and Slovak Olympiad III A, 2

Tags: geometry , construction , square , isosceles right triangle , geometric construction

Let a right isosceles triangle $APQ$ with the hypotenuse $AP$ be given in plane. Construct such a square $ABCD$ that the lines $BC, CD$ contain points $P, Q,$ respectively. Compute the length of side $AB = b$ in terms of $AQ=a$.

2019 Bundeswettbewerb Mathematik, 3

Tags: geometry , circumcircle , square

Let $ABCD$ be a square. Choose points $E$ on $BC$ and $F$ on $CD$ so that $\angle EAF=45^\circ$ and so that neither $E$ nor $F$ is a vertex of the square. The lines $AE$ and $AF$ intersect the circumcircle of the square in the points $G$ and $H$ distinct from $A$, respectively. Show that the lines $EF$ and $GH$ are parallel.

2016 BMT Spring, 14

Tags: equal circles , circles , square , geometry

Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?

2018 Ecuador Juniors, 3

Tags: geometry , square , geometric inequality , area

Let $ABCD$ be a square. Point $P, Q, R, S$ are chosen on the sides $AB$, $BC$, $CD$, $DA$, respectively, such that $AP + CR \ge AB \ge BQ + DS$. Prove that $$area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)$$ and determine all cases when equality holds.